Some Applications of The New Integral Transform For Partial Differential Equations

There are a lot of methods to solve partial differential equations, but for linear partial differential equations the most powerful method is an integral transformation method. Laplace transform is the most effective tool to solve some kinds of ordinary and partial differential equations. Actually an electric engineer Oliver Heaviside made Laplace transform popular by developing its operational calculus. After Laplace transform, in 1993 again an electrical engineer Watugula in [1] proposed a new integral transform named the Sumudu transform and used it for solving problems in control engineering, it is similar to the Laplace transform having the preservation property of unit and change of scale. After that, T. Elzaki [ 6 ] introduced a new integral transform named Elzaki transform and applied it for solving partial differential equations, Shaikh Sadikali has been applied Elzaki transform for solving integral equations of convolution type see in [ 5 ]. Likewise many integral transforms have been proposed which are similar to the Laplace transform, and each new transform claimed its own superiority over the Laplace transform. In this paper we considered a new integral transform named the Sadik transform [3], [4]. It is similar to the Laplace transform but the Laplace transform, the Sumudu transform, Elzaki transform and all integral transforms with kernel of an exponential type are particular cases of the Sadik transform. Due to the very general and unified nature of the Sadik transform, we can transport a problem of partial differential equations into the known transformation technique which is available in the literature through the Sadik transform.


Introduction
There are a lot of methods to solve partial differential equations, but for linear partial differential equations the most powerful method is an integral transformation method. Laplace transform is the most effective tool to solve some kinds of ordinary and partial differential equations. Actually an electric engineer Oliver Heaviside made Laplace transform popular by developing its operational calculus. After Laplace transform, in 1993 again an electrical engineer Watugula in [1] proposed a new integral transform named the Sumudu transform and used it for solving problems in control engineering, it is similar to the Laplace transform having the preservation property of unit and change of scale. After that, T. Elzaki [ 6 ] introduced a new integral transform named Elzaki transform and applied it for solving partial differential equations, Shaikh Sadikali has been applied Elzaki transform for solving integral equations of convolution type see in [ 5 ]. Likewise many integral transforms have been proposed which are similar to the Laplace transform, and each new transform claimed its own superiority over the Laplace transform. In this paper we considered a new integral transform named the Sadik transform [3], [4]. It is similar to the Laplace transform but the Laplace transform, the Sumudu transform, Elzaki transform and all integral transforms with kernel of an exponential type are particular cases of the Sadik transform. Due to the very general and unified nature of the Sadik transform, we can transport a problem of partial differential equations into the known transformation technique which is available in the literature through the Sadik transform.

Preliminaries
In this section we demonstrated basic definitions and some operational properties of Sadik transform.

Definition [1]
If When the integral of right side of (2) is converges.

Definition [3]
If, Where, v is complex variable, α is any non zero real numbers, and β is any real number. The beauty of this transform is that by changing the values of α and β we can convert a considered problem into the Laplace transform, the Sumudu transform, Elzaki transform and all other transforms whose kernels are of an exponential type. For instant suppose that α = 1 and β = 0 then it will be the Laplace transform, if α = −1 and β = 1 then it will be the Sumudu transform, and so on.

Operational Properties of the Sadik transform
then the followings are the operational properties of the Sadik transform of some standard functions, see in [3].
3) If f t cos at

5) Sadik transform of hyperbolic functions
Now suppose that ϕ x t , ( ) is a piecewise continuous function with exponential order such that their derivatives are also a piecewise continuous functions with exponential order then we can find its Sadik transform and we will use these properties for solving some partial differential equations in next section.

Proposition
If G x v , ,

Proof:
Integrating by parts twice we can get the required result easily, therefore we omit the proof.

Proposition
Where ϕ x x t ,

Proof
Using differentiation under integral sign, we can prove this result easily. In general we can easily established the following

Applications of Sadik Transform to Partial Differential Equations
In this section we demonstrate the applicability of new integral transform named Sadik transform to solve some examples of partial differential equations.
Example1. [6] If y x t , ( ) is a solution of the following first order partial differential equation

Now applying Sadik transform, we get
Using initial condition, we can rewrite this equation as

Hence required solution is
Since y is bounded therefore c=0. Applying the Sadik transform, we get It is second order ordinary differential equation, hence its particular integral is Anytime we can trans figurate above problem to any other known transform, which is available in the literature by fixing values of alpha and beta. Suppose that α β = = 1 0 and , then above problem is ready to solve by the Laplace transform.  , then above problem is ready to solve by such a integral transform particularly which is yet not exist in the literature or yet anybody wouldn't proposed. Therefore,