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

1 First order PDE

1.1 Linear PDE, the transport equation

1.2 Linear PDE

1.3 Linear PDE, initial value problem

1.4 Initial-boundary value problem

1.5 Linear PDE, the transport equation with initial conditions

1.6 General solution for a quasilinear first-order PDE

1.7 quasilinear first-order PDE, scalar conservation law

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

1.9 nonlinear first-order PDE, the Clairaut equation

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

1.11 Another example of nonlinear Clairaut equation

1.12 Recover a function from its gradient vector

1.13 General solution of a first order nonlinear PDE

1.14 Nonlinear first order PDE

2 Heat PDE

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2.17 Heat PDE infinite domain

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

3 Laplace PDE

3.1 Laplace PDE inside quarter-circle

3.2 Laplace PDE inside semi-circle

3.3 Laplace PDE inside rectangle

3.4 Laplace PDE inside rectangle

3.5 Laplace PDE inside rectangle

3.6 Laplace PDE inside rectangle

3.7 Laplace PDE inside rectangle

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

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

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

3.11 Laplace PDE example 18 from Maple help page

3.12 Laplace PDE on rectangle with one edge at infinity

3.13 Laplace PDE inside a disk, periodic boundary conditions

3.14 Dirichlet problem for the Laplace equation in upper half plan

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

3.16 Dirichlet problem for the Laplace equation in the first quadrant

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

3.18 Dirichlet problem for the Laplace equation in a rectangle

3.19 Laplace PDE outside a disk, periodic boundary conditions

3.20 Laplace equation in spherical coordinates

4 Poisson PDE

4.1 Dirichlet problem for the Poisson equation in a rectangle

5 Helmholtz PDE

5.1 Dirichlet problem for the Helmholtz equation in a rectangle

6 Wave PDE

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

6.2 Hyperbolic PDE with non-rational coefficients

6.3 Inhomogeneous hyperbolic PDE with constant coefficients

6.4 system of 2 inhomogeneous linear hyperbolic system with constant coefficients

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

6.26 non-linear wave PDE (Solitons)

7 Schrodinger PDE

7.1 Schrodinger PDE with zero potential

7.2 Schrodinger PDE with initial and boundary conditions

7.3 Initial value problem with Dirichlet boundary conditions

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

8 Beam PDE

8.1 Beam PDE with zero initial velocity

9 Burger’s PDE

9.1 viscous fluid flow with no initial conditions

9.2 viscous fluid flow with initial conditions

9.3 viscous fluid flow with initial conditions as UnitBox

10 Black Scholes PDE

10.1 classic Black Scholes model from finance

10.2 Boundary value problem for the Black Scholes equation

11 Korteweg-deVries PDE

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

12 Tricomi PDE

12.1 Boundary value problem for the Tricomi equation

13 Cauchy Riemann PDE’s

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

13.2 Cauchy Riemann PDE With extra term on right side

14 Hamilton-Jacobi PDE

14.1 Hamilton-Jacobi type PDE

15 Other second order PDE’s

15.1 A second order PDE

1.1 Linear PDE, the transport equation

1.2 Linear PDE

1.3 Linear PDE, initial value problem

1.4 Initial-boundary value problem

1.5 Linear PDE, the transport equation with initial conditions

1.6 General solution for a quasilinear first-order PDE

1.7 quasilinear first-order PDE, scalar conservation law

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

1.9 nonlinear first-order PDE, the Clairaut equation

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

1.11 Another example of nonlinear Clairaut equation

1.12 Recover a function from its gradient vector

1.13 General solution of a first order nonlinear PDE

1.14 Nonlinear first order PDE

2 Heat PDE

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

2.17 Heat PDE infinite domain

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

3 Laplace PDE

3.1 Laplace PDE inside quarter-circle

3.2 Laplace PDE inside semi-circle

3.3 Laplace PDE inside rectangle

3.4 Laplace PDE inside rectangle

3.5 Laplace PDE inside rectangle

3.6 Laplace PDE inside rectangle

3.7 Laplace PDE inside rectangle

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

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

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

3.11 Laplace PDE example 18 from Maple help page

3.12 Laplace PDE on rectangle with one edge at infinity

3.13 Laplace PDE inside a disk, periodic boundary conditions

3.14 Dirichlet problem for the Laplace equation in upper half plan

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

3.16 Dirichlet problem for the Laplace equation in the first quadrant

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

3.18 Dirichlet problem for the Laplace equation in a rectangle

3.19 Laplace PDE outside a disk, periodic boundary conditions

3.20 Laplace equation in spherical coordinates

4 Poisson PDE

4.1 Dirichlet problem for the Poisson equation in a rectangle

5 Helmholtz PDE

5.1 Dirichlet problem for the Helmholtz equation in a rectangle

6 Wave PDE

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

6.2 Hyperbolic PDE with non-rational coefficients

6.3 Inhomogeneous hyperbolic PDE with constant coefficients

6.4 system of 2 inhomogeneous linear hyperbolic system with constant coefficients

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

6.26 non-linear wave PDE (Solitons)

7 Schrodinger PDE

7.1 Schrodinger PDE with zero potential

7.2 Schrodinger PDE with initial and boundary conditions

7.3 Initial value problem with Dirichlet boundary conditions

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

8 Beam PDE

8.1 Beam PDE with zero initial velocity

9 Burger’s PDE

9.1 viscous fluid flow with no initial conditions

9.2 viscous fluid flow with initial conditions

9.3 viscous fluid flow with initial conditions as UnitBox

10 Black Scholes PDE

10.1 classic Black Scholes model from finance

10.2 Boundary value problem for the Black Scholes equation

11 Korteweg-deVries PDE

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

12 Tricomi PDE

12.1 Boundary value problem for the Tricomi equation

13 Cauchy Riemann PDE’s

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

13.2 Cauchy Riemann PDE With extra term on right side

14 Hamilton-Jacobi PDE

14.1 Hamilton-Jacobi type PDE

15 Other second order PDE’s

15.1 A second order PDE

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.

- Mathematica 11.3 (64 bit).
- 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%.

_________________________________________________________________________________

Taken from Mathematica Symbolic PDE document

Solve for

Result Solved

Result Solved

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Taken from Mathematica help pages

Solve for

Result Solved

Result Solved

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Taken from Mathematica help pages

Solve for

with initial value

Result Solved

Result Solved

_________________________________________________________________________________

Taken from Mathematica help pages

Solve for

with initial value and boundary value

Result Solved

Result Did not solve

_________________________________________________________________________________

Taken from Mathematica help pages

Solve for

With initial conditions

Result Solved

Result Solved

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Taken from Mathematica help pages

Solve for

Result Solved

Result Solved

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Taken from Mathematica Symbolic PDE document

Solve for

Result Solved, solution in implicit form

Result Solved

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Taken from Mathematica Symbolic PDE document

Solve for

With

Result Solved

Result Solved

_________________________________________________________________________________

Taken from Mathematica Symbolic PDE document

Solve for

Result Solved

Result Solved

_________________________________________________________________________________

Taken from Mathematica Symbolic PDE document

Solve for

With

Result Solved

Result Solved

_________________________________________________________________________________

Taken from Mathematica DSolve help pages

Solve for

Result Solved

Result Solved

_________________________________________________________________________________

Taken from Mathematica DSolve help pages

Solve for

Result Solved

Result Solved

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Taken from Maple pdsolve help pages

Solve for

Result Did not solve

Result Solved

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Taken from Maple pdsolve help pages, probem 5

Solve for

Result Solved

Result Solved

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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

Result Solved

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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

Result Solved

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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

Result Solved

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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

Result Solved

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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

Result Solved

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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

Result Did not solve

Result Solved

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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

Result Solved

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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

Result Solved

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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

Result Solved

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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

Result Solved

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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

Result Solved

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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

Result Solved

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Solve the heat equation

For and . The boundary conditions are

And initial conditions

Result Did not solve

Result Solved

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Solve the heat equation

For and . The boundary conditions is and And initial condition

Result Solved

Result Solved

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Solve the heat equation

For and . The boundary conditions is and And initial condition

Result Solved

Result Solved

_________________________________________________________________________________

Solve the heat equation

For and . The boundary conditions are

And initial condition

Result Did not solve

Result Solved

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Solve the heat equation

For and . The boundary conditions are

Initial condition is

Result Solved

Result Solved

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Solve the heat equation

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

Result Did not solve

Result Solved

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Taken from Maple PDE help pages

Solve the heat equation

For and . The boundary conditions are

Initial condition is

Result Solved

Result Solved

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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

Result Did not solve

Result Solved

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Solve the heat equation

For and . The boundary conditions are

Initial condition is

Result Did not solve

Result Solved

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added Feb 10, 2018.

Solve the heat equation

For and . The boundary conditions are

Initial condition is

Result Solved

Result Solved

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added Feb 10, 2018.

Solve the heat equation

For and . The boundary conditions are

Initial condition is

Result Solved

Result Solved

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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

Result Solved

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added March 8, 2018. Exam problem

Solve the heat equation

For and . The boundary conditions are

Initial condition is .

Result Solved

Result Solved

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added March 18, 2018.

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 .

Result Did not solve

Result Solved

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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 .

Result Did not solve

Result Solved

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added March 18, 2018.

Solve the heat equation

For and . The boundary conditions are

Initial condition is .

Result Did not solve

Result Solved

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added July 2, 2018.

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 .

Result Did not solve

Result Solved

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added July 2, 2018.

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 .

Result Did not solve

Result Solved

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added July 2, 2018.

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

Result Solved

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added July 2, 2018.

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

Result Solved

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

Solve the heat equation

For and . The boundary conditions are

Initial condition is .

Result Did not solve

Result Solved

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Taken from Maple PDE help pages

Solve the heat equation for

For and . The boundary conditions are

Initial condition is

Result Did not solve

Result Solved

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Taken from Maple help pages on PDE

Solve the heat equation for

For and and . The boundary conditions are

Initial condition is .

Result Did not solve

Result Solved

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Taken from Maple help pages on PDE

Solve the heat equation for

For and and . The boundary conditions are

Initial condition is .

Result Did not solve

Result Solved

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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

Result Did not solve

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Solve the heat equation for

Result Did not solve

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

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

With initial condition

Result Solved

Result Solved

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

With initial condition

Result Solved

Result Solved

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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

Result Solved

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From Mathematica DSolve help pages.

Solve the heat equation for on half the line and

With initial condition

And boundary conditions

Result Solved

Result Solved, but has unresolved inverse Laplace transforms

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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

Result Solved, but has unresolved inverse Laplace transforms

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From Mathematica DSolve help pages.

Solve the heat equation for on half the line and

With initial condition

And boundary conditions

Result Solved

Result Did not solve

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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

Result Did not solve

Result Did not solve

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Solve Laplace equation

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

Result Did not solve

Result Solved

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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

Result Solved

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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

Result Solved

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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

Result Did not solve

Result Solved

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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

Result Did not solve

Result Solved

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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

Result Did not solve

Result Solved

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Taken from Mathematica DSolve help pages.

Solve Laplace equation

inside a rectangle , with following boundary conditions

Result Solved

Result Solved

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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

Result Did not solve

Result Did not solve

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Solve Laplace equation

Inside circular annulus subject to the following boundary conditions

Result Solved

Result Did not solve

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Solve Laplace equation

With boundary conditions

Result Solved

Result Solved

_________________________________________________________________________________

Solve Laplace equation

With boundary conditions

Result Did not solve

Result Solved

_________________________________________________________________________________

Solve Laplace equation in polar coordinates inside a disk

Solve for

Boundary conditions

Result Solved

Result Solved

_________________________________________________________________________________

Taken from Mathematica DSolve help pages

Solve for

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

Result Solved

Result Did not solve

_________________________________________________________________________________

Taken from Mathematica DSolve help pages

Solve for

Boundary conditions .

Result Solved

Result Did not solve

_________________________________________________________________________________

Taken from Mathematica DSolve help pages

Solve for

Boundary conditions .

Result Solved

Result Did not solve

_________________________________________________________________________________

Taken from Mathematica DSolve help pages

Solve for

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

Result Solved

Result Did not solve

_________________________________________________________________________________

Taken from Mathematica DSolve help pages

Solve for

Boundary conditions .

Result Solved

Result Solved

_________________________________________________________________________________

Solve Laplace equation in polar coordinates outside a disk

Solve for

Boundary conditions

Result Did not solve

Result Solved

_________________________________________________________________________________

Taken from Maple pdsolve help pages

Solve for

Result Did not solve

Result Solved, but not verified

_________________________________________________________________________________

Taken from Mathematica DSolve help pages.

Solve for

Boundary conditions

Result Solved

Result Solved

_________________________________________________________________________________

Taken from Mathematica DSolve help pages.

Solve for

Boundary conditions

Result Solved

Result Solved

_________________________________________________________________________________

From Mathematica DSolve help pages (slightly modified)

Solve for with on real line

Result Solved

Result Solved

_________________________________________________________________________________

From Mathematica DSolve help pages

Solve for

Result Solved

Result Did not solve. Tried all HINTS

_________________________________________________________________________________

From Mathematica DSolve help pages

Solve for

Result Solved

Result Solved

_________________________________________________________________________________

From Mathematica DSolve help pages

Solve for

With initial conditions

Result Solved

Result Did not solve

_________________________________________________________________________________

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

Result Solved

_________________________________________________________________________________

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

Solve

With boundary condition

And initial conditions

Result Did not solve

Result Solved

_________________________________________________________________________________

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

With boundary condition

Result Did not solve

Result Solved

_________________________________________________________________________________

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

With boundary conditions

With initial conditions

Result Did not solve

Result Solved

_________________________________________________________________________________

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

With boundary conditions

With initial conditions

Result Did not solve

Result Solved

_________________________________________________________________________________

Added July 2, 2018.

Taken from Maple 2018.1 improvements to PDE’s document.

Solve

For and . With boundary conditions

With initial conditions

Where and

Result Solved

Result Solved

_________________________________________________________________________________

Added July 2, 2018.

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

Solve

For and . With boundary conditions

With initial conditions

Result Did not solve

Result Solved

_________________________________________________________________________________

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

With boundary conditions

With initial conditions

Result Did not solve

Result Solved

_________________________________________________________________________________

Solve for with and

With boundary conditions

With initial conditions

Result Solved

Result Solved

_________________________________________________________________________________

Solve

With boundary conditions

With initial conditions

Result Did not solve

Result Solved, But should not be included.

_________________________________________________________________________________

Solve

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

With and and with boundary conditions

With initial conditions

Result Did not solve due to adding dispersion term

Result Solved

_________________________________________________________________________________

Added March 9, 2018.

Solve

With boundary conditions

With initial conditions

Result Solved but sum should not include

Result Solved

_________________________________________________________________________________

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

Result Did not solve

Result Solved

_________________________________________________________________________________

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

Result Did not solve

Result Solved

_________________________________________________________________________________

Taken from Mathematica helps pages on DSolve

Solve for with and and .

Solve

With boundary conditions

With initial conditions

Result Solved

Result Solved

_________________________________________________________________________________

Taken from Mathematica helps pages on DSolve

Solve for with and .

With boundary conditions

With initial conditions

Result Solved

Result Did not solve

_________________________________________________________________________________

Solve for with and and

With boundary conditions

With initial conditions

Result Did not solve

Result Did not solve

_________________________________________________________________________________

Taken from Mathematica DSolve help pages.

Solve initial value wave PDE on infinite domain

With initial conditions

Result Solved

Result Solved

_________________________________________________________________________________

Taken from Mathematica DSolve help pages.

Solve initial value wave PDE on infinite domain

With initial conditions

Result Solved

Result Solved

_________________________________________________________________________________

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

Result Solved

_________________________________________________________________________________

Taken from Mathematica DSolve help pages.

Solve initial value wave PDE on infinite domain

With initial conditions

And boundary conditions

Result Solved

Result Did not solve

_________________________________________________________________________________

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

Solve

Result Solved. build a special solution.

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

_________________________________________________________________________________

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

Solve

With boundary conditions

Result Solved

Result Solved

_________________________________________________________________________________

Solve for

With boundary conditions

And initial conditions

Result Solved

Result Did not solve

_________________________________________________________________________________

Taken from Mathematica DSolve help pages

Solve for

With boundary conditions

And initial conditions where

Result Solved

Result Solved

_________________________________________________________________________________

Taken from Mathematica DSolve help pages

Solve for

With boundary conditions

Result Solved

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

_________________________________________________________________________________

Added January 20, 2018.

Solve

With boundary conditions

And initial conditions

Result Solved

Result Solved

_________________________________________________________________________________

From Mathematica symbolic PDE document.

Solve for

Result Solved

Result Solved

_________________________________________________________________________________

From Mathematica symbolic PDE document.

Solve for

With initial conditions

Result Solved

Result Solved, but has unresolved integrals

_________________________________________________________________________________

From Mathematica DSolve help pages.

Solve for

With initial conditions

Result Solved

Result Solved, but has unresolved integrals

_________________________________________________________________________________

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

Reference https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation

Result Solved

Result Did not solve

_________________________________________________________________________________

From Mathematica DSolve help pages.

Solve for

With boundary condition

Reference https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_equation

Result Solved

Result Did not solve

_________________________________________________________________________________

From Mathematica symbolic PDE document.

Solve for

Reference https://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation

Result Solved

Result Solved

_________________________________________________________________________________

From Mathematica DSolve helps pages.

Solve for

With boundary conditions

Result Solved

Result Solved

_________________________________________________________________________________

From Mathematica DSolve helps pages.

Solve for

With boundary conditions

Result Solved

Result Did not solve

_________________________________________________________________________________

Solve for

Result Did not Solve

Result Solved

_________________________________________________________________________________

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

Result Did not Solve

Result Solved

_________________________________________________________________________________

Taken from Maple pdsolve help pages, problem 4.

Solve for

Result Did not Solve

Result Solved