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## Analytical solutions to some textbooks PDE’s using computer algebera systems

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

### 1 Introduction and lookup table

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

### 2 Problem 2.3.3, Richard Haberman applied partial differential equations, 5th edition, (heat PDE, 1D)

#### 2.1 part a

##### 2.1.1 Maple

comment Solution is correct.

##### 2.1.2 Mathematica

comment Solution is correct.

#### 2.2 part b

##### 2.2.1 Maple

comment Verified OK against my analytical solution.

comment Ok

#### 2.3 part c

##### 2.3.1 Maple

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

##### 2.3.2 Mathematica

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

#### 2.4 part d

##### 2.4.1 Maple

comment Verified correct solution, compared to my hand solution.

##### 2.4.2 Mathematica

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

### 3 Problem 2.3.7, Richard Haberman applied partial differential equations, 5th edition (heat PDE, 1D)

#### 3.1 Maple

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.

#### 3.2 Mathematica

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

### 4 Problem 2.3.8, Richard Haberman applied partial differential equations, 5th edition (heat PDE, 1D)

comment Ok.

#### 4.2 Mathematica

comment Did not solve. Input returned.

### 5 Problem 2.4.1, Richard Haberman applied partial differential equations, 5th edition (heat PDE, 1D)

#### 5.1 part a

##### 5.1.1 Maple

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

##### 5.1.2 Mathematica

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

#### 5.2 part b

##### 5.2.1 Maple

comment Did not solve. Returned ().

##### 5.2.2 Mathematica

comment Ok

The above is verified using Maple

#### 5.3 part c

##### 5.3.1 Maple

comment Partially correct. Overlooked the zero eigenvalue.

##### 5.3.2 Mathematica

comment Correct solution. Mathematica accounted for the zero eigenvalue.

#### 5.4 part d

##### 5.4.1 Maple

comment Did not solve. Returned ().

##### 5.4.2 Mathematica

comment Ok

Verified by Maple:

### 6 Problem 2.4.2, Richard Haberman applied partial differential equations, 5th edition (heat PDE, 1D)

#### 6.1 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.

#### 6.2 Mathematica

comment Correct solution.

### 7 Problem 2.5.1, Richard Haberman applied partial differential equations, 5th edition (Laplace PDE, 2D)

#### 7.1 part a

##### 7.1.1 Maple

comment Did not solve. Returned ().

##### 7.1.2 Mathematica

comment Solved Ok.

#### 7.2 part b

##### 7.2.1 Maple

comment Did not solve. Returned ().

##### 7.2.2 Mathematica

comment Solved Ok.

#### 7.3 part c

##### 7.3.1 Maple

comment Did not solve. Returned ().

##### 7.3.2 Mathematica

comment Could not solve.

The solution is

#### 7.4 part d

##### 7.4.1 Maple

comment Did not solve. Returned ().

##### 7.4.2 Mathematica

comment Could not solve.

#### 7.5 part e

##### 7.5.1 Maple

comment Did not solve. Returned ().

##### 7.5.2 Mathematica

comment Could not solve.

### 8 Problem 2.5.5 part(c), Richard Haberman applied partial differential equations, 5th edition (heat PDE, semi-disk)

#### 8.1 Maple

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

#### 8.2 Mathematica

comment Did not solve.

### 9 Problem 2.5.8 part(b), Richard Haberman applied partial differential equations, 5th edition (Laplace on circular annulus)

#### 9.1 Maple

comment Did not solve.

#### 9.2 Mathematica

comment Did not solve.

### 10 page 115, David J Logan textbook, applied PDE (Wave PDE 1D nonhomogeneous)

Falling cable lying on a table that is suddenly removed.

With initial conditions

And boundary condition

comment Solved.

#### 10.2 Mathematica

comment Solved.

Verify Mathematica solution using Maple

### 11 page 76, David J Logan textbook, applied PDE (heat PDE 1D)

With initial conditions

And boundary condition

comment Solved.

comment Solved.

### 12 page 28, David J Logan textbook, applied PDE (Wave PDE 1D)

With boundary conditions

comment Solved.

#### 12.2 Mathematica

comment Could not solve.

### 13 page 130, David J Logan textbook, applied PDE (Wave PDE 1D)

Solve

With initial conditions

And boundary conditions

#### 13.1 Maple

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.

#### 13.2 Mathematica

comment Did not solve.

### 14 page 131, David J Logan textbook, applied PDE (heat PDE 1D, periodic BC)

Solve

With initial conditions

And boundary conditions

comment Solved.

#### 14.2 Mathematica

comment Did not solve.

### 15 page 149, David J Logan textbook, applied PDE (wave PDE 1D, nonhomogeneous)

Solve

With initial conditions

And boundary conditions

comment Solved.

#### 15.2 Mathematica

comment Did not solve. Gives only trivial solution.

### 16 page 213, David J Logan textbook, 3rd ed, applied PDE (nonhomogeneous wave PDE 1D)

Solve

With initial conditions

And boundary conditions

comment Solved.

#### 16.2 Mathematica

comment Did not solve.

### 17 page 30, David J Logan textbook, applied PDE (Schrodinger PDE with zero potential)

Solve

With boundary conditions

comment Solved.

comment Solved.

### 18 Problem 18, from Maple help page (Laplace PDE, 1D)

Solve

With

comment Solved.

#### 18.2 Mathematica

comment Solved.

Verify Mathematica solution using Maple

### 19 PDE 2 from Maple PDF help, page 14 (wave PDE 1D)

Solve

With initial conditions

And boundary condition

comment Solved.

comment Solved.

### 20 PDE 3 from Maple PDF help, page 14 (heat PDE 1D)

Solve

With initial conditions

And boundary condition

comment Solved.

comment Solved.

### 21 PDE 4 from Maple PDF help, page 15 (heat PDE 1D)

Solve

With initial conditions

And boundary condition

comment Solved.

comment Solved.

### 22 heat absorption radiation in bounded domain

Solve

With initial conditions

And boundary condition

comment Solved.

#### 22.2 Mathematica

comment Did not solve.

### 23 heat PDE, infinite domain with initial conditions, nonhomogeneous

Solve

With initial conditions

comment Solved.

comment Solved

### 24 PDE 23, page 23, from Maple pdsolve help PDF file

Solve

With boundary conditions

comment Solved.

comment Solved

### 25 heat PDE on 1D with boundary conditions from

Solve

With boundary conditions

And initial conditions

#### 25.1 Maple

comment Did not solve.

#### 25.2 Mathematica

comment Did not solve.