Solving Fundamental Solution of Non-Homogeneous Heat Equation with Dirichlet Boundary Conditions

In this study, we developed a solution of nonhomogeneous heat equation with Dirichlet boundary conditions. moreover, the non-homogeneous heat equation with constant coefficient. since heat equation has a simple form, we would like to start from the heat equation to find the exact solution of the partial differential equation with constant coefficient. to emphasize our main results, we also consider some important way of solving of partial differential equation specially solving heat equation with Dirichlet boundary conditions. the main results of our paper are quite general in nature and yield some interesting solution of non-homogeneous heat equation with Dirichlet boundary conditions and it is used for problems of mathematical modeling and mathematical physics.


Introduction
Differential equations are used to construct more models of reality and these modeling suggests that some solutions of the differential equations with variable coefficients using different methods There are more methods of solving different researchers applied to solve heat equation by different method in this paper we find a solution of nonhomogeneous heat equation with Dirichlet Boundary conditions.
Many physical problems such as wave equation, heat equation, Poisson equation and Laplace equation are modeled by differential equations which are an example of partial differential equations. some partial differential equations have numerical solution and exact solution in regular shape domain but in this paper, we will try to solve exact solution of non-homogeneous heat equation. Then in this paper we try to solve the exact solution of nonhomogeneous heat equation with Dirichlet Boundary conditions.
Heat equation is a superposition principle of solutions and therefore from a stock of simple solutions it is possible to build solutions to complex problems. Thus, the temperature distribution in a body can be considered to be due to the additive influence of the various internal,external and boundary agents affecting the heat flow. There are in fact special solutions to the heat equation which are sufficiently fundamental that solutions to very broad categories of heat conduction problems can be written immediately in terms of these fundamental solutions to the differential equation.
We recall, an equation containing the derivatives or differentials of one or more dependent variables in relation to one or more independent variables is called a partial differential equation (abbreviated to PDE).
Note that the order of a partial differential equation is the degree of the highest order derivatives in the equation. For instance, if there are two independent variables ( , ), a partial differential equation of second order has the general form [1].

Preliminaries
The non-homogeneous heat equation arises when studying heat equation problems with a heat source we can now solve this equation. The ideas in the proof are very important to know about the solution of non-homogeneous heat equation.
Theorem1.The solution of the inhomogeneous heat equation With initial condition Is given by Now consider the homogeneous system; One easily see that Comparing the two results we see that the solution to the non-homogeneous equation with zero initial value can be represented as a summation of the solution of the solution of a family of homogenous equation with nonzero initial value With ( ; ) satisfies the homogenous equation (2.10) with initial time an initial value ( ).This is the Duhamel's principle.
By this principle we can write down the solution Theorem 2:( solution of non-homogenous heat equation) Let ∈ 1 2 ( > 0, −∞ < < ∞ ) and have compact support then Proof a) From eq (1.12) we have Assumption we can differentiate inside the integrals; (2.17)

Bulletin of Mathematical Sciences and Applications vol. 22 3
The integral is well defined, as is singularity at = 0 we write (by using improper integral as → 0 the lower boundary of the integral starts from ) Combining the above results, we can present the formula for the solution in the general case in the whole space; The required solution is As an application of the Theorem above, we consider the effect of a heat source localized at one point on the bar. The solution will be expressed in terms of the special function known as the incomplete gamma function and defined by In particular, if −1 < < 0 > 0, we can write

Remark 2: boundary value problem
In equation (1), 0 ( ) represented a source of heat that is applied at the point , ∀ > 0 Solution: Applying the above theorem with ( , ) = 0 ( ) we find Integrating against 0 ( ) in the integral yields To simplify the exponential function we make a change of variables: and differentiating both sides we get Or it can be write by rearranging as =  We assume that has the following from Since we are dealing with Dirchilet boundary conditions then we take for ( , ) = . Then the function has to be the solution of the following problem The eigenvalues and the corresponding eigenfunctions for the associated Sturm-Louville problem.
Are given by Hence, the formal solution of the above problem is given by the formal Fourier series (3.11) Differentiating (formally) with respect to and twice with respect to , and substituting into the equation we get This implies that Moreover at, t= 0 (3.14) So that

Conclusion
This paper thus, consisted of a brief overview of what a solution of heat equation is, and what is it used for based on the work has been done on the paper, it can be constructed the solution of nonhomogeneous heat equation with Dirichlet boundary conditions. A solution of non-homogeneous heat equation was constructed using superposition principles through combining simple solutions of the no-homogenous heat equation. Therefore, the solution of non-homogeneous heat equation derived and to derive this we have used different concepts as theorem and remarks of non-homogeneous equations.
However, in this study we considered a solution for non-homogeneous heat equation with Dirichlet boundary conditions and as we see it simple and easy to derive. Moreover, this procedure can be possibly applied for homogeneous heat equations with Dirichlet boundary conditions.