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## Solving partial differential equations in Maple and Mathematica

August 14, 2018   Compiled on August 14, 2018 at 9:13pm

### Contents

This report gives the result of running a number of partial differential equations in Mathematica and Maple.

The following systems were used at this time.

1. Mathematica 11.3 (64 bit).
2. Maple 2018.1 (64 bit) with Physics version MapleCloud 84.

No time limit was used.

All possible options, assumptions and HINTS were tried to obtain a solution. The command DSolve was used in Mathematica and the command pdsolve in Maple.

It is possible I missed some option, assumption or HINT, which could help make the CAS able to solve a given PDE now marked as unsolved. Will correct such a case if found. I have verified some, but not all, solutions returned by Maple or Mathematica.

Number of problems is [ 122 ]. Mathematica solved 82 or 67.21%. Maple solved 102 or 83.61%.

 Table 1: Breakdown of results for each PDE # PDE description Mathematica Maple 1 First order PDE Linear PDE, the transport equation Solved Solved 2 First order PDE Linear PDE Solved Solved 3 First order PDE Linear PDE, initial value problem Solved Solved 4 First order PDE Initial-boundary value problem Solved Did not solve 5 First order PDE Linear PDE, the transport equation with initial conditions Solved Solved 6 First order PDE General solution for a quasilinear first-order PDE Solved Solved 7 First order PDE quasilinear first-order PDE, scalar conservation law Solved, solution in implicit form Solved 8 First order PDE quasilinear first-order PDE, scalar conservation law with initial value Solved Solved 9 First order PDE nonlinear first-order PDE, the Clairaut equation Solved Solved 10 First order PDE nonlinear first-order PDE, the Clairaut equation with initial value Solved Solved 11 First order PDE Another example of nonlinear Clairaut equation Solved Solved 12 First order PDE Recover a function from its gradient vector Solved Solved 13 First order PDE General solution of a first order nonlinear PDE Did not solve Solved 14 First order PDE Nonlinear first order PDE Solved Solved 15 Heat PDE on bar, homogeneous Dirichlet boundary conditions, No source. Solved Solved 16 Heat PDE on bar, homogeneous Dirichlet boundary conditions, No source. Solved Solved 17 Heat PDE on bar, homogeneous Dirichlet boundary conditions, No source. Solved Solved 18 Heat PDE on bar, homogeneous Dirichlet boundary conditions, No source. Solved Solved 19 Heat PDE on bar, homogeneous Neumann boundary conditions, No source. Solved Solved 20 Heat PDE on bar, homogeneous Dirichlet boundary conditions with heat sink Did not solve Solved 21 Heat PDE on bar, homogeneous Neumann boundary conditions, No source Solved Solved 22 Heat PDE on bar, homogeneous Neumann boundary conditions, No source Solved Solved 23 Heat PDE on bar, homogeneous Neumann boundary conditions, No source Solved Solved 24 Heat PDE on bar, homogeneous Neumann boundary conditions, No source Solved Solved 25 Heat PDE on bar, homogeneous Neumann on left and Dirichlet on right, No source Solved Solved 26 Heat PDE on bar, semi-infinite domain, No source Solved Solved 27 Heat PDE on bar, periodic boundary conditions, No source Did not solve Solved 28 Heat PDE on bar, semi-infinite domain, zero initial condition, No source Solved Solved 29 Heat PDE on bar, semi-infinite domain, non-zero initial condition, No source Solved Solved 30 Heat PDE on bar, heat absorption radiation in bounded domain, No source Did not solve Solved 31 Heat PDE infinite domain Solved Solved 32 Heat PDE on bar, with domain from -1 to +1, no source Did not solve Solved 33 Heat PDE on bar, Dirichlet nonhomogeneous BC, no source term Solved Solved 34 Heat PDE on bar, nonhomogeneous Dirichlet BC, with constant source term Did not solve Solved 35 Heat PDE on bar, homogeneous Dirichlet BC, non zero initial conditions, with extra term Did not solve Solved 36 Heat PDE on bar with initial conditions sum of sine terms, homogeneous Dirichlet BC, no source Solved Solved 37 Heat PDE on bar, homogeneous Dirichlet BC, initial condition is piecewise function, no source Solved Solved 38 Heat PDE on bar, inhomogeneous Dirichlet BC, initial condition is piecewise function, no source Solved Solved 39 Heat PDE on bar, inhomogeneous Dirichlet BC which depends on time. Zero initial condition, no source Solved Solved 40 Heat PDE on bar, homogeneous Neumann boundary conditions, non zero initial conditions, with source as sin function that depends on space only. Did not solve Solved 41 Heat PDE on bar, homogeneous Neumann boundary conditions, nonzero initial conditions, with source that depends on time only Did not solve Solved 42 Heat PDE on bar, homogeneous Neumann boundary conditions, nonzero initial conditions, with source that depends on time and space Did not solve Solved 43 Heat PDE on bar, non-homogeneous, time dependent, Neumann boundary conditions, with source that depends on time and space Did not solve Solved 44 Heat PDE on bar, non-homogeneous, not time dependent Neumann boundary conditions, No source term Did not solve Solved 45 Heat PDE on bar, homogeneous Neumann boundary conditions, Source term that depends on both time and space Solved Solved 46 Heat PDE on bar, homogeneous Neumann boundary conditions, Source term that depends on both time and space Solved Solved 47 Heat PDE on bar, Dirichlet boundary conditions that depends on time with source that depends on space only Did not solve Solved 48 Heat PDE on bar, homogeneous Dirichlet boundary conditions, with source that depends on time and space Did not solve Solved 49 Heat/Diffusion PDE in 2D, inside rectangle with initial and boundary conditions Did not solve Solved 50 Heat/Diffusion PDE in 2D, inside rectangle with initial and boundary conditions with heat loss Did not solve Solved 51 Heat PDE inside disk, with no dependency. initial and boundary conditions given Solved Did not solve 52 Heat PDE on whole line with no intial nor boundary conditions specified Did not solve Solved, returning a solution that is not the most general one 53 Heat PDE in 1D on the whole real line with initial position specified Solved Solved 54 Heat PDE in 1D on the whole real line, with linear adevction Solved Solved 55 Heat PDE in 1D on the whole real line with initial position as UnitBox Solved Solved 56 Heat PDE on half the line with non-zero initial conditions and Dirichlet boundary conditions Solved Solved, but has unresolved inverse Laplace transforms 57 Heat PDE on half the line with zero initial conditions and time dependent boundary conditions Solved Solved, but has unresolved inverse Laplace transforms 58 Initial value problem for the heat PDE with a Neumann condition on the half-line Solved Did not solve 59 Laplace PDE inside quarter-circle Did not solve Did not solve 60 Laplace PDE inside semi-circle Did not solve Solved 61 Laplace PDE inside rectangle Solved Solved 62 Laplace PDE inside rectangle Solved Solved 63 Laplace PDE inside rectangle Did not solve Solved 64 Laplace PDE inside rectangle Did not solve Solved 65 Laplace PDE inside rectangle Did not solve Solved 66 Laplace PDE inside rectangle, top/bottom edges non-zero Solved Solved 67 Laplace PDE inside circular annulus, Neumann boundary conditions using unspecified functions Did not solve Did not solve 68 Laplace PDE inside circular annulus, Dirichlet boundary conditions using specified functions Solved Did not solve 69 Laplace PDE example 18 from Maple help page Solved Solved 70 Laplace PDE on rectangle with one edge at infinity Did not solve Solved 71 Laplace PDE inside a disk, periodic boundary conditions Solved Solved 72 Dirichlet problem for the Laplace equation in upper half plan Solved Did not solve 73 Dirichlet problem for the Laplace equation in right half-plane: Solved Did not solve 74 Dirichlet problem for the Laplace equation in the first quadrant Solved Did not solve 75 Neumann problem for the Laplace equation in the upper half-plane Solved Did not solve 76 Dirichlet problem for the Laplace equation in a rectangle Solved Solved 77 Laplace PDE outside a disk, periodic boundary conditions Did not solve Solved 78 Laplace equation in spherical coordinates Did not solve Solved, but not verified 79 Dirichlet problem for the Poisson equation in a rectangle Solved Solved 80 Dirichlet problem for the Helmholtz equation in a rectangle Solved Solved 81 General solution for a second-order hyperbolic PDE on real line Solved Solved 82 Hyperbolic PDE with non-rational coefficients Solved Did not solve. Tried all HINTS 83 Inhomogeneous hyperbolic PDE with constant coefficients Solved Solved 84 system of 2 inhomogeneous linear hyperbolic system with constant coefficients Solved Did not solve 85 Wave PDE on string (finite domain) with zero initial position and velocity, and with source term Solved Solved 86 Wave PDE on string, one end fixed, another free, both initial conduitions non zero, and source that depends on time and space Did not solve Solved 87 Wave PDE on string (finite domain), fixed ends, no initial conduitions give and no source Did not solve Solved 88 Wave PDE on string (finite domain), one fixed end, one free end, initial position not zero, initial velocity zero, no source Did not solve Solved 89 Wave PDE on string (finite domain), both ends fixed end, initial conditions zero, with source as generic function that depends on time and space Did not solve Solved 90 Wave PDE on string (finite domain), both ends fixed, initial conditions both not zero, No source Solved Solved 91 Wave PDE on string (finite domain), both ends fixed end, initial conditions both not zero, and with constant source Did not solve Solved 92 Wave PDE on string (finite domain), both ends fixed end, with source Did not solve Solved 93 Wave PDE on semi-infinite domain, with one end having a moving boundary condition Solved Solved 94 Telegraphy PDE, a wave PDE on string, both ends fixed with damping Did not solve Solved, But should not be included. 95 Wave PDE, on string, both ends fixed. Initial velocity zero. Dispersion term present Did not solve due to adding dispersion term Solved 96 Wave PDE on string with fixed ends, non-zero initial position Solved but sum should not include Solved 97 Wave PDE homogeneous in square, given initial position but with zero initial velocity Did not solve Solved 98 Wave PDE homogeneous in square with damping. Given zero initial position but with non-zero initial velocity Did not solve Solved 99 Wave PDE inside rectangle. All 4 edges are fixed and given non-zero initial position with zero initial velocity Solved Solved 100 Wave PDE inside disk. fixed edge of disk, no dependency, with initial position and velocity given Solved Did not solve 101 Wave PDE inside disk. fixed edge of disk, with dependency, zero initial velocity Did not solve Did not solve 102 Wave PDE on infinite domain with initial conditions specified, no source Solved Solved 103 Wave PDE on infinite domain with initial conditions specified, with source term Solved Solved 104 Wave PDE initial value with a Dirichlet condition on the half-line Solved Solved 105 Wave PDE Initial value problem with a Neumann condition on the half-line Solved Did not solve 106 non-linear wave PDE (Solitons) Solved. build a special solution. Solved. Returning a solution that is not the most general one 107 Schrodinger PDE with zero potential Solved Solved 108 Schrodinger PDE with initial and boundary conditions Solved Did not solve 109 Initial value problem with Dirichlet boundary conditions Solved Solved 110 Solve a Schrodinger equation with potential over the whole real line Solved Did not solve. Maple does not support in boundary conditions 111 Beam PDE with zero initial velocity Solved Solved 112 viscous fluid flow with no initial conditions Solved Solved 113 viscous fluid flow with initial conditions Solved Solved, but has unresolved integrals 114 viscous fluid flow with initial conditions as UnitBox Solved Solved, but has unresolved integrals 115 classic Black Scholes model from finance Solved Did not solve 116 Boundary value problem for the Black Scholes equation Solved Did not solve 117 Korteweg-deVries (waves on shallow water surfaces) with no initial conditions Solved Solved 118 Boundary value problem for the Tricomi equation Solved Solved 119 Cauchy Riemann PDE with Prescribe the values of and on the axis Solved Did not solve 120 Cauchy Riemann PDE With extra term on right side Did not Solve Solved 121 Hamilton-Jacobi type PDE Did not Solve Solved 122 A second order PDE Did not Solve Solved

### 1 First order PDE

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#### 1.1 Linear PDE, the transport equation

problem number 1

Taken from Mathematica Symbolic PDE document

Solve for

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 1.2 Linear PDE

problem number 2

Taken from Mathematica help pages

Solve for

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 1.3 Linear PDE, initial value problem

problem number 3

Taken from Mathematica help pages

Solve for

with initial value

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 1.4 Initial-boundary value problem

problem number 4

Taken from Mathematica help pages

Solve for

with initial value and boundary value

Result Solved

##### Maple

Result Did not solve

_________________________________________________________________________________

#### 1.5 Linear PDE, the transport equation with initial conditions

problem number 5

Taken from Mathematica help pages

Solve for

With initial conditions

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 1.6 General solution for a quasilinear first-order PDE

problem number 6

Taken from Mathematica help pages

Solve for

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 1.7 quasilinear first-order PDE, scalar conservation law

problem number 7

Taken from Mathematica Symbolic PDE document

Solve for

##### Mathematica

Result Solved, solution in implicit form

##### Maple

Result Solved

_________________________________________________________________________________

#### 1.8 quasilinear first-order PDE, scalar conservation law with initial value

problem number 8

Taken from Mathematica Symbolic PDE document

Solve for

With

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 1.9 nonlinear first-order PDE, the Clairaut equation

problem number 9

Taken from Mathematica Symbolic PDE document

Solve for

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 1.10 nonlinear first-order PDE, the Clairaut equation with initial value

problem number 10

Taken from Mathematica Symbolic PDE document

Solve for

With

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 1.11 Another example of nonlinear Clairaut equation

problem number 11

Taken from Mathematica DSolve help pages

Solve for

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 1.12 Recover a function from its gradient vector

problem number 12

Taken from Mathematica DSolve help pages

Solve for

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 1.13 General solution of a first order nonlinear PDE

problem number 13

Taken from Maple pdsolve help pages

Solve for

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 1.14 Nonlinear first order PDE

problem number 14

Taken from Maple pdsolve help pages, probem 5

Solve for

Result Solved

Result Solved

### 2 Heat PDE

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#### 2.1 Heat PDE on bar, homogeneous Dirichlet boundary conditions, No source.

problem number 15

This is problem 2.3.3, part (a) from Richard Haberman applied partial differential equations, 5th edition.

Consider the heat equation

Subject to boundary conditions and with the temperature initially

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.2 Heat PDE on bar, homogeneous Dirichlet boundary conditions, No source.

problem number 16

This is problem 2.3.3, part (b) from Richard Haberman applied partial differential equations, 5th edition.

Consider the heat equation

Subject to boundary conditions and with the temperature initially

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.3 Heat PDE on bar, homogeneous Dirichlet boundary conditions, No source.

problem number 17

This is problem 2.3.3, part (c) from Richard Haberman applied partial differential equations, 5th edition.

Consider the heat equation

Subject to boundary conditions and with the temperature initially

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.4 Heat PDE on bar, homogeneous Dirichlet boundary conditions, No source.

problem number 18

This is problem 2.3.3, part (d) from Richard Haberman applied partial differential equations, 5th edition.

Consider the heat equation

Subject to boundary conditions and with the temperature initially

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.5 Heat PDE on bar, homogeneous Neumann boundary conditions, No source.

problem number 19

This is problem 2.3.7, from Richard Haberman applied partial differential equations, 5th edition.

Consider the heat equation

Subject to boundary conditions with the temperature initially

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.6 Heat PDE on bar, homogeneous Dirichlet boundary conditions with heat sink

problem number 20

This is problem 2.3.8, from Richard Haberman applied partial differential equations, 5th edition.

Consider the heat equation

This corresponds to a one-dimentional rod either with heat loss through the lateral sides with outside temperature zero degrees () or with insulated sides with a heat sink propertional to the temperature.

Suppose the boundary conditions are , solve with the temperature initially if

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.7 Heat PDE on bar, homogeneous Neumann boundary conditions, No source

problem number 21

This is problem 2.4.1 part(a) from Richard Haberman applied partial differential equations, 5th edition.

Consider the heat equation

The boundary conditions are with the temperature initially

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.8 Heat PDE on bar, homogeneous Neumann boundary conditions, No source

problem number 22

This is problem 2.4.1 part(b) from Richard Haberman applied partial differential equations, 5th edition.

Solve the heat equation

The boundary conditions are with the temperature initially

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.9 Heat PDE on bar, homogeneous Neumann boundary conditions, No source

problem number 23

This is problem 2.4.1 part(c) from Richard Haberman applied partial differential equations, 5th edition.

Solve the heat equation

The boundary conditions are with the temperature initially

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.10 Heat PDE on bar, homogeneous Neumann boundary conditions, No source

problem number 24

This is problem 2.4.1 part(d) from Richard Haberman applied partial differential equations, 5th edition.

Solve the heat equation

The boundary conditions are and with the temperature initially

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.11 Heat PDE on bar, homogeneous Neumann on left and Dirichlet on right, No source

problem number 25

This is problem 2.4.2 from Richard Haberman applied partial differential equations, 5th edition.

Solve the heat equation

The boundary conditions are with the temperature initially

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.12 Heat PDE on bar, semi-infinite domain, No source

problem number 26

This is problem at page 76 from David J Logan text book.

Solve the heat equation

The boundary conditions are and initial conditions

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.13 Heat PDE on bar, periodic boundary conditions, No source

problem number 27

Solve the heat equation

For and . The boundary conditions are

And initial conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.14 Heat PDE on bar, semi-infinite domain, zero initial condition, No source

problem number 28

Solve the heat equation

For and . The boundary conditions is and And initial condition

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.15 Heat PDE on bar, semi-infinite domain, non-zero initial condition, No source

problem number 29

Solve the heat equation

For and . The boundary conditions is and And initial condition

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.16 Heat PDE on bar, heat absorption radiation in bounded domain, No source

problem number 30

Solve the heat equation

For and . The boundary conditions are

And initial condition

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.17 Heat PDE infinite domain

problem number 31

Solve the heat equation

For and . The boundary conditions are

Initial condition is

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.18 Heat PDE on bar, with domain from -1 to +1, no source

problem number 32

Solve the heat equation

For and . The boundary conditions are zero at both ends. Initial condition is

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.19 Heat PDE on bar, Dirichlet nonhomogeneous BC, no source term

problem number 33

Taken from Maple PDE help pages

Solve the heat equation

For and . The boundary conditions are

Initial condition is

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.20 Heat PDE on bar, nonhomogeneous Dirichlet BC, with constant source term

problem number 34

This is problem 8.2.1 par(d) from Richard Haberman applied partial differential equations 5th edition.

Solve the heat equation

For and . The boundary conditions are

Initial condition is

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.21 Heat PDE on bar, homogeneous Dirichlet BC, non zero initial conditions, with extra term

problem number 35

Solve the heat equation

For and . The boundary conditions are

Initial condition is

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.22 Heat PDE on bar with initial conditions sum of sine terms, homogeneous Dirichlet BC, no source

problem number 36

Solve the heat equation

For and . The boundary conditions are

Initial condition is

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.23 Heat PDE on bar, homogeneous Dirichlet BC, initial condition is piecewise function, no source

problem number 37

Solve the heat equation

For and . The boundary conditions are

Initial condition is

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.24 Heat PDE on bar, inhomogeneous Dirichlet BC, initial condition is piecewise function, no source

problem number 38

Added July 2, 2018, taken from Maple 2018.1 improvement to PDE document.

Solve the heat equation

For and . The boundary conditions are

Initial condition is

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.25 Heat PDE on bar, inhomogeneous Dirichlet BC which depends on time. Zero initial condition, no source

problem number 39

added March 8, 2018. Exam problem

Solve the heat equation

For and . The boundary conditions are

Initial condition is .

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.26 Heat PDE on bar, homogeneous Neumann boundary conditions, non zero initial conditions, with source as sin function that depends on space only.

problem number 40

This is problem 8.2.1, part(f) from Richard Haberman applied partial differential equations 5th edition.

Solve the heat equation

For and . The boundary conditions are

Initial condition is .

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.27 Heat PDE on bar, homogeneous Neumann boundary conditions, nonzero initial conditions, with source that depends on time only

problem number 41

Added July 2, 2018. Taken from Maple 2018.1 document, originally exercise 6.25 from Pinchover and Rubinstein.

Solve the heat equation

For and . The boundary conditions are

Initial condition is .

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.28 Heat PDE on bar, homogeneous Neumann boundary conditions, nonzero initial conditions, with source that depends on time and space

problem number 42

Solve the heat equation

For and . The boundary conditions are

Initial condition is .

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.29 Heat PDE on bar, non-homogeneous, time dependent, Neumann boundary conditions, with source that depends on time and space

problem number 43

Pinchover and Rubinstein’s exercise 6.17. Taken from Maple document for new improvements in Maple 2018.1

Solve the heat equation

For and . The boundary conditions are

Initial condition is .

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.30 Heat PDE on bar, non-homogeneous, not time dependent Neumann boundary conditions, No source term

problem number 44

Second example from Maple document for new improvements in Maple 2018.1

Solve the heat equation

For and . The boundary conditions are

Initial condition is .

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.31 Heat PDE on bar, homogeneous Neumann boundary conditions, Source term that depends on both time and space

problem number 45

4th example from Maple document for new improvements in Maple 2018.1, originally taken from Pinchover and Rubinstein’s exercise 6.23 .

Solve the heat equation on bar

Where for and . The boundary conditions are

Initial condition is where .

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.32 Heat PDE on bar, homogeneous Neumann boundary conditions, Source term that depends on both time and space

problem number 46

Taken from Maple document for new improvements in Maple 2018.1, originally taken from Pinchover and Rubinstein’s exercise 6.21

Solve the heat equation on bar

Where for and . The boundary conditions are

Initial condition is where .

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.33 Heat PDE on bar, Dirichlet boundary conditions that depends on time with source that depends on space only

problem number 47

added March 28, 2018. A problem from my PDE animation page.

Solve the heat equation

For and . The boundary conditions are

Initial condition is .

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.34 Heat PDE on bar, homogeneous Dirichlet boundary conditions, with source that depends on time and space

problem number 48

Taken from Maple PDE help pages

Solve the heat equation for

For and . The boundary conditions are

Initial condition is

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.35 Heat/Diffusion PDE in 2D, inside rectangle with initial and boundary conditions

problem number 49

Taken from Maple help pages on PDE

Solve the heat equation for

For and and . The boundary conditions are

Initial condition is .

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.36 Heat/Diffusion PDE in 2D, inside rectangle with initial and boundary conditions with heat loss

problem number 50

Taken from Maple help pages on PDE

Solve the heat equation for

For and and . The boundary conditions are

Initial condition is .

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.37 Heat PDE inside disk, with no dependency. initial and boundary conditions given

problem number 51

Taken from Mathematica DSolve help pages

Solve the heat equation in polar coordinates for

For and . The boundary conditions are

Initial condition is .

Result Solved

##### Maple

Result Did not solve

_________________________________________________________________________________

#### 2.38 Heat PDE on whole line with no intial nor boundary conditions specified

problem number 52

Solve the heat equation for

##### Mathematica

Result Did not solve

##### Maple

Result Solved, returning a solution that is not the most general one

_________________________________________________________________________________

#### 2.39 Heat PDE in 1D on the whole real line with initial position specified

problem number 53

From Mathematica DSolve help pages. Solve the heat equation for on real line with

With initial condition

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.40 Heat PDE in 1D on the whole real line, with linear adevction

problem number 54

From Mathematica DSolve help pages. Solve the heat equation for on real line with

With initial condition

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.41 Heat PDE in 1D on the whole real line with initial position as UnitBox

problem number 55

From Mathematica DSolve help pages. Solve the heat equation for on real line with

With initial condition

Where UnitBox is equal to 1 if and zero otherwise.

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 2.42 Heat PDE on half the line with non-zero initial conditions and Dirichlet boundary conditions

problem number 56

From Mathematica DSolve help pages.

Solve the heat equation for on half the line and

With initial condition

And boundary conditions

Result Solved

##### Maple

Result Solved, but has unresolved inverse Laplace transforms

_________________________________________________________________________________

#### 2.43 Heat PDE on half the line with zero initial conditions and time dependent boundary conditions

problem number 57

Solve the heat equation for on half the line and

With initial condition

And boundary conditions

The last condition above means it is bounded at infinity.

Result Solved

##### Maple

Result Solved, but has unresolved inverse Laplace transforms

_________________________________________________________________________________

#### 2.44 Initial value problem for the heat PDE with a Neumann condition on the half-line

problem number 58

From Mathematica DSolve help pages.

Solve the heat equation for on half the line and

With initial condition

And boundary conditions

Result Solved

##### Maple

Result Did not solve

### 3 Laplace PDE

_________________________________________________________________________________

#### 3.1 Laplace PDE inside quarter-circle

problem number 59

This is problem 2.5.5 part (c) from Richard Haberman applied partial differential equations, 5th edition

Solve Laplace equation

Inside quarter circle of radius 1 with and , with following boundary conditions

##### Mathematica

Result Did not solve

##### Maple

Result Did not solve

_________________________________________________________________________________

#### 3.2 Laplace PDE inside semi-circle

problem number 60

Solve Laplace equation

Inside semi-circle of radius 1 with and , with following boundary conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 3.3 Laplace PDE inside rectangle

problem number 61

This is problem 2.5.1 part (a) from Richard Haberman applied partial differential equations, 5th edition

Solve Laplace equation

inside a rectangle , with following boundary conditions

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 3.4 Laplace PDE inside rectangle

problem number 62

This is problem 2.5.1 part (b) from Richard Haberman applied partial differential equations, 5th edition

Solve Laplace equation

inside a rectangle , with following boundary conditions

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 3.5 Laplace PDE inside rectangle

problem number 63

This is problem 2.5.1 part (c) from Richard Haberman applied partial differential equations, 5th edition

Solve Laplace equation

inside a rectangle , with following boundary conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 3.6 Laplace PDE inside rectangle

problem number 64

This is problem 2.5.1 part (d) from Richard Haberman applied partial differential equations, 5th edition

Solve Laplace equation

inside a rectangle , with following boundary conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 3.7 Laplace PDE inside rectangle

problem number 65

This is problem 2.5.1 part (e) from Richard Haberman applied partial differential equations, 5th edition

Solve Laplace equation

inside a rectangle , with following boundary conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 3.8 Laplace PDE inside rectangle, top/bottom edges non-zero

problem number 66

Taken from Mathematica DSolve help pages.

Solve Laplace equation

inside a rectangle , with following boundary conditions

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 3.9 Laplace PDE inside circular annulus, Neumann boundary conditions using unspecified functions

problem number 67

This is problem 2.5.8 part (b) from Richard Haberman applied partial differential equations, 5th edition

Solve Laplace equation

Inside circular annulus subject to the following boundary conditions

##### Mathematica

Result Did not solve

##### Maple

Result Did not solve

_________________________________________________________________________________

#### 3.10 Laplace PDE inside circular annulus, Dirichlet boundary conditions using specified functions

problem number 68

Solve Laplace equation

Inside circular annulus subject to the following boundary conditions

Result Solved

##### Maple

Result Did not solve

_________________________________________________________________________________

#### 3.11 Laplace PDE example 18 from Maple help page

problem number 69

Solve Laplace equation

With boundary conditions

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 3.12 Laplace PDE on rectangle with one edge at infinity

problem number 70

Solve Laplace equation

With boundary conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 3.13 Laplace PDE inside a disk, periodic boundary conditions

problem number 71

Solve Laplace equation in polar coordinates inside a disk

Solve for

Boundary conditions

Result Solved

##### Maple

Result Solved

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#### 3.14 Dirichlet problem for the Laplace equation in upper half plan

problem number 72

Taken from Mathematica DSolve help pages

Solve for

Boundary conditions for and otherwise. This is called UnitBox in Mathematica.

Result Solved

##### Maple

Result Did not solve

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#### 3.15 Dirichlet problem for the Laplace equation in right half-plane:

problem number 73

Taken from Mathematica DSolve help pages

Solve for

Boundary conditions .

Result Solved

##### Maple

Result Did not solve

_________________________________________________________________________________

#### 3.16 Dirichlet problem for the Laplace equation in the first quadrant

problem number 74

Taken from Mathematica DSolve help pages

Solve for

Boundary conditions .

Result Solved

##### Maple

Result Did not solve

_________________________________________________________________________________

#### 3.17 Neumann problem for the Laplace equation in the upper half-plane

problem number 75

Taken from Mathematica DSolve help pages

Solve for

Boundary conditions where is for and otherwise. This is called UnitBox in Mathematica.

Result Solved

##### Maple

Result Did not solve

_________________________________________________________________________________

#### 3.18 Dirichlet problem for the Laplace equation in a rectangle

problem number 76

Taken from Mathematica DSolve help pages

Solve for

Boundary conditions .

Result Solved

##### Maple

Result Solved

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#### 3.19 Laplace PDE outside a disk, periodic boundary conditions

problem number 77

Solve Laplace equation in polar coordinates outside a disk

Solve for

Boundary conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

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#### 3.20 Laplace equation in spherical coordinates

problem number 78

Taken from Maple pdsolve help pages

Solve for

##### Mathematica

Result Did not solve

##### Maple

Result Solved, but not verified

### 4 Poisson PDE

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#### 4.1 Dirichlet problem for the Poisson equation in a rectangle

problem number 79

Taken from Mathematica DSolve help pages.

Solve for

Boundary conditions

Result Solved

Result Solved

### 5 Helmholtz PDE

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#### 5.1 Dirichlet problem for the Helmholtz equation in a rectangle

problem number 80

Taken from Mathematica DSolve help pages.

Solve for

Boundary conditions

Result Solved

Result Solved

### 6 Wave PDE

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#### 6.1 General solution for a second-order hyperbolic PDE on real line

problem number 81

From Mathematica DSolve help pages (slightly modified)

Solve for with on real line

Result Solved

##### Maple

Result Solved

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#### 6.2 Hyperbolic PDE with non-rational coefficients

problem number 82

From Mathematica DSolve help pages

Solve for

Result Solved

##### Maple

Result Did not solve. Tried all HINTS

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#### 6.3 Inhomogeneous hyperbolic PDE with constant coefficients

problem number 83

From Mathematica DSolve help pages

Solve for

Result Solved

##### Maple

Result Solved

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#### 6.4 system of 2 inhomogeneous linear hyperbolic system with constant coefficients

problem number 84

From Mathematica DSolve help pages

Solve for

With initial conditions

Result Solved

##### Maple

Result Did not solve

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#### 6.5 Wave PDE on string (finite domain) with zero initial position and velocity, and with source term

problem number 85

This is problem at page 115, David J Logan textbook, applied PDE textbook.

Falling cable lying on a table that is suddenly removed.

With boundary condition

And initial conditions

Result Solved

##### Maple

Result Solved

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#### 6.6 Wave PDE on string, one end fixed, another free, both initial conduitions non zero, and source that depends on time and space

problem number 86

Added July 2, 2018. Taken from Maple 2018.1 improvement to PDE document.

Solve

With boundary condition

And initial conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

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#### 6.7 Wave PDE on string (finite domain), fixed ends, no initial conduitions give and no source

problem number 87

This is problem at page 28, David J Logan textbook, applied PDE textbook.

With boundary condition

##### Mathematica

Result Did not solve

##### Maple

Result Solved

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#### 6.8 Wave PDE on string (finite domain), one fixed end, one free end, initial position not zero, initial velocity zero, no source

problem number 88

This is problem at page 130, David J Logan textbook, applied PDE textbook.

With boundary conditions

With initial conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

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#### 6.9 Wave PDE on string (finite domain), both ends fixed end, initial conditions zero, with source as generic function that depends on time and space

problem number 89

This is problem at page 149, David J Logan textbook, applied PDE textbook.

With boundary conditions

With initial conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

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#### 6.10 Wave PDE on string (finite domain), both ends fixed, initial conditions both not zero, No source

problem number 90

Taken from Maple 2018.1 improvements to PDE’s document.

Solve

For and . With boundary conditions

With initial conditions

Where and

Result Solved

##### Maple

Result Solved

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#### 6.11 Wave PDE on string (finite domain), both ends fixed end, initial conditions both not zero, and with constant source

problem number 91

Third example, from Maple 2018.1 improvements to PDE’s document.

Solve

For and . With boundary conditions

With initial conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

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#### 6.12 Wave PDE on string (finite domain), both ends fixed end, with source

problem number 92

This is problem at page 213, David J Logan textbook, applied PDE textbook.

With boundary conditions

With initial conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

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#### 6.13 Wave PDE on semi-infinite domain, with one end having a moving boundary condition

problem number 93

Solve for with and

With boundary conditions

With initial conditions

Result Solved

##### Maple

Result Solved

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#### 6.14 Telegraphy PDE, a wave PDE on string, both ends fixed with damping

problem number 94

Solve

With boundary conditions

With initial conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved, But should not be included.

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#### 6.15 Wave PDE, on string, both ends fixed. Initial velocity zero. Dispersion term present

problem number 95

Solve

Dispersion term causes the shape of the original wave to distort with time.

With and and with boundary conditions

With initial conditions

##### Mathematica

Result Did not solve due to adding dispersion term

##### Maple

Result Solved

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#### 6.16 Wave PDE on string with fixed ends, non-zero initial position

problem number 96

Solve

With boundary conditions

With initial conditions

##### Mathematica

Result Solved but sum should not include

##### Maple

Result Solved

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#### 6.17 Wave PDE homogeneous in square, given initial position but with zero initial velocity

problem number 97

Taken from Maple PDE help pages. This wave PDE inside square with free to move on left edge and right edge, and top and bottom edges are fixed. It has zero initial velocity, but given a non-zero initial position. Where and and .

Solve

With boundary conditions

With initial conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

_________________________________________________________________________________

#### 6.18 Wave PDE homogeneous in square with damping. Given zero initial position but with non-zero initial velocity

problem number 98

Taken from Maple PDE help pages. This wave PDE inside square with damping present.

Membrane is free to move on the right edge and also on top edge. But fixed at left edge and bottom edge.

It has zero initial position, but given a non-zero initial velocity. Where and and .

Solve

With boundary conditions

With initial conditions

##### Mathematica

Result Did not solve

##### Maple

Result Solved

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#### 6.19 Wave PDE inside rectangle. All 4 edges are fixed and given non-zero initial position with zero initial velocity

problem number 99

Taken from Mathematica helps pages on DSolve

Solve for with and and .

Solve

With boundary conditions

With initial conditions

Result Solved

##### Maple

Result Solved

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#### 6.20 Wave PDE inside disk. fixed edge of disk, no dependency, with initial position and velocity given

problem number 100

Taken from Mathematica helps pages on DSolve

Solve for with and .

With boundary conditions

With initial conditions

Result Solved

##### Maple

Result Did not solve

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#### 6.21 Wave PDE inside disk. fixed edge of disk, with dependency, zero initial velocity

problem number 101

Solve for with and and

With boundary conditions

With initial conditions

##### Mathematica

Result Did not solve

##### Maple

Result Did not solve

_________________________________________________________________________________

#### 6.22 Wave PDE on infinite domain with initial conditions specified, no source

problem number 102

Taken from Mathematica DSolve help pages.

Solve initial value wave PDE on infinite domain

With initial conditions

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 6.23 Wave PDE on infinite domain with initial conditions specified, with source term

problem number 103

Taken from Mathematica DSolve help pages.

Solve initial value wave PDE on infinite domain

With initial conditions

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 6.24 Wave PDE initial value with a Dirichlet condition on the half-line

problem number 104

Taken from Mathematica DSolve help pages.

Solve for initial value wave PDE on infinite domain with and .

With initial conditions

And boundary conditions

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 6.25 Wave PDE Initial value problem with a Neumann condition on the half-line

problem number 105

Taken from Mathematica DSolve help pages.

Solve initial value wave PDE on infinite domain

With initial conditions

And boundary conditions

Result Solved

##### Maple

Result Did not solve

_________________________________________________________________________________

#### 6.26 non-linear wave PDE (Solitons)

problem number 106

This was first solved analytically by (Krvskal, Zabrsky 1965).

Solve

##### Mathematica

Result Solved. build a special solution.

##### Maple

Result Solved. Returning a solution that is not the most general one

### 7 Schrodinger PDE

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#### 7.1 Schrodinger PDE with zero potential

problem number 107

From page 30, David J Logan textbook, applied PDE textbook.

Solve

With boundary conditions

Result Solved

##### Maple

Result Solved

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#### 7.2 Schrodinger PDE with initial and boundary conditions

problem number 108

Solve for

With boundary conditions

And initial conditions

Result Solved

##### Maple

Result Did not solve

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#### 7.3 Initial value problem with Dirichlet boundary conditions

problem number 109

Taken from Mathematica DSolve help pages

Solve for

With boundary conditions

And initial conditions where

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 7.4 Solve a Schrodinger equation with potential over the whole real line

problem number 110

Taken from Mathematica DSolve help pages

Solve for

With boundary conditions

Result Solved

##### Maple

Result Did not solve. Maple does not support in boundary conditions

### 8 Beam PDE

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#### 8.1 Beam PDE with zero initial velocity

problem number 111

Solve

With boundary conditions

And initial conditions

Result Solved

Result Solved

### 9 Burger’s PDE

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#### 9.1 viscous fluid flow with no initial conditions

problem number 112

From Mathematica symbolic PDE document.

Solve for

Result Solved

##### Maple

Result Solved

_________________________________________________________________________________

#### 9.2 viscous fluid flow with initial conditions

problem number 113

From Mathematica symbolic PDE document.

Solve for

With initial conditions

Result Solved

##### Maple

Result Solved, but has unresolved integrals

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#### 9.3 viscous fluid flow with initial conditions as UnitBox

problem number 114

From Mathematica DSolve help pages.

Solve for

With initial conditions

Result Solved

##### Maple

Result Solved, but has unresolved integrals

### 10 Black Scholes PDE

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#### 10.1 classic Black Scholes model from finance

problem number 115

From Mathematica symbolic PDE document.

Solve for where is the price of the option as a function of stock price and time . is the risk-free interest rate, and is the volatility of the stock.

With boundary condition

Result Solved

##### Maple

Result Did not solve

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#### 10.2 Boundary value problem for the Black Scholes equation

problem number 116

From Mathematica DSolve help pages.

Solve for

With boundary condition

Result Solved

##### Maple

Result Did not solve

### 11 Korteweg-deVries PDE

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#### 11.1 Korteweg-deVries (waves on shallow water surfaces) with no initial conditions

problem number 117

From Mathematica symbolic PDE document.

Solve for

Result Solved

Result Solved

### 12 Tricomi PDE

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#### 12.1 Boundary value problem for the Tricomi equation

problem number 118

From Mathematica DSolve helps pages.

Solve for

With boundary conditions

Result Solved

Result Solved

### 13 Cauchy Riemann PDE’s

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#### 13.1 Cauchy Riemann PDE with Prescribe the values of and on the axis

problem number 119

From Mathematica DSolve helps pages.

Solve for

With boundary conditions

Result Solved

##### Maple

Result Did not solve

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#### 13.2 Cauchy Riemann PDE With extra term on right side

problem number 120

Solve for

##### Mathematica

Result Did not Solve

Result Solved

### 14 Hamilton-Jacobi PDE

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#### 14.1 Hamilton-Jacobi type PDE

problem number 121

Taken from Maple pdsolve help pages, which is taken from Landau, L.D. and Lifshitz, E.M. Translated by Sykes, J.B. and Bell, J.S. Mechanics. Oxford: Pergamon Press, 1969

Solve for

##### Mathematica

Result Did not Solve

Result Solved

### 15 Other second order PDE’s

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#### 15.1 A second order PDE

problem number 122

Taken from Maple pdsolve help pages, problem 4.

Solve for

##### Mathematica

Result Did not Solve

Result Solved