Abstract
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.
1. Introduction
Double integral transform and their characteristics and theories are nevertheless new and below studies [1–3], in which the preceding research treated some components of them along with definitions, simple theories, and the answer of normal and partial differential equations [4–16]; additionally, some researchers addressed these transforms and combine them with exclusive mathematical method such as differential transform approach, homotopy perturbation technique, Adomian decomposition method, and variational iteration method [7–16] so that we can solve the linear and nonlinear fractional differential equations.
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 of two variables and is denoted by and defined as Clearly, double Laplace–Sumudu transform is a linear integral transformation as shown below: where and are constants.
Definition 2. The inverse double Laplace–Sumudu transform is defined by the following form:
2. Double Laplace–Sumudu Transform of Basic Functions
(1)Let then (2)Let then
If and are positive integral, then (3)Let then
Similarly,
Consequently,
(4) Let
Recall that
Therefore,
(5) Let then
where is the modified Bessel function of order zero.
(6) Let then
2.1. Existence Condition for the Double Laplace–Sumudu Transform
If is an exponential order, then and as , and if a positive constant such that then and we write as Or, equivalently, The function is called an exponential order as , and clearly, it does not grow faster than as .
Theorem 3. If a function is a continuous function in every finite interval and of exponential order , then the double Laplace–Sumudu transform of exists for all and provided and
Proof. From the Definition 1., we have Then, from Eq. (16) we have or .
2.2. Basic Derivative Properties of the Double Laplace–Sumudu Transform
If , then
Proof.
Using integration by parts, let then
Proof.
Using integration by parts, let then .
Similarly, we can prove
Theorem 4. If , then
where is the Heaviside unit step function defined by
Proof. We have, by Definition 1.,
that is, by putting
2.3. Convolution Theorem of Double Laplace–Sumudu Transform
Definition 5. The convolution of and is denoted by and defined by
Theorem 6. (convolution theorem) If and then
Proof. From the definition 1., we have which is, using the Heaviside unit step function, that is, by Theorem 4 gives
3. 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.
3.1. Volterra Integral Equation
Consider the linear Volterra integral equation as form where is the unknown function, is a constant, and and 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 for equation (34), we obtain the solution of equation (32).
We illustrate the above method by simple examples. (a)Solve the equationwhere and are constant.
Applying the double Laplace–Sumudu transform (DLST) of equation (36), we get
Consequently,
Taking for equation (38), we obtain the solution of equation (36). (b)Solve the equationwhere is a constant.
Applying (DLST) of equation (40), we get Or
Taking for equation (42), we obtain the solution of equation (40). (c)Solve the equation
Applying (DLST) of equation (44), we get
Simplifying and taking for equation (45), we obtain
3.2. Volterra Integro-Partial Differential Equations
Consider the linear Volterra integro-partial differential equation as form with the conditions: where is the unknown function, is a constant, and and are two known functions.
Applying (DLST) to both sides of (47) and single (LT) and (ST) for equation (48) and simplification, we get
Applying to (49), we obtain the solution of (47) in the form
We illustrate the above method by a simple example. (d)Solve the equation
By substituting in (47), we have got with the conditions:
Substituting in (50) and simplifying, we get the solution of (51)
3.3. Partial Integro-Differential Equation
Consider the linear partial integro-differential equation as form with the conditions:
Applying (DLST) to both sides of (55) and single (LT) and (ST) for equation (56) and simplification, we get
Applying to (57), we obtain the solution of (55) in the form
We illustrate the above method by a simple example. (e)Solve the equation:
By substituting in (55), we have got with the conditions:
Substituting in (58) and simplifying, we get a solution of (59)
4. 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.