The Solution of Partial Differential Equations - Maths Assignment Help

Assignment Task

Select one of the following problems involving the solution of partial differential equations.

For numerical calculations, you may choose particular values for the unspecified parameters. 
(1) Consider the diffusion of heat in a two-dimensional circular region of radius a governed by the heat equation with a uniform diffusivity x- Develop a formulation to determine the eigenvalues and eigenfunctions corresponding to purely axisymmetric response, assuming that the temperature on the outer perimeter at r = a is maintained at zero. Demonstrate the orthogonality property of the eigenfunctions. Then, solve the initial value problem for the temperature T(r,t), assuming the initial distribution T(r,0)=T0(1— r2 / a2), where To is a specified constant. Investigate the convergence characteristics of the solution. Finally, discuss the modifications that would be necessary in order to solve this diffusion problem for a case in which the initial temperature distribution is not purely axisymmetric. 

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