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Solving partial differential equations in Maple and Mathematica

April 10, 2018   Compiled on April 10, 2018 at 12:32am

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). Windows 7.
2. Maple 2018.0 (64 bit). Windows 7 with ”Physics Updates.maple” version March 28, 2018.

The PC used is an Intel i7-3930k running at 3.20 GHz with 16 GB memory. 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 [ 113 ]. Mathematica solved 78 or 69.03%. Maple solved 89 or 78.76%.

 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, No source, nonhomogeneous BC Solved Solved 34 Heat PDE on bar, with source, nonhomogeneous BC Did not solve Solved 35 Heat PDE on bar with extra term Did not solve Solved 36 Heat PDE on bar with initial conditions sum of sine terms, no source Solved Solved 37 Heat PDE on bar, with initial conditions as piecewise function, no source Solved Solved 38 Heat PDE on bar, with time dependent B.C, no source Solved Did not solve 39 Heat PDE on bar, homogeneous Neumann boundary conditions, with source as sin function Did not solve Solved 40 Heat PDE on bar, homogeneous Neumann boundary conditions, with source that depends on time and space Did not solve Solved 41 Heat PDE on bar, Dirichlet boundary conditions that depends on time with source that depends on space only Did not solve Did not solve 42 Heat PDE on bar, homogeneous Dirichlet boundary conditions, with source that depends on time and space Did not solve Solved 43 Heat/Diffusion PDE in 2D, inside rectangle with initial and boundary conditions Did not solve Solved 44 Heat/Diffusion PDE in 2D, inside rectangle with initial and boundary conditions with heat loss Did not solve Solved 45 Heat PDE inside disk, with no dependency. initial and boundary conditions given Solved Did not solve 46 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 47 Heat PDE in 1D on the whole real line with initial position specified Solved Solved 48 Heat PDE in 1D on the whole real line with extra term Solved Solved 49 Heat PDE in 1D on the whole real line with initial position as UnitBox Solved Solved 50 Heat PDE on half the line with non-zero initial conditions and Dirichlet boundary conditions Solved Solved, but has unresolved inverse Laplace transforms 51 Heat PDE on half the line with zero initial conditions and time dependent boundary conditions Solved Solved, but has unresolved inverse Laplace transforms 52 Initial value problem for the heat PDE with a Neumann condition on the half-line Solved Did not solve 53 Laplace PDE inside quarter-circle Did not solve Did not solve 54 Laplace PDE inside semi-circle Did not solve Solved 55 Laplace PDE inside rectangle Solved Solved 56 Laplace PDE inside rectangle Solved Solved 57 Laplace PDE inside rectangle Did not solve Solved 58 Laplace PDE inside rectangle Did not solve Solved 59 Laplace PDE inside rectangle Did not solve Solved 60 Laplace PDE inside rectangle, top/bottom edges non-zero Solved Solved 61 Laplace PDE inside circular annulus, Neumann boundary conditions using unspecified functions Did not solve Did not solve 62 Laplace PDE inside circular annulus, Dirichlet boundary conditions using specified functions Solved Did not solve 63 Laplace PDE example 18 from Maple help page Solved Solved 64 Laplace PDE on rectangle with one edge at infinity Did not solve Solved 65 Laplace PDE inside a disk, periodic boundary conditions Solved Solved 66 Dirichlet problem for the Laplace equation in upper half plan Solved Did not solve 67 Dirichlet problem for the Laplace equation in right half-plane: Solved Did not solve 68 Dirichlet problem for the Laplace equation in the first quadrant Solved Did not solve 69 Neumann problem for the Laplace equation in the upper half-plane Solved Did not solve 70 Dirichlet problem for the Laplace equation in a rectangle Solved Solved 71 Laplace PDE outside a disk, periodic boundary conditions Did not solve Solved 72 Laplace equation in spherical coordinates Did not solve Solved, but not verified 73 Dirichlet problem for the Poisson equation in a rectangle Solved Solved 74 Dirichlet problem for the Helmholtz equation in a rectangle Solved Did not solve 75 General solution for a second-order hyperbolic PDE on real line Solved Solved 76 Hyperbolic PDE with non-rational coefficients Solved Did not solve. Tried all HINTS 77 Inhomogeneous hyperbolic PDE with constant coefficients Solved Solved 78 system of 2 inhomogeneous linear hyperbolic system with constant coefficients Solved Did not solve 79 Wave PDE on string (finite domain) with source Solved Solved 80 Wave PDE on string (finite domain), fixed ends Did not solve Solved 81 Wave PDE on string (finite domain), one fixed end, one free end Did not solve Solved 82 Wave PDE on string (finite domain), both ends fixed end, with source Did not solve Solved 83 Wave PDE on string (finite domain), both ends fixed end, with source Did not solve Solved 84 Wave PDE on semi-infinite domain, with one end having a moving boundary condition Solved Solved 85 Telegraphy PDE, a wave PDE on string, both ends fixed with damping Did not solve Solved, But should not be included. 86 Wave PDE, on string, both ends fixed. Initial velocity zero. Dispersion term present Did not solve due to adding dispersion term Solved 87 Wave PDE on string with fixed ends, non-zero initial position Solved but sum should not include Solved, but sum should not include 88 Wave PDE homogeneous in square, given initial position but with zero initial velocity Did not solve Solved 89 Wave PDE homogeneous in square with damping. Given zero initial position but with non-zero initial velocity Did not solve Solved 90 Wave PDE inside rectangle. All 4 edges are fixed and given non-zero initial position with zero initial velocity Solved Solved 91 Wave PDE inside disk. fixed edge of disk, no dependency, with initial position and velocity given Solved Did not solve 92 Wave PDE inside disk. fixed edge of disk, with dependency, zero initial velocity Did not solve Did not solve 93 Wave PDE on infinite domain with initial conditions specified, no source Solved Solved 94 Wave PDE on infinite domain with initial conditions specified, with source term Solved Solved 95 Wave PDE initial value with a Dirichlet condition on the half-line Solved Solved 96 Wave PDE Initial value problem with a Neumann condition on the half-line Solved Did not solve 97 non-linear wave PDE (Solitons) Solved. build a special solution. Solved. Returning a solution that is not the most general one 98 Schrodinger PDE with zero potential Solved Solved 99 Schrodinger PDE with initial and boundary conditions Solved Did not solve 100 Initial value problem with Dirichlet boundary conditions Solved Solved 101 Solve a Schrodinger equation with potential over the whole real line Solved Did not solve. Maple does not support in boundary conditions 102 Beam PDE with zero initial velocity Solved Solved 103 viscous fluid flow with no initial conditions Solved Solved 104 viscous fluid flow with initial conditions Solved Solved, but has unresolved integrals 105 viscous fluid flow with initial conditions as UnitBox Solved Solved, but has unresolved integrals 106 classic Black Scholes model from finance Solved Did not solve 107 Boundary value problem for the Black-Scholes equation Solved Did not solve 108 Korteweg-deVries (waves on shallow water surfaces) with no initial conditions Solved Solved 109 Boundary value problem for the Tricomi equation Solved Did not solve 110 Cauchy-Riemann PDE with Prescribe the values of and on the axis Solved Did not solve 111 Cauchy-Riemann PDE With extra term on right side Did not Solve Solved 112 Hamilton-Jacobi type PDE Did not Solve Solved 113 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

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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

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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

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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

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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

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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

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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

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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

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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

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1.11 Another example of nonlinear Clairaut equation

problem number 11

Taken from Mathematica DSolve help pages

Solve for

Result Solved

Maple

Result Solved

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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, No source, nonhomogeneous BC

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, with source, nonhomogeneous BC

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 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, no source

problem number 36

Solve the heat equation

For and . The boundary conditions are

Initial condition is

Result Solved

Maple

Result Solved

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2.23 Heat PDE on bar, with initial conditions as 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, with time dependent B.C, no source

problem number 38

added March 8, 2018. Exam problem

Solve the heat equation

For and . The boundary conditions are

Initial condition is .

Result Solved

Maple

Result Did not solve

_________________________________________________________________________________

2.25 Heat PDE on bar, homogeneous Neumann boundary conditions, with source as sin function

problem number 39

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

_________________________________________________________________________________

2.26 Heat PDE on bar, homogeneous Neumann boundary conditions, with source that depends on time and space

problem number 40

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, Dirichlet boundary conditions that depends on time with source that depends on space only

problem number 41

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 Did not solve

_________________________________________________________________________________

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

problem number 42

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

_________________________________________________________________________________

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

problem number 43

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

_________________________________________________________________________________

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

problem number 44

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

_________________________________________________________________________________

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

problem number 45

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

_________________________________________________________________________________

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

problem number 46

Solve the heat equation for

Mathematica

Result Did not solve

Maple

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

_________________________________________________________________________________

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

problem number 47

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

With initial condition

Result Solved

Maple

Result Solved

_________________________________________________________________________________

2.34 Heat PDE in 1D on the whole real line with extra term

problem number 48

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

With initial condition

Result Solved

Maple

Result Solved

_________________________________________________________________________________

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

problem number 49

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

_________________________________________________________________________________

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

problem number 50

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

_________________________________________________________________________________

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

problem number 51

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

_________________________________________________________________________________

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

problem number 52

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

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3.1 Laplace PDE inside quarter-circle

problem number 53

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 54

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 55

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 56

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 57

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 58

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 59

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 60

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 61

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 62

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 63

Solve Laplace equation

With boundary conditions

Result Solved

Maple

Result Solved

_________________________________________________________________________________

3.12 Laplace PDE on rectangle with one edge at infinity

problem number 64

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 65

Solve Laplace equation in polar coordinates inside a disk

Solve for

Boundary conditions

Result Solved

Maple

Result Solved

_________________________________________________________________________________

3.14 Dirichlet problem for the Laplace equation in upper half plan

problem number 66

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

_________________________________________________________________________________

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

problem number 67

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 68

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 69

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 70

Taken from Mathematica DSolve help pages

Solve for

Boundary conditions .

Result Solved

Maple

Result Solved

_________________________________________________________________________________

3.19 Laplace PDE outside a disk, periodic boundary conditions

problem number 71

Solve Laplace equation in polar coordinates outside a disk

Solve for

Boundary conditions

Mathematica

Result Did not solve

Maple

Result Solved

_________________________________________________________________________________

3.20 Laplace equation in spherical coordinates

problem number 72

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 73

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 74

Taken from Mathematica DSolve help pages.

Solve for

Boundary conditions

Result Solved

Maple

Result Did not solve

6 Wave PDE

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

problem number 75

From Mathematica DSolve help pages (slightly modified)

Solve for with on real line

Result Solved

Maple

Result Solved

_________________________________________________________________________________

6.2 Hyperbolic PDE with non-rational coefficients

problem number 76

From Mathematica DSolve help pages

Solve for

Result Solved

Maple

Result Did not solve. Tried all HINTS

_________________________________________________________________________________

6.3 Inhomogeneous hyperbolic PDE with constant coefficients

problem number 77

From Mathematica DSolve help pages

Solve for

Result Solved

Maple

Result Solved

_________________________________________________________________________________

6.4 system of 2 inhomogeneous linear hyperbolic system with constant coefficients

problem number 78

From Mathematica DSolve help pages

Solve for

With initial conditions

Result Solved

Maple

Result Did not solve

_________________________________________________________________________________

6.5 Wave PDE on string (finite domain) with source

problem number 79

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

_________________________________________________________________________________

6.6 Wave PDE on string (finite domain), fixed ends

problem number 80

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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6.7 Wave PDE on string (finite domain), one fixed end, one free end

problem number 81

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

problem number 82

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

problem number 83

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

problem number 84

Solve for with and

With boundary conditions

With initial conditions

Result Solved

Maple

Result Solved

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

problem number 85

Solve

With boundary conditions

With initial conditions

Mathematica

Result Did not solve

Maple

Result Solved, But should not be included.

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

problem number 86

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

problem number 87

Solve

With boundary conditions

With initial conditions

Mathematica

Result Solved but sum should not include

Maple

Result Solved, but sum should not include

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

problem number 88

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

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

problem number 89

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

problem number 90

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

problem number 91

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

problem number 92

Solve for with and and

With boundary conditions

With initial conditions

Mathematica

Result Did not solve

Maple

Result Did not solve

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

problem number 93

Taken from Mathematica DSolve help pages.

Solve initial value wave PDE on infinite domain

With initial conditions

Result Solved

Maple

Result Solved

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

problem number 94

Taken from Mathematica DSolve help pages.

Solve initial value wave PDE on infinite domain

With initial conditions

Result Solved

Maple

Result Solved

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

problem number 95

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

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

problem number 96

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

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6.23 non-linear wave PDE (Solitons)

problem number 97

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 98

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 99

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 100

Taken from Mathematica DSolve help pages

Solve for

With boundary conditions

And initial conditions where

Result Solved

Maple

Result Solved

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

problem number 101

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 102

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 103

From Mathematica symbolic PDE document.

Solve for

Result Solved

Maple

Result Solved

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9.2 viscous fluid flow with initial conditions

problem number 104

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 105

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 106

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 107

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 108

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 109

From Mathematica DSolve helps pages.

Solve for

With boundary conditions

Result Solved

Maple

Result Did not solve

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 110

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 111

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 112

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 113

Taken from Maple pdsolve help pages, problem 4.

Solve for

Mathematica

Result Did not Solve

Result Solved