On The Analytical Solutions of Nonhomogeneous Heat equations On The Analytical Solutions of Nonhomogeneous Heat equations

: In this paper, we consider the analytical solution of Nonhomogeneous mixed problem of Heat equation. Namely, we use separation of variable and Duhamel’s principle to derive the formula of the general solution to this problem. Moreover, we find the solution for a special case of this problem as an application to our result.


Introduction
Over the last decades, the solutions of partial differential equations have been considered by many authors; see for instance [1][2][3][4]. The nonhomogeneous heat equation with n space variables takes the form [1]: It is well known that heat equation is a special case of a class of equations called parabolic type equations. Heat equation can model heat conduction, diffusion, etc. In this paper, we will study the analytical solution of the case when . (With non-zero initialboundary conditions)

Mixed problem: Separation of variables
Consider the following mixed problem [2]: Since the equation and the initial and boundary conditions are linear, we may split the problem into three simpler problems and then use superposition's to get a solution of the original problem. We have Suppose u 1, u 2 and u 3 are solutions of (I), (II) and (III) respectively, u 1 +u 2 +u 3 is a solution of (2.1). We can see that (III) can be reduced to (I) and (II).

Indeed, letting
We see that U (0, t) = Ø (t) and U (L, t) = (t). Let u be a solution of (III) and define V(x,t) = u (x,t) -U (x,t). Then V (x,t) is a solution of the following problem We have u (x,t)= V (x,t) is a solution of (III). Therefore, if we can solve (I) and (II), we can solve (I) first. We can solve this problem by using separation of variables. Solution of (I) We first find a solution of the equation in the form u (x,t) = X(x) T (t) satisfying the boundary conditions u (0,t)= u (L,t)= 0. Then we use superpositions to find a solution of the original problem by matching the initial condition. Substitute the function in this special form to the equation and the boundary condition,

Rewrite the equation in the form
(Because the left hand side is a function of while the right hand side is a function of x). Therefore we have the following two problems

On The Analytical Solutions of Nonhomogeneous Heat equations Mustafa A. Sabri
To find an non-trivial solution of (A) we must have and then we may find that for k = 1,2,... and the solution of (A) is in the form where is a constant to be determined. Substitute into (B) we can solve it and the solution is , where is a constant. Hence we have Where A k is a constant to be determined. Using superpositions, we obtain It is easy to see that solves the last problem. Therefore (

On The Analytical Solutions of Nonhomogeneous Heat equations Mustafa A. Sabri
Therefore (W is a solution of (3.1)). The boundary conditions can be easily checked.

Application
In this section, we solve the following two mixed problems by separation of variables and Duhamel's principle.

Conclusion
In this paper, we have used both the separation of variables method and Duhamel's principle to solve the non-homogeneous heat equation with nonzero initial-boundary conditions. We conclude that this technique is robust and it can be used efficiently to obtain the solution on time-depended problems, such as heat equation.