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July 20, 2017 compiled on — Thursday July 20, 2017 at 11:26 AM

This is a collection of PDE problems solved analytically using CAS. Current systems used are Maple 2017 and Mathematica 11.1.1. Problems are mostly taken from textbooks and help pages. A number of these I solved by hand as well for HW’s.

For each problem, the CAS commands used are given, and a comment is given about the solution produced. This is work in progress.

It is clear from the results obtained, that CAS is still not very strong at solving PDE’s analytically (compared to solving ODE’s for example). All the PDE’s that were tried below are basic ones, which all can be solved by hand analytically at the level of undergraduate or first year graduate level, but a number of them could not be solved, or the solution given was incomplete or wrong.

Notation used in the table below: T1 is textbook Richard Haberman applied partial differential equations 5th edition. T2 is David J. Logan’s Applied Partial Differential Equations.

ToDo: Need to make summary of missing features or class of PDE’s CAS can’t solve.

In Maple the command pdsolve is used and in Mathematica the command DSolve is used.

Summary lookup table of result

Problem |
Mathematica |
Maple |

T1, Problem 2.3.3(a) (heat PDE, 1D) |
Solved |
Solved |

T1, Problem 2.3.3(b) (heat PDE, 1D) |
Solved |
Solved |

T1, Problem 2.3.3(c) (heat PDE, 1D) |
Solved |
Solved |

T1, Problem 2.3.3(d) (heat PDE, 1D) |
Solved |
Solved |

T1, Problem 2.3.7 (heat PDE, 1D) |
Solved |
Partially correct. Missing zero eigenvalue |

T1, Problem 2.3.8 (heat PDE, 1D) |
Did not solve |
Solved |

T1, Problem 2.4.1(a) (heat PDE, 1D) |
Solved |
Partially correct. Missing zero eigenvalue |

T1, Problem 2.4.1(b) (heat PDE, 1D) |
Solved |
Did not solve |

T1, Problem 2.4.1(c) (heat PDE, 1D) |
Solved |
Partially correct. Missing zero eigenvalue |

T1, Problem 2.4.1(d) (heat PDE, 1D) |
Solved |
Did not solve |

T1, Problem 2.4.2 (heat PDE, 1D) |
Solved |
Partially correct. Used instead of ? |

T1, Problem 2.5.1(a) (Laplace PDE, 2D) |
Solved |
Did not solve |

T1, Problem 2.5.1(b) (Laplace PDE, 2D) |
Solved |
Did not solve |

T1, Problem 2.5.1(c) (Laplace PDE, 2D) |
Did not solve |
Did not solve |

T1, Problem 2.5.1(d) (Laplace PDE, 2D) |
Did not solve |
Did not solve |

T1, Problem 2.5.1(e) (Laplace PDE, 2D) |
Did not solve |
Did not solve |

T1, Problem 2.5.5(c) (Laplace PDE,quarter circle) |
Did not solve |
Error in input. Need to find why |

T1, Problem 2.5.8(b) (Laplace on circular annulus) | Did not solve | Did not solve |

T2, page 115 (Wave PDE 1D nonhomogeneous) |
Solved |
Solved |

T2, page 76 (heat PDE 1D) |
Solved |
Solved |

T2, page 28 (Wave PDE 1D) |
Did not solve |
Solved |

T2, page 130 (Wave PDE 1D) |
Did not solve |
Partially correct. Used instead of ? as above. |

T2, page 131 (heat PDE 1D, periodic BC) |
Did not solve |
Solved |

T2, page 149 (wave PDE 1D, nonhomogeneous) |
Did not solve |
Solved |

T2, page 213 (wave PDE 1D, nonhomogeneous) |
Did not solve |
Solved |

T2, page 30 (Schrodinger PDE with zero potentia) |
Solved |
Solved |

Problem 18 (Maple help page, p20, Laplace PDE, 1D) |
Solved |
Solved |

PDE 2 (Maple help p14, wave PDE 1D) |
Solved |
Solved |

PDE 3 (Maple help p14, heat PDE 1D) |
Solved |
Solved |

PDE 4 (Maple help p15, heat PDE 1D) |
Solved |
Solved |

heat absorption radiation in bounded domain |
Did not solve |
Solved |

heat PDE infinite domain with initial conditions, nonhomogeneous |
Solved |
Solved |

PDE 23, page 23, from Maple pdsolve help PDF file |
Solved |
Solved |

heat PDE on 1D with boundary conditions from |
Did not solve |
Did not solve |

comment Solution is correct.

comment Solution is correct.

comment Verified OK against my analytical solution.

comment Ok

comment I did not solve this by hand. Assuming solution is correct.

comment Solved, and gives same solution as Maple, assuming correct solution.

comment Verified correct solution, compared to my hand solution.

comment Solved, form is little different from Maple and my solution, but verified to be same by numerically evaluating at random times and locations.

comment Answer is partially correct. This is a problem I found in Maple’s solutions when both ends of the heat PDE are insulated. Maple overlooks the zero eigenvalue and seems to assume eigenvalues are all greater than zero. See my analytical solution. As the book also shows in the answer it gives.

comment Solved. Mathematica did better here than Maple as it accounted for the zero eigenvalue.

comment Ok.

comment Did not solve. Input returned.

comment Partially correct. Same problem as mentioned earlier. Maple overlooks the zero eigenvalue. The correct answer is

comment Mathematica gives the correct answer. It accounts for the zero eigenvalue.

comment Did not solve. Returned ().

comment Ok

The above is verified using Maple

comment Partially correct. Overlooked the zero eigenvalue.

comment Correct solution. Mathematica accounted for the zero eigenvalue.

comment Did not solve. Returned ().

comment Ok

Verified by Maple:

comment This seems to be wrong solution. It should be and not . Otherwise, the sum should start from zero. See Mathematica solution below which also matches my hand solution.

comment Correct solution.

comment Did not solve. Returned ().

comment Solved Ok.

comment Did not solve. Returned ().

comment Solved Ok.

comment Did not solve. Returned ().

comment Could not solve.

The solution is

comment Did not solve. Returned ().

comment Could not solve.

comment Did not solve. Returned ().

comment Could not solve.

comment I get an error which I do not understand. Here it is

comment Did not solve.

comment Did not solve.

comment Did not solve.

Falling cable lying on a table that is suddenly removed.

With initial conditions

And boundary condition

comment Solved.

comment Solved.

Verify Mathematica solution using Maple

With initial conditions

And boundary condition

comment Solved.

comment Solved.

With boundary conditions

comment Solved.

comment Could not solve.

Solve

With initial conditions

And boundary conditions

comment Solved. But sum is wrong. It should be and not . If is to be used, then the sum index should start from zero and not one.

comment Did not solve.

Solve

With initial conditions

And boundary conditions

comment Solved.

comment Did not solve.

Solve

With initial conditions

And boundary conditions

comment Solved.

comment Did not solve. Gives only trivial solution.

Solve

With initial conditions

And boundary conditions

comment Solved.

comment Did not solve.

Solve

With boundary conditions

comment Solved.

comment Solved.

Solve

With

comment Solved.

comment Solved.

Verify Mathematica solution using Maple

Solve

With initial conditions

And boundary condition

comment Solved.

comment Solved.

Solve

With initial conditions

And boundary condition

comment Solved.

comment Solved.

Solve

With initial conditions

And boundary condition

comment Solved.

comment Solved.

Solve

With initial conditions

And boundary condition

comment Solved.

comment Did not solve.

Solve

With initial conditions

comment Solved.

comment Solved

Solve

With boundary conditions

comment Solved.

comment Solved

Solve

With boundary conditions

And initial conditions

comment Did not solve.

comment Did not solve.