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On the convergence of double Elzaki transform

In this research, we have studied the convergence properties of Double Elzaki transformation and the results have been presented in the form of theorems on convergence, absolute convergence and uniform convergence of Double Elzaki transformation. The Double Elzaki transform of double Integral has also been discussed for integral evaluation. Finally, we have solved a Volterra integro-partial differential equation by using Double Elzaki transformation.


Introduction
*Integral transforms are valuable for the simplification that they bring about, most often in dealing with differential equation subject to particular boundary conditions. Proper choice of the class of transformation usually makes it possible to convert not only the derivatives in an intractable differential equation but also the boundary values into terms of an algebraic equation that can be easily solved. The solution obtained is, of course, the transform of the solution of the original differential equation, and it is necessary to invert this transform to complete the operation.
Integral transform, mathematical operator that produces a new function ( ) by integrating the product of an existing function ( ) and a so-called kernel function ( , ) between suitable limits. The process, which is called transformation, is symbolized by the equation ( ) = ∫ ( , ) ( ) . Several transforms are commonly named for the mathematicians who introduced them. In the Laplace transform, the kernel is − and the limits of integration are zero and plus infinity, in the Fourier transform, the kernel is (2π) −1/2 e − and the limits are minus and plus infinity. In Schiff (2013), The Laplace transform of f as whenever the limit exists (as a finite number). When it does, the above integral is said to converge. If the limit does not exist, the integral is said to diverge and there is no Laplace transform defined for f. The notation ( ) will also be used to denote the Laplace transform of f, and the integral is the ordinary Riemann (improper) integral. The parameter belongs to some domain on the real line or in the complex plane.
In Belgacem et al. (2003) and Watugala (1993), a new integral transform, called the Sumudu transform defined for functions of exponential order. We consider functions in the set A, defined by For a given function in the set A, the constant M must be finite, while 1 and 2 need not simultaneously exist, and each may be infinite. Instead of being used as a power to the exponential as in the case of the Laplace transform, the variable in the Sumudu transform is used to factor the variable t in the argument of the function f. Specifically, for ( ) in A, the Sumudu transform is defined by Belgacem (2007) presented the fundamental properties, analytical investigation of the samudu transform and applications to integral equations. In Belgacem (2007) and Belgacem and Karaballi (2006), all existing samudu shifting theorems and recurrence results have been generalized, also presented applications to convolution type integral equations with focus on production problems and inverse Sumudu transform of a singular function satisfies the Tauberian theorem, where the dirac delta function fails.
In Belgacem (2006Belgacem ( , 2009, Laplace transform definition is implemented without resorting to Adomian decomposition nor Homotopy perturbation methods. He also applied the natural transform to Maxwell's equations and obtained the transient electric and magnetic field solution. In Belgacem and Silambarasan (2017b), The Samudu transform is applied to arbitrary powers Dumont bimodular Jacobi elliptic functions for arbitrary powers. Belgacem (2010) applied the Samudu transform applications to Bessel's functions and equations. In Belgacem and Silambarasan (2017a), the Samudu transform integral equation is solved by continuous integration by parts, to obtain its definition for trigonometric functions.
Belgacem and Al-Shemas (2014) proposed ideas towards the mathematical investigations of the environmental fitness effects on populations dispersal and persistence. Goswami and Belgacem (2012) giave a sufficient condition to guarantee the solution of the constant coefficient fractional differential equations by Samudu transform.
The Elzaki Transform is a new integral transform introduced by Elzaki (2011a). Elzaki Transform is modified form of Laplace Transform. Elzaki transform is well applied to initial value problems with variable coefficients and solving integral equations of convolution type. Elzaki Transform is also used to find solution of system of partial differential equations (Elzaki, 2011b). Typically, Fourier, Laplace and Sumudu transforms are the convenient mathematical tools for solving differential equations.
Elzaki Transformation is defined for the function of exponential order. Consider a function in the set S defined as For a given function f(t) in the set S, the constant M must be finite, number k 1 , k 2 may be finite or infinite. The Elzaki Transform denoted by the operator E is defined as The variable v in this transform is used to factorize the variable t in the argument of the function f. The purpose of this study is to show the applicability of this interesting new transform and its efficiency in solving some convergence theorems.

Convergence theorem of double Elzaki integral
In this section, we prove the convergence theorem of double Elzaki integral. Theorem 2.1: Let ( , ) be a function of two variables continuous in the positive quard of the xt-plane. If the integral converges at = and = then integral converges for < , < .
for the proof we will use the following lemmas.
converges at s= then the integral converges for < Proof: Consider the set Now let 1 → 0. Both terms on the right which depend on 1 approach a limit and The given theorem is proved if the integral on the right converges. Now by using the "Limit test" for convergence (Widder, 2005). For this we have converges for ≤ and integral converges at = then the integral (6) converges for < Proof: Therefore ( Now let R 2 → ∞. If s < s o , the first term on the right approaches zero. The given theorem is proved if the integral on the right converges. Now by using the Limit test for convergence (Widder, 2005). We consider h (x, s)dx converges for p < p o .
The proof of the Theorem 2.1 is as follows where converges for < . Therefore, the integral in RHS of (10) converges for < , < . Hence the integral Therefore the integral (1) converges absolutely for (p ≤ p o s ≤ s o ).

Uniform convergence
In this section we prove the uniform convergence of double Elzaki Transform.

Double Elzaki transform of double integral
We now find the double Elzaki transform of double integral.

Application of double Elzaki transform in Volterra integro-partial differential equation
We use the double Elzaki transform to solve the problem which is already solved in Moghadam and Saeedi (2010) using Differential transform method.
Example 5.1: Consider the following Volterra Integro Partial Differential Equation.
The single Elzaki transforms of equation ( Now by using double inverse Elzaki transform, we obtain solution of (20) as follows, ( , ) = + .

Conclusion
We have proved the convergence, absolute convergence and uniform convergence of double Elzaki transform. Besides these, we obtained double Elzaki transform of double integral and use it to solve Volterra integro-partial differential equation.