Solution of Integral Differential Equations by New Double Integral Transform (Laplace–Sumudu Transform)

The primary purpose of this research is to demonstrate an efficient replacement double transform named the Laplace–Sumudu transform (DLST) to unravel integral differential equations. The theorems handling fashionable properties of the Laplace– Sumudu transform are proved; the convolution theorem with an evidence is mentioned; then, via the usage of these outcomes, the solution of integral differential equations is built.

In this paper, we are ready to spotlight the way during which the Laplace-Sumudu transform is blend to solve the integral differential equations.
A wide range of linear integral differential equations are considered which include the Volterra integral equation (Section 3.1), the Volterra integro-partial differential equation (Section 3.2), and the partial integro-differential equation (Section 3.3). Definition 1. The double Laplace-Sumudu transform of the function ϕðx, tÞ of two variables x > 0 and t > 0 is denoted by L x S t ½ϕðx, tÞ = ϕðρ, σÞ and defined as Clearly, double Laplace-Sumudu transform is a linear integral transformation as shown below: where γ and η are constants.

Convolution Theorem of Double Laplace-Sumudu Transform
Definition 5. The convolution of ϕðx, tÞ and ψðx, tÞ is denoted by ðϕ * * ψÞðx, tÞ and defined by Theorem 6. (convolution theorem) If L x S t ½ϕðx, tÞ = ϕðρ, σÞ and L x S t ½ψðx, tÞ = ψðρ, σÞ, then Proof. From the definition 1., we have which is, using the Heaviside unit step function, that is, by Theorem 4 gives

Application of Laplace-Sumudu Transform (DLST) of Integral Differential Equations
In this section, we apply the double Laplace-Sumudu transform (DLST) method to linear integral differential equations.

Volterra Integral Equation.
Consider the linear Volterra integral equation as form where ϕðx, tÞ is the unknown function, λ is a constant, and gðx, tÞ and ψðx, tÞ are two known functions. Applying the double Laplace-Sumudu transform (DLST) with linearity to both sides of equation (32) and using Theorem 6 (convolution theorem), we get Consequently, Taking L −1 x S −1 t ½ϕðρ, σÞ for equation (34), we obtain the solution ϕðx, tÞ of equation (32).
We illustrate the above method by simple examples.
(a) Solve the equation where a and λ are constant.

Conclusion
In this paper, the Laplace-Sumudu transform approach for solving integral differential equations is studied. We provided the theorems and popular properties for this new double transform and furnished some examples. The examples show that the Laplace-Sumudu transform approach is powerful in solving the equations of taken into consideration type, and a couple of advanced problems in linear and nonlinear partial differential equations and nonlinear integral differential equations could be discussed during a later paper.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.