The inhomogeneous heat equation is described as follows:
Ut - kUxx = f(x,t). In order to solve this inhomogeneous PDE, Are there any restrictions on the type of the function f(x,t)? Even / Odd or neither?
Thanks
Kulthoum
Hello again!
If you have homogeneous boundary conditions,the Steklov's theorem works and you can seek for a solution of your problem in the form of Fourier series with respect to eigenfunctions of the corresponding Sturm-Liouville problem. In this case you also have to expand the right hand side of your heat equation into Fourier series with respect to eigenfunctions of this Sturm-Liouville problem. There are some restrictions on function f(x,t) in order to insure existence and uniqueness of this problem's solution. Roughly speaking, f(x,t) should be continuous in 0
Thanks Dr Dmitry. My question is ..This f(X, t) is of which type ? Even or odd or neither?
It does not matter, function f may be even, odd or neither, i.e. from this point of view function f may be of any type. However, function f should satisfy conditions (on its smoothness), which are listed in Vladimirov's book.
Dear Kulthoum Ismail,
Your PDE is linear and Non-homogeneous.The non- homogeneous function f(x,t) must be known real valued continuous function on given domain.It may be any function.There are so many methods for solving this equation under suitable initial and/ or boundary conditions.Some of them are:
I) Method of Separation (Oldest and simplest method)
II) Method of integral Transform
III) Adomian Decomposition method
IV) Perturbation Method
But the non- homogeneous function f is nonlinear in U then we have to put certain conditions on f. It is different issue.
Best luck!!!
DNYAN
dear Kulthoum Ismail
There are many method for solving these equations. One of the most common methods is Exponential Time Differencing.
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