# Approximate Solutions of Multidimensional Wave Problems Using an Effective Approach

Authors: Muhammad Nadeem; Qura tul Ain; Yahya Alsayaad
Journal: Journal of Function Spaces
 (2023)

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## Abstract

The main goal of this paper is to introduce a new scheme for the approximate solution of 1D, 2D, and 3D wave equations. The recurrence relation is very important to deal with the approximate solution of differential problems. We construct a scheme with the help of the Laplace-Carson integral transform (LcIT) and the homotopy perturbation method (HPM), called Laplace-Carson homotopy integral transform method (LcHITM). LcIT produces the recurrence relation and destructs the restriction of variables whereas HPM gives the successive iteration of the relation using the initial conditions. The convergence analysis is provided to study the wave equation with multiple dimensions. Some numerical examples are considered to show the efficiency of this scheme. Graphical representation and plot distribution between the approximate and the exact solution predict the high rate of convergence of this approach.


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## Body

## 1. Introduction

Numerous physical phenomena in real world are modeled using partial differential equations (PDEs) in a variety of applied science fields including fluid dynamics, mathematical biology, quantum physics, chemical kinetics, and linear optics [1–3]. There are various perturbation approaches that can be used to analytically solve the PDEs. Although the calculations for these strategies are pretty straightforward, their limitations are predicated on the assumption of small parameters. As a result, many researchers are searching for novel methods to get around these restrictions. Various researchers and scientists have studied multiple novel methods for getting the analytical solution that are reasonably close to the precise solutions such as homotopy analysis method [4], modified extended tanh method [5], new Kudryashov’s method [6], Chun-Hui He’s iteration method [7], subequation method [8], exp-function method [9], modified exponential rational method [10], homotopy asymptotic method [11], modified extended tanh expansion [12], fractal variational iteration transform method [13], Laplace homotopy perturbation transform method [14], residual power series (RPS) method [15], and Adomian decomposition method [16]. In the past, many experts and researchers established the application of the homotopy perturbation method (HPM) [17, 18] in various physical problems, because this approach consistently transforms the challenging issue into a straightforward resolution. The method yields a physical problem, because this approach consistently transforms the challenging issue into a straightforward resolution. The method yields a very rapid convergence of the solution perturbation theory and showed the ability to be a very strong mathematical tool.The wave equation, which describes the wave propagation phenomenon, is a partial differential equation for a scalar function. It is influenced by time and one or more spatial factors. The wave equations perform an important role in different area of engineering, physics, and scientific applications. Wazwaz [19] studied linear and nonlinear problems in bounded and unbounded domains using the variational iteration method. Ghasemi et al. [20] employed the homotopy perturbation method to derive the numerical solution of two-dimensional nonlinear differential equation. Keskin and Oturanc [21] applied reduced differential transform method to various wave equations. Ullah et al. [22] proposed optimal homotopy asymptotic method to obtain the analytic series solution of wave equations. Adwan et al. [23] presented the numerical solution of multidimensional wave equations and showed the accuracy of proposed techniques. Jleli et al. [24] studied the framework of the homotopy perturbation transform method for analytic treatment of wave equations. Mullen and Belytschko [25] provided the finite element scheme for the examination of two-dimensional wave equation and considered some semidiscretizations. These schemes have many limitations and assumptions in finding the approximate solution of the problems. To overcome these limitations and restriction of variable, we introduce a new iterative strategy for the approximate solution of multidimensional wave problem.The variational iteration method (VIM), Laplace transform, and homotopy analysis method (HAM) have some limitations such as VIM involves the integration and produces the constant of integration, Laplace transform involves the convolution theorem, and HAM also considered some assumptions. The Laplace-Carson integral transform is very easy to implement to the differential problems. The purpose of this paper is to applyLcHITM with the combination of the Laplace-Carson integral transform and the HPM for wave problems of different dimensions. Less computations, fast convergence, and significant results make this scheme unique and different than other approaches of literature. This strategy derives the series of solution with fast convergence and yields the approximate solution very close to the precise solution. This approach is more useful and reliable for the solution of these problems. This paper is introduced as follows: in Section 2, we give a brief detail of the Laplace-Carson integral transform. In Section 3, we present the formulation of LcHITM for solving multidimensional problems. We provide the convergence analysis in Section 4. Some numerical applications are demonstrated to show the effectiveness in Section 5, and eventually, we discuss the conclusion in Section 6.


## 2. Preliminary Definitions ofLcIT

In this section, we describe a few fundamental characteristics and concepts ofLcIT that are very helpful in the formulation of this scheme.Definition 1.
Letϑϕ be a function precise for σ≥0; then,
(1)Lϑϕ=Fs=∫0∞ϑϕe−σϕdϕis called the Laplace transform.Definition 2.
TheLcIT of a function ϑϕ is defined as [26]
(2)Lcϑϕ=Rσ=σ∫0∞ϑϕe−σϕdϕ,ϕ≥0,k1≤σ≤k2,where Lc represents the symbol of LcIT, k1 and k2 are constants, and σ is the independent variable of the transformed function ϕ. Conversely, since Rσ is the LcIT of function ϑϕ, then
(3)Lc−1Rσ=ϑϕ.
Lc−1 is called inverse LcIT.Proposition 3.
LetLcϑ1ϕ=R1σ and Lcϑ2ϕ=R2σ; then [27]
(4)Lcau1ϕ+bu2ϕ=aSϑ1ϕ+bSϑ2ϕ,⇒Lcau1ϕ+bu2ϕ=aR1σ+bR2σ.Proposition 4.
IfAϑϕ=Rσ, the differential properties are defined as [26, 27]
(5)Lcϑ′ϕ=σRσ−σϑ0,Lcϑ′′ϕ=σ2Rσ−σ2ϑ0−σϑ′0,Lcϑmϕ=σmRσ−σmϑ0−σm−1ϑ′0−⋯−σϑm−10.


## 3. Formulation ofLcHITM

In this segment, we formulate the strategy ofLcHITM for finding the approximate solutions of 1D, 2D, and 3D wave equation flows. We observe that this strategy is independent of integration and any hypothesis during the formulation of this scheme. We consider a differential problem such that
(6)ϑ′′ς,ϕ=ϑς,ϕ+gϑ+gς,ϕ,with initial condition
(7)ϑς,0=a1,ϑϕς,0=a2,whereϑ denotes the function in region of time ϕ, gϑ is considered as a nonlinear term, and gς,ϕ is source term arbitrary constant a. Employing LcIT on Equation (6), it yields
(8)Lcϑ′′ς,ϕ=Lcϑς,ϕ+gϑ+gς,ϕ.Using proposition (5) of LcIT, we obtain
(9)σ2Rσ−σ2ϑς,0−σϑ′ς,0=Lcϑς,ϕ+gϑ+gς,ϕ.Hence,Rσ is evaluated such as
(10)Rσ=ϑς,0+ϑ′ς,0σ+1σ2Lcϑς,ϕ+gϑ+gς,ϕ.Operating inverseLcIT on Equation (10), we get
(11)ϑς,ϕ=ϑς,0+ϕϑ′ς,0+Lc−11σ2Lcϑς,ϕ+gϑ+gς,ϕ.Using initial conditions, we get
(12)ϑς,ϕ=a1+ϕa2+Lc−11σ2Lcϑς,ϕ+gϑ+gς,ϕ.Using proposition (4), we obtain
(13)ϑς,ϕ=a1+ϕa2+Lc−11σ2Lcgς,ϕ+Lc−11σ2Lcϑς,ϕ+gϑ.This implies that(14)ϑς,ϕ=Gς,ϕ+Lc−11σ2Lcϑς,ϕ+gϑ.Equation (14) is called the formulation of LcHITM of Equation (6) and
(15)Gς,ϕ=a1+ϕa2+Lc−11σ2Lcgς,ϕ.We introduce HPM in such a way that(16)ϑϕ=∑i=0∞piϑin=ϑ0+p1ϑ1+p2ϑ2+⋯,and nonlinear termsgϑ are evaluated by considering an algorithm:
(17)gϑ=∑i=0∞piHiϑ=H0+p1H1+p2H2+⋯,whereHn polynomials are derived as
(18)Hnϑ0+ϑ1+⋯+ϑn=1n!∂n∂png∑i=0∞piϑip=0,n=0,1,2,⋯.Use Equations (16)–(18) in Equation (14) to compare the identical power of p such as
(19)p0:ϑ0ς,ϕ=Gς,ϕ,p1:ϑ1ς,ϕ=Lc−11σ2Lcϑ0ς,ϕ+H0ϑ,p2:ϑ2ς,ϕ=Lc−11σ2Lcϑ1ς,ϕ+H1ϑ,p3:ϑ3ς,ϕ=Lc−11σ2Lcϑ2ς,ϕ+H2ϑ,⋮.On proceeding this process, this yields(20)ϑς,ϕ=ϑ0+ϑ1+ϑ2+⋯=∑i=0∞ϑi.Thus, Equation (20) is the approximate result of the differential problem (6).


## 4. Convergence Analysis

LetP and Q be Banach spaces where X:P⟶Q is a nonlinear mapping. If the series produced by HPM is
(21)ϑnP,ς=Xϑn−1P,ς=∑i=0n−1ϑiP,ς,n=1,2,3⋯,the following conditions must be true:
(1)
ϑnP,ς−ϑP,ς≤φnϑP,ς−ϑP,ς(2)
ϑnP,ς is forever in the neighbourhood of ϑP,x meaning ϑnP,ς∈BϑP,ς,r=ϑ∗P,ς/ϑ∗P,ς−ϑP,ς(3)
limn⟶∞ϑnP,x=ϑP,ςProof.
(1)
We demonstrate condition (1) by recognition onn, such as ϑ1−ϑ=Gϑ0−ϑ, and the Banach fixed point theorem states that X has a fixed point ϑ, i.e., Xϑ=ϑ; therefore,
(22)ϑ1−ϑ=Gϑ0−ϑ=Gϑ0−Gϑ≤φϑ0−ϑ=φϑP,ς−ϑ,where X is a nonlinear mapping. Consider that ϑn−1−ϑ≤φn−1ϑP,0−ϑP,x is an induction hypothesis; then,
(23)ϑn−ϑ=Gϑn−1−Gϑ≤φϑn−1−ϑ≤φφn−1ϑP,ς−ϑ(2)
Our initial challenge is to demonstrate theϑP,ς∈BϑP,ς,r, which is attained by replacing m. Thus, m=1,ϑP,ς−ϑP,ς=ϑP,0−ϑP,ς≤r with ϑP,0 as an initial condition. Consider that ϑP,x−ϑP,ς≤r for m−2 is an induction theory, so
(24)ϑP,ς−ϑP,ς=ϑm−2P,ς−fmPΓδ−m+1xδ−m≤ϑm−1P,ς−ϑP,ς+fmPΓδ−m+1xδ−m=r
Now,∀n≥1, using (1) we get
(25)ϑn−ϑ≤φnϑP,ς−ϑ≤φnr≤r.(3)
Using condition (2) andlimn⟶∞φn=0, it provides that limn⟶∞ϑn−ϑ=0; hence,
(26)limn⟶∞ϑn=ϑThus, ϑ converges.


## 5. Numerical Applications

We illustrate some numerical applications to check the validity and authenticity ofLcHITM. We observe that this strategy is extremely convenient to utilize and generate the series of convergence much easier than other schemes. We also study the physical behaviors of these surface solutions. The error distribution is obtained graphically to show that the results obtained by LcHITM are very close to the precise results.


### 5.1. Example 1

Suppose a one-dimensional wave equation(27)∂2ϑ∂ϕ2=∂2ϑ∂ς2−3ϑ,with the initial condition
(28)ϑς,0=0,ϑϕς,0=2cosς,and boundary condition
(29)ϑ0,ϕ=sin2ϕ,ϑςπ,ϕ=−sin2ϕ.UsingLcIT on Equation (27), we obtain Rσ such as
(30)Rσ=ϑς,0+ϑ′ς,0σ+1σ2A∂2ϑ∂ς2−3ϑ.Using inverseLcIT, it yields
(31)ϑς,ϕ=ϑς,0+ϕϑϕς,0+Lc−11σ2Lc∂2ϑ∂ς2−3ϑ.Now, apply HPM to obtain He’s polynomials(32)∑i=0∞piϑiς,ϕ=2ϕcosς+Lc−11σ2Lc∑i=0∞pi∂2ϑi∂ς2−3∑i=0∞piϑ.Evaluating similar components ofp, we obtain
(33)p0:ϑ0ς,ϕ=ϑς,0=2ϕcosς,p1:ϑ1ς,ϕ=Lc−11σ2Lc∂2ϑ0∂ς2−3ϑ0=−2ϕ33!cosς,p2:ϑ2ς,ϕ=Lc−11σ2Lc∂2ϑ1∂ς2−3ϑ1=2ϕ55!cosς,p3:ϑ3ς,ϕ=Lc−11σ2Lc∂2ϑ2∂ς2−3ϑ2=−2ϕ77!cosς,p4:ϑ4ς,ϕ=Lc−11σ2Lc∂2ϑ3∂ς2−3ϑ3=2ϕ99!cosς,⋮.In the similar way, we can consider the approximate series such as(34)ϑς,ϕ=ϑ0ς,ϕ+ϑ1ς,ϕ+ϑ2ς,ϕ+ϑ3ς,ϕ+ϑ4ς,ϕ+⋯,ς,ϕ=cosς2ϕ−2ϕ33!+2ϕ55!−2ϕ77!+2ϕ99!+⋯,which can approach to
(35)ϑς,ϕ=cosςsin2ϕ.Figure1 contains two diagrams: (a) the LcHITM results of ϑς,ϕ and (b) the exact results of ϑς,ϕ at −2≤ς≤2 and 0≤ϕ≤0.5 for 1D wave problem. Figure 2 represents the graphical error of 1D wave equation between the approximate and the precise solutions at 0≤ς≤20 with ϕ=0.5. We observe that the current approach demonstrates the strong agreement with the precise answer to the problem (5.1) only after a few iterations. The rate of convergence shows that LcHITM is a relatable approach for ϑς,ϕ. It states that we can effectively model any surface in accordance with the desired physical processes appearing in science and engineering.Figure 1
Surface solutions of 1D wave equation.
(a)
Surface plot for approximate results(b)
Surface plot for precise resultsFigure 2
Graphical error between the approximate and the precise results ofϑς,ϕ.


### 5.2. Example 2

Suppose a two-dimensional wave equation(36)∂2ϑ∂ϕ2=2∂2ϑ∂ς2+∂2ϑ∂ξ2+6ϕ+2ς+4ξ,with the initial condition
(37)ϑς,ξ,0=0,ϑϕς,ξ,0=2sinςsinξ,and boundary condition
(38)ϑ0,ξ,ϕ=ϕ3+2ϕ2ξ,ϑςπ,ξ,ϕ=ϕ3+πϕ2+2ϕ2ξ,ϑς,0,ϕ=ϕ3+ϕ2ς,ϑςς,π,ϕ=ϕ3+2πϕ2+ϕ2ς.ApplyLcIT on
(39)Lc∂2ϑ∂ϕ2=Lc2∂2ϑ∂ς2+∂2ϑ∂ξ2+6ϕ+2ς+4ξ.Using the property functions ofLcIT, we obtain
(40)σ2Rσ−ϑς,0−ϑ′ς,0σ=Lc2∂2ϑ∂ς2+∂2ϑ∂ξ2+6ϕ+2ς+4ξ,σ2Rσ−ϑς,0−ϑ′ς,0σ=Lc2∂2ϑ∂ς2+∂2ϑ∂ξ2+6Lcϕ+2ςLc1+4ξLc1.Hence,Rσ is evaluated. Using LcIT on Equation (36), we obtain Rσ such as
(41)Rσ=6σ3+2ςσ2+4ξσ2+ϑς,0+ϑ′ς,0σ+1σ2Lc2∂2ϑ∂ς2+∂2ϑ∂ξ2.Using inverseLcIT, it yields
(42)ϑς,ξ,ϕ=ϕ3+ςϕ2+2ξϕ2+ϑς,0+ϕϑϕς,0+Lc−11σ2Lc2∂2ϑ∂ς2+∂2ϑ∂ξ2.Now, apply HPM to obtain He’s polynomials(43)∑i=0∞piϑiς,ξ,ϕ=ϕ3+ςϕ2+2ξϕ2+2ϕsinςsinξ+Lc−11σ2Lc2∑i=0∞pi∂2ϑi∂ς2+∑i=0∞pi∂2ϑi∂ξ2.Evaluating similar components ofp, we obtain
(44)p0:ϑ0ς,ξ,ϕ=ϑς,0=ϕ3+ςϕ2+2ξϕ2+2ϕsinςsinξ,p1:ϑ1ς,ξ,ϕ=Lc−11σ2Lc∂2ϑ0∂ς2+∂2ϑ0∂ξ2=−2ϕ33!sinςsinξ,p2:ϑ2ς,ξ,ϕ=Lc−11σ2Lc∂2ϑ1∂ς2+∂2ϑ1∂ξ2=2ϕ55!sinςsinξ,p3:ϑ3ς,ξ,ϕ=Lc−11σ2Lc∂2ϑ2∂ς2+∂2ϑ2∂ξ2=−2ϕ77!sinςsinξ,p4:ϑ4ς,ξ,ϕ=Lc−11σ2Lc∂2ϑ3∂ς2+∂2ϑ3∂ξ2=2ϕ99!sinςsinξ,⋮.In the similar way, we can consider the approximate series such as(45)ϑς,ξ,ϕ=ϑ0ς,ξ,ϕ+ϑ1ς,ξ,ϕ+ϑ2ς,ξ,ϕ+ϑ3ς,ξ,ϕ+ϑ4ς,ξ,ϕ+⋯,ϑς,ξ,ϕ=ϕ3+ςϕ2+2ξϕ2+sinςsinξ2ϕ−2ϕ33!+2ϕ55!−2ϕ77!+2ϕ99!+⋯,which can approach to
(46)ϑς,ξ,ϕ=ϕ3+ςϕ2+2ξϕ2+sinςsinξsin2ϕ.Figure3 contains two diagrams: (a) the LcHITM results of ϑς,ξ,ϕ and (b) the exact results of ϑς,ξ,ϕ at −1≤ς≤1 and 0≤ϕ≤0.1 with ξ=0.5 for 2D wave problem. Figure 4 represents the graphical error of 2D wave equation between the approximate and the precise solutions at 0≤ς≤20 with ξ=0.01 and ϕ=0.01. We observe that current approach demonstrates the strong agreement with the precise answer to the problem (5.2) only after a few iterations. The rate of convergence shows that LcHITM is a reliable approach for ϑς,ξ,ϕ. It states that we can effectively model any surface in accordance with the desired physical processes appearing in nature.Figure 3
Surface solutions of 2D wave equation.
(a)
Surface plot for approximate results(b)
Surface plot for precise resultsFigure 4
Graphical error between the approximate and the precise results ofϑς,ξ,ϕ.


### 5.3. Example 3

Consider the three-dimensional wave problem(47)∂2ϑ∂ϕ2=ς218∂2ϑ∂ς2+ξ218∂2ϑ∂ξ2+η218∂2ϑ∂η2−ϑ,with the initial condition
(48)ϑς,ξ,η,0=0,ϑϕς,ξ,η,0=ς4ξ4η4,and boundary condition
(49)ϑ0,ξ,η,ϕ=0,ϑ1,ξ,η,ϕ=ξ4η4sinhϕ,ϑς,0,η,ϕ=0,ϑς,1,η,ϕ=ς4η4sinhϕ,ϑς,ξ,0,ϕ=0,ϑς,ξ,1,ϕ=ς4ξ4sinhϕ.UsingLcIT on Equation (47), we obtain Rσ such as
(50)Rσ=ϑς,0+ϑ′ς,0σ+1σ2Lcς218∂2ϑ∂ς2+ξ218∂2ϑ∂ξ2+η218∂2ϑ∂η2−ϑ.Using inverseLcIT, it yields
(51)ϑς,ξ,η,ϕ=ϑς,0+ϕϑϕς,0+Lc−11σ2Lcς218∂2ϑ∂ς2+ξ218∂2ϑ∂ξ2+η218∂2ϑ∂η2−ϑ.Now, apply HPM to obtain He’s polynomials(52)∑i=0∞piϑς,ξ,η,ϕ=ϕς4ξ4η4+Lc−11σ2Lc∑i=0∞piς218∂2ϑi∂ς2+∑i=0∞piξ218∂2ϑi∂ξ2+∑i=0∞piη218∂2ϑi∂η2−∑i=0∞piϑ.Evaluating similar components ofp, we obtain
(53)p0:ϑ0ς,ξ,η,ϕ=ϑς,ξ,η,0=ϕς4ξ4η4,p1:ϑ1ς,ξ,ϕ=Lc−11σ2Lcς218∂2ϑ0∂ς2+ξ218∂2ϑ0∂ξ2+η218∂2ϑ0∂η2−ϑ0=ϕ33!ς4ξ4η4,p2:ϑ2ς,ξ,ϕ=Lc−11σ2Lcς218∂2ϑ1∂ς2+ξ218∂2ϑ1∂ξ2+η218∂2ϑ1∂η2−ϑ1=ϕ55!ς4ξ4η4,p3:ϑ3ς,ξ,ϕ=Lc−11σ2Lcς218∂2ϑ2∂ς2+ξ218∂2ϑ2∂ξ2+η218∂2ϑ2∂η2−ϑ2=ϕ77!ς4ξ4η4,p4:ϑ4ς,ξ,ϕ=Lc−11σ2Lcς218∂2ϑ3∂ς2+ξ218∂2ϑ3∂ξ2+η218∂2ϑ3∂η2−ϑ3=ϕ99!ς4ξ4η4,⋮.In the similar way, we can consider the approximate series such as(54)ϑς,ξ,η,ϕ=ϑ0ς,ξ,η,ϕ+ϑ1ς,ξ,η,ϕ+ϑ2ς,ξ,η,ϕ+ϑ3ς,ξ,η,ϕ+ϑ4ς,ξ,η,ϕ+⋯,ϑς,ξ,η,ϕ=ς4ξ4η4ϕ+ϕ33!+ϕ55!+ϕ77!+ϕ99!+⋯,which can approach to
(55)ϑς,ξ,η,ϕ=ς4ξ4η4sinhϕ.Figure5 contains two diagrams: (a) the LcHITM results of ϑς,ξ,η,ϕ and (b) the exact results of ϑς,ξ,η,ϕ at −5≤ς≤5 and 0≤ϕ≤0.05 with ξ=0.5 and η=0.5 for 3D wave problem. Figure 6 represents the graphical error of 3D wave equation between the approximate and the precise solutions at 0≤ς≤10 with ξ=0.5, ς=0.5, and ϕ=0.1. We observe that the current approach demonstrates the strong agreement with the precise answer to the problem (5.3) only after a few iterations. The rate of convergence shows that LcHITM is a reliable approach for ϑς,ξ,η,ϕ. It states that we can effectively model any surface in accordance with the desired physical processes appearing in nature.Figure 5
Surface solutions of 3D wave equation.
(a)
Surface plot for approximate results(b)
Surface plot for precise resultsFigure 6
Graphical error between the approximate and the precise results ofϑς,ξ,η,ϕ.


## 5.1. Example 1

Suppose a one-dimensional wave equation(27)∂2ϑ∂ϕ2=∂2ϑ∂ς2−3ϑ,with the initial condition
(28)ϑς,0=0,ϑϕς,0=2cosς,and boundary condition
(29)ϑ0,ϕ=sin2ϕ,ϑςπ,ϕ=−sin2ϕ.UsingLcIT on Equation (27), we obtain Rσ such as
(30)Rσ=ϑς,0+ϑ′ς,0σ+1σ2A∂2ϑ∂ς2−3ϑ.Using inverseLcIT, it yields
(31)ϑς,ϕ=ϑς,0+ϕϑϕς,0+Lc−11σ2Lc∂2ϑ∂ς2−3ϑ.Now, apply HPM to obtain He’s polynomials(32)∑i=0∞piϑiς,ϕ=2ϕcosς+Lc−11σ2Lc∑i=0∞pi∂2ϑi∂ς2−3∑i=0∞piϑ.Evaluating similar components ofp, we obtain
(33)p0:ϑ0ς,ϕ=ϑς,0=2ϕcosς,p1:ϑ1ς,ϕ=Lc−11σ2Lc∂2ϑ0∂ς2−3ϑ0=−2ϕ33!cosς,p2:ϑ2ς,ϕ=Lc−11σ2Lc∂2ϑ1∂ς2−3ϑ1=2ϕ55!cosς,p3:ϑ3ς,ϕ=Lc−11σ2Lc∂2ϑ2∂ς2−3ϑ2=−2ϕ77!cosς,p4:ϑ4ς,ϕ=Lc−11σ2Lc∂2ϑ3∂ς2−3ϑ3=2ϕ99!cosς,⋮.In the similar way, we can consider the approximate series such as(34)ϑς,ϕ=ϑ0ς,ϕ+ϑ1ς,ϕ+ϑ2ς,ϕ+ϑ3ς,ϕ+ϑ4ς,ϕ+⋯,ς,ϕ=cosς2ϕ−2ϕ33!+2ϕ55!−2ϕ77!+2ϕ99!+⋯,which can approach to
(35)ϑς,ϕ=cosςsin2ϕ.Figure1 contains two diagrams: (a) the LcHITM results of ϑς,ϕ and (b) the exact results of ϑς,ϕ at −2≤ς≤2 and 0≤ϕ≤0.5 for 1D wave problem. Figure 2 represents the graphical error of 1D wave equation between the approximate and the precise solutions at 0≤ς≤20 with ϕ=0.5. We observe that the current approach demonstrates the strong agreement with the precise answer to the problem (5.1) only after a few iterations. The rate of convergence shows that LcHITM is a relatable approach for ϑς,ϕ. It states that we can effectively model any surface in accordance with the desired physical processes appearing in science and engineering.Figure 1
Surface solutions of 1D wave equation.
(a)
Surface plot for approximate results(b)
Surface plot for precise resultsFigure 2
Graphical error between the approximate and the precise results ofϑς,ϕ.


## 5.2. Example 2

Suppose a two-dimensional wave equation(36)∂2ϑ∂ϕ2=2∂2ϑ∂ς2+∂2ϑ∂ξ2+6ϕ+2ς+4ξ,with the initial condition
(37)ϑς,ξ,0=0,ϑϕς,ξ,0=2sinςsinξ,and boundary condition
(38)ϑ0,ξ,ϕ=ϕ3+2ϕ2ξ,ϑςπ,ξ,ϕ=ϕ3+πϕ2+2ϕ2ξ,ϑς,0,ϕ=ϕ3+ϕ2ς,ϑςς,π,ϕ=ϕ3+2πϕ2+ϕ2ς.ApplyLcIT on
(39)Lc∂2ϑ∂ϕ2=Lc2∂2ϑ∂ς2+∂2ϑ∂ξ2+6ϕ+2ς+4ξ.Using the property functions ofLcIT, we obtain
(40)σ2Rσ−ϑς,0−ϑ′ς,0σ=Lc2∂2ϑ∂ς2+∂2ϑ∂ξ2+6ϕ+2ς+4ξ,σ2Rσ−ϑς,0−ϑ′ς,0σ=Lc2∂2ϑ∂ς2+∂2ϑ∂ξ2+6Lcϕ+2ςLc1+4ξLc1.Hence,Rσ is evaluated. Using LcIT on Equation (36), we obtain Rσ such as
(41)Rσ=6σ3+2ςσ2+4ξσ2+ϑς,0+ϑ′ς,0σ+1σ2Lc2∂2ϑ∂ς2+∂2ϑ∂ξ2.Using inverseLcIT, it yields
(42)ϑς,ξ,ϕ=ϕ3+ςϕ2+2ξϕ2+ϑς,0+ϕϑϕς,0+Lc−11σ2Lc2∂2ϑ∂ς2+∂2ϑ∂ξ2.Now, apply HPM to obtain He’s polynomials(43)∑i=0∞piϑiς,ξ,ϕ=ϕ3+ςϕ2+2ξϕ2+2ϕsinςsinξ+Lc−11σ2Lc2∑i=0∞pi∂2ϑi∂ς2+∑i=0∞pi∂2ϑi∂ξ2.Evaluating similar components ofp, we obtain
(44)p0:ϑ0ς,ξ,ϕ=ϑς,0=ϕ3+ςϕ2+2ξϕ2+2ϕsinςsinξ,p1:ϑ1ς,ξ,ϕ=Lc−11σ2Lc∂2ϑ0∂ς2+∂2ϑ0∂ξ2=−2ϕ33!sinςsinξ,p2:ϑ2ς,ξ,ϕ=Lc−11σ2Lc∂2ϑ1∂ς2+∂2ϑ1∂ξ2=2ϕ55!sinςsinξ,p3:ϑ3ς,ξ,ϕ=Lc−11σ2Lc∂2ϑ2∂ς2+∂2ϑ2∂ξ2=−2ϕ77!sinςsinξ,p4:ϑ4ς,ξ,ϕ=Lc−11σ2Lc∂2ϑ3∂ς2+∂2ϑ3∂ξ2=2ϕ99!sinςsinξ,⋮.In the similar way, we can consider the approximate series such as(45)ϑς,ξ,ϕ=ϑ0ς,ξ,ϕ+ϑ1ς,ξ,ϕ+ϑ2ς,ξ,ϕ+ϑ3ς,ξ,ϕ+ϑ4ς,ξ,ϕ+⋯,ϑς,ξ,ϕ=ϕ3+ςϕ2+2ξϕ2+sinςsinξ2ϕ−2ϕ33!+2ϕ55!−2ϕ77!+2ϕ99!+⋯,which can approach to
(46)ϑς,ξ,ϕ=ϕ3+ςϕ2+2ξϕ2+sinςsinξsin2ϕ.Figure3 contains two diagrams: (a) the LcHITM results of ϑς,ξ,ϕ and (b) the exact results of ϑς,ξ,ϕ at −1≤ς≤1 and 0≤ϕ≤0.1 with ξ=0.5 for 2D wave problem. Figure 4 represents the graphical error of 2D wave equation between the approximate and the precise solutions at 0≤ς≤20 with ξ=0.01 and ϕ=0.01. We observe that current approach demonstrates the strong agreement with the precise answer to the problem (5.2) only after a few iterations. The rate of convergence shows that LcHITM is a reliable approach for ϑς,ξ,ϕ. It states that we can effectively model any surface in accordance with the desired physical processes appearing in nature.Figure 3
Surface solutions of 2D wave equation.
(a)
Surface plot for approximate results(b)
Surface plot for precise resultsFigure 4
Graphical error between the approximate and the precise results ofϑς,ξ,ϕ.


## 5.3. Example 3

Consider the three-dimensional wave problem(47)∂2ϑ∂ϕ2=ς218∂2ϑ∂ς2+ξ218∂2ϑ∂ξ2+η218∂2ϑ∂η2−ϑ,with the initial condition
(48)ϑς,ξ,η,0=0,ϑϕς,ξ,η,0=ς4ξ4η4,and boundary condition
(49)ϑ0,ξ,η,ϕ=0,ϑ1,ξ,η,ϕ=ξ4η4sinhϕ,ϑς,0,η,ϕ=0,ϑς,1,η,ϕ=ς4η4sinhϕ,ϑς,ξ,0,ϕ=0,ϑς,ξ,1,ϕ=ς4ξ4sinhϕ.UsingLcIT on Equation (47), we obtain Rσ such as
(50)Rσ=ϑς,0+ϑ′ς,0σ+1σ2Lcς218∂2ϑ∂ς2+ξ218∂2ϑ∂ξ2+η218∂2ϑ∂η2−ϑ.Using inverseLcIT, it yields
(51)ϑς,ξ,η,ϕ=ϑς,0+ϕϑϕς,0+Lc−11σ2Lcς218∂2ϑ∂ς2+ξ218∂2ϑ∂ξ2+η218∂2ϑ∂η2−ϑ.Now, apply HPM to obtain He’s polynomials(52)∑i=0∞piϑς,ξ,η,ϕ=ϕς4ξ4η4+Lc−11σ2Lc∑i=0∞piς218∂2ϑi∂ς2+∑i=0∞piξ218∂2ϑi∂ξ2+∑i=0∞piη218∂2ϑi∂η2−∑i=0∞piϑ.Evaluating similar components ofp, we obtain
(53)p0:ϑ0ς,ξ,η,ϕ=ϑς,ξ,η,0=ϕς4ξ4η4,p1:ϑ1ς,ξ,ϕ=Lc−11σ2Lcς218∂2ϑ0∂ς2+ξ218∂2ϑ0∂ξ2+η218∂2ϑ0∂η2−ϑ0=ϕ33!ς4ξ4η4,p2:ϑ2ς,ξ,ϕ=Lc−11σ2Lcς218∂2ϑ1∂ς2+ξ218∂2ϑ1∂ξ2+η218∂2ϑ1∂η2−ϑ1=ϕ55!ς4ξ4η4,p3:ϑ3ς,ξ,ϕ=Lc−11σ2Lcς218∂2ϑ2∂ς2+ξ218∂2ϑ2∂ξ2+η218∂2ϑ2∂η2−ϑ2=ϕ77!ς4ξ4η4,p4:ϑ4ς,ξ,ϕ=Lc−11σ2Lcς218∂2ϑ3∂ς2+ξ218∂2ϑ3∂ξ2+η218∂2ϑ3∂η2−ϑ3=ϕ99!ς4ξ4η4,⋮.In the similar way, we can consider the approximate series such as(54)ϑς,ξ,η,ϕ=ϑ0ς,ξ,η,ϕ+ϑ1ς,ξ,η,ϕ+ϑ2ς,ξ,η,ϕ+ϑ3ς,ξ,η,ϕ+ϑ4ς,ξ,η,ϕ+⋯,ϑς,ξ,η,ϕ=ς4ξ4η4ϕ+ϕ33!+ϕ55!+ϕ77!+ϕ99!+⋯,which can approach to
(55)ϑς,ξ,η,ϕ=ς4ξ4η4sinhϕ.Figure5 contains two diagrams: (a) the LcHITM results of ϑς,ξ,η,ϕ and (b) the exact results of ϑς,ξ,η,ϕ at −5≤ς≤5 and 0≤ϕ≤0.05 with ξ=0.5 and η=0.5 for 3D wave problem. Figure 6 represents the graphical error of 3D wave equation between the approximate and the precise solutions at 0≤ς≤10 with ξ=0.5, ς=0.5, and ϕ=0.1. We observe that the current approach demonstrates the strong agreement with the precise answer to the problem (5.3) only after a few iterations. The rate of convergence shows that LcHITM is a reliable approach for ϑς,ξ,η,ϕ. It states that we can effectively model any surface in accordance with the desired physical processes appearing in nature.Figure 5
Surface solutions of 3D wave equation.
(a)
Surface plot for approximate results(b)
Surface plot for precise resultsFigure 6
Graphical error between the approximate and the precise results ofϑς,ξ,η,ϕ.


## 6. Conclusion

In this paper, we construct a new scheme known as the Laplace-Carson homotopy integral transform method (LcHITM) for obtaining the approximate solution of 1D, 2D, and 3D wave equations. The main advantage of LcIT is that the recurrence relation produces the iteration without any assumption of a small parameter. HPM helps to produce successive iterations in the recurrence relation. The obtained results show that this approach is very simple to utilize and derive the series solution in the convergence form. Some graphical results are demonstrated to show the physical nature of these wave problems. The graphical error of plot distortion shows that LcHITM has the best agreement with the exact solution. We encourage the readers can extend this scheme for the numerical solution of a nonlinear coupled system of fractional order in science and engineering for their future work.



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