#### Abstract

This study focuses on solving three-dimensional non-linear Klein-Gordon equations of four variables by using the quadruple Laplace transform method coupled with the iterative method. This study was designed in order to show the quadruple Laplace transform with an iterative method for solving three-dimensional nonlinear Klein-Gordon equations. The quadruple Laplace transform with the iterative method was aimed at getting analytical solutions of three-dimensional nonlinear Klein-Gordon equations. Exact solutions obtained through the iterative method have been analytically evaluated and presented in the form of a table and graph. The analytical solutions of these equations have been given in terms of convergent series with a simply calculable system, and the nonlinear terms in equations can easily be solved by the iterative method. Illustrative examples are also provided to demonstrate the applicability and efficiency of the method. The result renders the applicability and efficiency of the applied method. Finally, the quadruple Laplace transform and iterative method is an excellent method for the solution of nonlinear Klein-Gordon equations.

#### 1. Introduction

Partial differential equation has big importance in mathematics and other fields of science. Therefore, it is very important to know methods to solve such partial differential equations. As [1], the investigation of the exact solutions of nonlinear partial differential equations plays an important role in the study of nonlinear physical phenomena. In order to better understand these non-linear phenomena, many mathematicians and physical scientists make efforts to seek more exact solutions to them [2].

Nonlinear systems of partial differential equations (PDEs) play an important role in various areas of modern physics and engineering such as fluid mechanics, optical fibers, and quantum mechanics. Additionally, nonlinear partial differential equations arise various areas of nonlinear science such as hydrodynamics, plasma physics, molecular biology, quantum mechanics, nonlinear optics, surface water waves, etc. As mentioned in the work of [3].

Moreover, nonlinear partial differential equations (NLPDEs) and fractional differential equations (FDEs) are of massive concern for engineers, physicists, and mathematicians as the generality of physical systems is intrinsically nonlinear in nature. The solitary solutions of nonlinear PDEs and FDEs can demonstrate quite a few phenomenon in physics and may also provide farther understanding of the physical aspects of problems [4] and see also in the work of [5].

The Klein-Gordon equation is a relativistic version of the Schrodinger equation describing free particles, which was proposed by Oskar Klein and Walter Gordon in 1926. It has many applications in physics and engineering such as quantum field theory of relativistic physics, dispersive wave phenomena, plasma physics, and nonlinear optics. Various methods have been developed to get approximate and numerical solutions of linear Klein-Gordon (LKG) and nonlinear Klein-Gordon (NLKG) equations. [6] used the Adomain decomposition method (ADM) by Adomain in 1994 for solving LKG and NLKG equations. [7] used the variational iteration method to obtain an approximate solution of the NLKG equation. Khuri used a modified Laplace decomposition method proposed by [8] to solve the KG equation. As mentioned in the work of [9], the Laplace decomposition method and the Adomain decomposition method used to solve NLKG equations.

Consequently, one of the consummate known methods to solve partial differential equations is the integral transform method. In mathematics, physics, engineering, and throughout the sciences, the Laplace transform is a widely used integral transform method. P. S. Laplace (1749–1827) introduced the idea of the Laplace transform in 1782, in his celebrated study of probability theory and celestial mechanics. Laplace’s classic treatise on LaTh’eorie Analytique des Probabilities (analytical theory of probability) contained some basic results of the Laplace transform which is one of the longest-lived and most commonly used linear integral transforms available in the mathematical literature. This has effectively been used in finding the solutions of linear differential, difference, and integral equations as explained by [10, 11].

In mathematics, partial differential equations (PDEs) are equations that contain unknown multivariate functions and their partial derivatives. PDEs are used to formulate problems involving functions of several variables and are either solved by hand or used to create a relevant computer model. The Laplace transform method is any of several methods for solving ordinary and partial differential equations in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation and the convolution into multiplication leads one to the solution as it is simple to solve algebraic equations. Mainly, the advantage of the Laplace transform method is that it evaluates exact solutions while the iteration method mainly leads to approximate solutions. The method of Laplace transform has been used intensively for solving different linear and nonlinear equations as [12].

The Laplace transform method (LTM) is the basic scheme for finding exact solutions of linear and nonlinear differential equations. This method has many applications in different fields of science and engineering. LTM helps to convert differential equations and the convolution part is changed into multiplications which easily leads one to the solution as LTM reduces a linear/non-linear differential equation to simple and solvable equations. So, these equations are evaluated by the basic definition of algebra and by applying the inverse Laplace transform to the original differential equation. Nowadays, different kinds of differential equations such as linear, nonlinear, functional, integral, and fractional differential equations [13–16] solved by using the double Laplace transform method to solve a one-dimensional boundary value problem from the extension of the single Laplace transform.

In addition, different researchers apply the concept of double Laplace transform method to study triple Laplace transform in order to solve two dimensional nonlinear partial differential equations getting up in different fields of physics, engineering, and other related study areas. Different biographer nearly new triple Laplace transform method to find different differential equations as explained from the works of [17–19]. Khan et al. applied the concept of triple Laplace transform and the work is very effective for solving such kinds of differential equations involving triple integrals, see from [20, 21].

In the same way, an iterative method (IM) is an applicable method in a branch of computational mathematics in order to get the solution of nonlinear differential equations. An iterative process direct to a series of procedures which is helpful to get an analytical solution which are approximated in a suitable condition (see [22]). An iterative method shows an effective and rapid convergent series solution as described by [13]. Our main intent is to apply quadruple Laplace transform coupled with an iterative method to find analytical solutions of three-dimensional nonlinear Klein-Gordon equations dependent on initial and boundary conditions and bluster terms that disappear from the iteration process. Finally, we get the required analytical solution.

This study mainly considers three-dimensional nonlinear Klein-Gordon equations with four variables:

With initial conditions as follows:

Boundary conditions are as follows:where and are constants with necessary initial and boundary conditions. Where are positive and are real numbers. is a nonlinear term and is a source function.

At present, there is a very little work available on the triple Laplace transform of three positive variables and their properties. So, the major objective of this work is to study the quadruple Laplace transform and its properties with examples of three-dimensional nonlinear Klein-Gordon equations using the quadruple Laplace transform coupled with an iterative method. The main focus of this work is relating to QLT coupled with IM to examine exact solutions to three-dimensional 3D nonlinear KGEs using initial and boundary value problems to establish the advantages and effectiveness of the method in applied mathematics and mathematical physics.

The organization of this article is as follows: section 1 is about the introduction, in section 2 the definition of quadruple Laplace transform, properties of quadruple Laplace transform and iterative method and its convergence conditions are mentioned, section 3 contains analytical, graphical, tabular results and discussion and section 5 about conclusion.

Recalling the following definition for the purpose of integration of important concepts.

*Definition 1. *Let be a piecewise continuously differentiable function in every finite interval and is absolutely integrable function on the whole real line. The Fourier integral formula of is defined bywhere a complex number is taken as the variable of the transform.

#### 2. Definition of Quadruple Laplace Transform

A continuous function of four variables quadruple Laplace transform is defined by the quadruple integral in the form

Provided the integral exists.

*Definition 2. *(see [23]). For a continuous function of four variables, then the quadruple

Laplace transform is defined as follows:With by considering are Laplace variables.

Theorem 1. *Let is a function, then the inverse quadruple Laplace transformis given by the complex quadruple integral formula as follows:where must be an analytic function for each in the given region defined by the inequality where , , , and are real constants to be chosen suitably based on [23]. To verify this we recall the Fourier integral formula given by (4) which expresses the representation of a function defined on , in the form*

This can be set as the concept of Heaviside functionwhere is a positive fixed number. Now (9) becomes

##### 2.1. Using a Change of Variables

and (11) can be equivalently written in the form

From (12) we get the formal definition of the Laplace transform of :where, is the kernel of the transform and is the transform variable which is a complex number. Thus, (12) can be used to give the formal definition of the inverse Laplace transform

Hence, from (14) one can deduce that

Further, [18] key rules of quadruple Laplace transforms of functions of four variables are given as follows:(1)For first order partial derivatives of functions of four variables is as follows:(2)For second order partial derivative of functions of four variables is as follows:

##### 2.2. Existence of Quadruple Laplace Transform

*Definition 3. *A function is said to be of exponential order on and . If a positive constant exists for every and and writeSuch a function is simply called an exponential order as based on the exponential order as grow more faster as described by [14].

Theorem 2. *(Existence): For a continuous function in every finite intervals and and of exponential order , then the quadruple Laplace transform of exists for all and provided that and .*

*Proof. *We haveFor and . It follows from this (18) thatorThis proves the existence of quadruple Laplace transform. This result can be regarded as the limiting property of the quadruple Laplace transform, clearly,is not quadruple Laplace transform of any function because does not tend to zero as . On the other hand, cannot have quadruple Laplace transform even through it is continuous but is not of the exponential order because Then as [14].

Theorem 3. *(Uniqueness): For any two continuous functions and defined for 0 and having Laplace transforms, and , respectively, ifas described on [23].*

*Proof. *Based on the definition of the inverse Laplace transform, if are sufficiently large, then the integral expression,By hypothesis, we have that ,

Then replacing this in the previous expression, we have the following:Use the above relation, , (18) can be rewritten asThis completes the required proof. And this proves the uniqueness of the quadruple Laplace transform.

##### 2.3. Main Properties of Quadruple Laplace Transform

The properties of quadruple Laplace transform are mentioned and proved in this section. Note that these properties follow from those of the double Laplace transform described by [17] from the source [24], properties of the triple Laplace transform from [17]. Some properties of quadruple transform as follows:

*Property 1. *(Linearity of quadruple Laplace transform): As presented on [10] for triple Laplace transform and [23] for quadruple Laplace transform linearity property. If and are two functions with variables of *x*, *m*, and *t* such thatThen, with and as constants.

*Property 2. *Linearity of inverse of quadruple Laplace transform: Based on property (1), the inverse quadruple Laplace transformis defined by the quadruple integral formula as folllows:It shows that satisfies the linearity propertywith and as constants.

is also a linear transformation.

*Property 3. *(Change of scale property): as described on [10]. Let thenwhere are nonzero constants.

*Proof. *From the definition of quadruple Laplace transformApplying change of variableswhere are nonzero constants.

*Property 4. *(First shifting property): As explained by [10]: Let

*Proof. *By definition of quadruple Laplace transform (19) can be solved asThe interior integral with respect to *t* in (36) can be solved next to proper substitution as(36) becomesThe next inner integral with respect to in (38) can be solved after proper substitution asAfter substituting the value from (39) in (37), we can getOn the same procedure integrating (40) with respect to *m* and can be solved after proper substitution asOn the last process of integration of (41) with respect to *x* and can be solved after proper substitution asAs the required proof of (35).

##### 2.4. An Iterative Method

In computational mathematics, an iterative method with a powerful set of rules that must be followed when solving a particular problem such as linear and nonlinear ordinary and partial differential equations. The analysis takes place by means of numerical examples. This method in solving the above types of equations is used to want linearization which verifies the capacity, exactness, and directness of the given method in contrast with some of the other methods as explained in the work of [25].

Consider the following general functional equation, as described by [22]:

From the Banach space where is a nonlinear operator as is a common function.

We are cast about a solution in (43) having the sequence:

The disintegrate form of the nonlinear operator is given as follows:

From (43)–(45) is equivalent to

Now, define

Consider the sequence given by

We define the recurrence relation:

Then,

And hence,

If is contraction, i.e., then and the series

The Banach fixed point theorem shows (43) has unique, absolute, and uniform converges to a solution as explained by [26]. Consequently, the comparative solution of (43) is disposed as follows:

##### 2.5. Taylor Series and New Iterative Method

Consider the sequence given by (47) as shown on [1].

Then

Similarly,

Generally,

Now in equation (28) to (39) we have,

Hence, the simplified form of (57) is given by

Hence, (58) is the Taylor series expansion of around , and this shows that the new iterative method is equivalent to Taylor’s expansion around as described by [22].

##### 2.6. The New Iterative Method and Its Convergence

Theorem 4. *For a continuously differentiable function a neighborhood of and**For each the series converges absolutely as described by [22].*

*Proof. *The recurrence relationConsider and based on (48) and (56) and the idea of theorem (4), we observe thatTakeNote that and

Moreover, , hence, is the series of positive real numbers. Note that. In general, which yields that bounded and convergent. Consequently, by comparison test is absolutely convergent. So, the new iterative method proves the existence of analytical solution.

#### 3. Numerical Solutions

The aim of this study is to obtain analytical solution of three-dimensional NLKGE by using coupled QLTIM. This is done by extending the work of [9] that was used to solve NLKGE by using coupled DLTIM. So, the definitions, theorems, and some derivations related to quadruple Laplace transform was mentioned in the preceding section would be applied here. Consider the three dimensional nonlinear Klein-Gordon equation of four variables given in (1).

With initial conditions

Boundary conditionsWhere are positive and are real numbers is a nonlinear term and is a source function. In order to solve such problem we first decompose the source function into and Where is the term in (1) which leads to express in a simple algebra when applying the inverse quadruple Laplace transform. Whereas is the nonlinear term of (1) helps to reject a complicated term in the process of iteration. Perform quadruple Laplace transform in opposite sides of (1) by using (17), we obtain,Where , and are Laplace transforms of functions of four variables. Note that the existence of Laplace transform of functions of four variables is shown in section (2.1). Next, perform triple Laplace transform of both ICs and BCs of equation (46) and equation (47), yields Here and are the Laplace transforms of functions of three variables. Based on the concept of [17]. Then, plugging these ICs and BCs into (67) and using the properties of quadruple Laplace transform as follows, we obtainWhere,

Applying the definition of inverse quadruple Laplace transform on (70) we obtain

Now start the iterative process. Consider