Homotopy Perturbation Method for Solving Wave-Like Nonlinear Equations with Initial-Boundary Conditions

Abstract

The homotopy perturbation method is employed to obtain approximate analytical solutions of the wave-like nonlinear equations with initial-boundary conditions. An efficient way of choosing the auxiliary operator is presented. The results demonstrate reliability and efficiency of the method.

1. Introduction

In this paper, we consider the equation 𝑢_𝑡𝑡−𝑐²𝑢_𝑥𝑥=𝑓𝑥,𝑡,𝑢,𝑢_𝑥,𝑢_𝑦,𝑢_𝑥𝑡+𝑔(𝑥,𝑡),0<𝑥<∞ (1.1) with initial conditions 𝑢(𝑥,0)=𝜑(𝑥),𝑢_𝑡(𝑥,0)=𝜓(𝑥), (1.2) and boundary condition 𝑢(0,𝑡)=ℎ(𝑡), (1.3) where 𝑓, 𝑔, 𝜑, 𝜓, and ℎ are known functions.

Problems like (1.1)-(1.2)-(1.3) model many problems in classical and quantum mechanics, solitons, and matter physics [1, 2]. If 𝑓 is a function of 𝑢 only, we obtain a Klein-Gordon or sine-Gordon-type equations.

In the last decade, some various approximate methods have been developed, such as the homotopy perturbation method (HPM) [3–13] and Adomian’s decomposition method (ADM) [14–20] to solve linear and nonlinear differential equations.

Unlike the various approximation techniques for solving nonlinear wave type problems, which are usually valid for initial value problems (without boundary conditions) or some special type of problems (homogenous, etc.), our technique is applicable for all initial-boundary problems of type (1.1)-(1.2)-(1.3). Chowdhury and Hashim [9] applied the HPM for solving Klein-Gordon and sine-Gordon equations, with initial conditions (1.2). El-Sayed [19] and Wazwaz and Gorguis [20] used ADM for solving wave-like and heat-like problems. Their approaches cannot be applied for all wave-like equations with initial-boundary conditions since the operator 𝐿=𝑢_𝑡𝑡 cannot control the boundary condition (1.3)—see Example 2.1 below.

The central idea here is that the problem 𝑢_𝑡𝑡−𝑐²𝑢_𝑥𝑥(),𝑢()(𝑥)=𝑔𝑥,𝑡𝑥,0=𝜑,𝑢_𝑡(𝑥,0)=𝜓(𝑥),𝑢(0,𝑡)=ℎ(𝑡), (1.4) has a unique solution (see, e.g., [21]) and therefore there exists an inverse of the operator 𝐿=𝑢_𝑡𝑡−𝑐²𝑢_𝑥𝑥.

The main idea of HPM is to introduce a homotopy parameter, say 𝑝, which takes values from 0 to 1. When 𝑝=0 the equation usually reduces to a sufficiently simplified form (linear or very easy nonlinear). As 𝑝 increases to 1, the equation goes through a sequence of “deformations” (homotopics) and at 𝑝=1 takes the original form of the equation.

We rewrite (1.1) as 𝐿𝑢=𝑓𝑥,𝑡,𝑢,𝑢_𝑥,𝑢_𝑦,𝑢_𝑥𝑦+𝑔(𝑥,𝑡), (1.5) and construct the following homotopy: 𝐿𝑢−𝐿𝑣₀+𝑝𝐿𝑣₀−𝑓𝑥,𝑡,𝑢,𝑢_𝑥,𝑢_𝑦,𝑢_𝑥𝑦−𝑔(𝑥,𝑡)=0. (1.6) Usually we take 𝑣₀ as a solution of the problem (1.1)-(1.2)-(1.3) with 𝑓=0 or simply 𝑣₀=0. Assume that the solution of (1.6) is in the form 𝑢=𝑢₀+𝑝𝑢₁+𝑝𝑢₂+⋯, (1.7) and substituting (1.7) into (1.6) and equating terms of the same powers of 𝑝 we obtain a system of equations for 𝑢₀,𝑢₁,𝑢₂,…. Solving these system of equations we obtain a solution in the form 𝑢=𝑢₀+𝑢₁+𝑢₂+⋯. (1.8)

2. Applications

The HPM and ADM offer excellent choices for obtaining the closed-form analytical solutions of wave-like equations. Chowdhury and Hashim [9], El-Sayed [19], and Wazwaz and Gorguis [20] recently showed how the HPM and ADM can be applied to find an analytic approximate solution of wave-like equations with initial conditions (1.2). They mainly used the operator 𝐿=𝑢_𝑡𝑡 for solving the wave-like problems. But in case of inhomogeneous nonlinear or even linear equations with initial-boundary conditions, these approaches have some difficulties. If we construct the standard homotopy with 𝐿=𝑢_𝑡𝑡, for solving wave-like equations with initial-boundary conditions, usually in the second or even in the first stage of HPM, we obtain an “overdetermined” or very difficult problem. To explain these difficulties we consider the following example.

Example 2.1. Consider the linear problem 𝑢_𝑡𝑡−𝑢_𝑥𝑥𝑢()=6𝑥,𝑥,0=−𝑥³,𝑢_𝑡(𝑥,0)=0,𝑢(0,𝑡)=sin²𝑡. (2.1) The exact solution is 𝑢=−𝑥³+sin²(𝑡−𝑥)for𝑥<𝑡=−𝑥³for𝑥>𝑡. (2.2) Using our HPM technique we can easily find this solution. Indeed, let us take 𝐿=𝑢_𝑡𝑡−𝑢_𝑥𝑥, 𝑣₀=0 and construct the homotopy 𝐿𝑢−𝐿𝑣₀+𝑝𝐿𝑣₀−6𝑥=0. (2.3) Now substituting (1.7) into 𝑢 and equating the coefficients of like powers of 𝑝, we get a system of linear equations 𝐿𝑢₀=0,𝑢₀(𝑥,0)=−𝑥³,𝑢_0𝑡(𝑥,0)=0,𝑢₀(0,𝑡)=sin²𝑡,𝐿𝑢₁=6𝑥,𝑢₁(𝑥,0)=0,𝑢_1𝑡(𝑥,0)=0,𝑢₁(0,𝑡)=0,𝐿𝑢₂=0,𝑢₂(𝑥,0)=0,𝑢_2𝑡(𝑥,0)=0,𝑢₂(0,𝑡)=0,…. (2.4) Solving correspondingly we obtain 𝑢₀1=−2(𝑥+𝑡)³+(𝑥−𝑡)³=−3𝑡²𝑥−𝑥³for𝑥≥𝑡=−3𝑡²𝑥−𝑥³+sin²()𝑢𝑡−𝑥for𝑥<𝑡,₁=12^𝑡₀^{𝑥+𝑡−𝑠}_{𝑥−𝑡+𝑠}6𝑦𝑑𝑦𝑑𝑠=3𝑡²𝑢𝑥,for𝑥>𝑡,₁=12₀^{𝑡−𝑥𝑥+𝑡−𝑠}_{𝑡−𝑥−𝑠}16𝑦𝑑𝑦𝑑𝑠+2^𝑡_𝑡−𝑥^{𝑥+𝑡−𝑠}_{𝑥−𝑡+𝑠}6𝑦𝑑𝑦𝑑𝑠=3𝑡²𝑥,for𝑥<𝑡. (2.5)𝑢₂=0,…. Thus we obtain an exact solution 𝑢_exact=𝑢₁+𝑢₂.

Now let us show that this problem can not be solved when 𝐿 is taken as 𝐿=𝑢_𝑡𝑡.

Indeed if we choose 𝑣₀=0 or 𝑣₀=−𝑥³ (which seems most natural and appropriate) and 𝐿=𝑢_𝑡𝑡 and construct the homotopy 𝐿𝑢−𝐿𝑣₀+𝑝𝐿𝑣₀−𝑢_𝑥𝑥−6𝑥=0, (2.6)

we obtain an overdetermined problem for 𝑢₀ in the form 𝐿𝑢₀=0,𝑢₀(𝑥,0)=−𝑥³,𝑢_0𝑡(𝑥,0)=0,𝑢₀(0,𝑡)=sin²𝑡 (2.7)

which has no solution.

If in the first stage we consider the boundary conditions 𝑢₀(𝑥,0)=−𝑥³, 𝑢_0𝑡(𝑥,0)=0, 𝑢₀(0,𝑡)=0, we obtain 𝑢₀=−𝑥³ and in the second stage we need to solve the problem𝐿𝑢₁=𝑢_0𝑥𝑥+6𝑥,𝑢₁(𝑥,0)=0,𝑢_1𝑡(𝑥,0)=0,𝑢₁(0,𝑡)=sin²𝑡, (2.8)

which is overdetermined again and has no solution.

Example 2.2. Consider the problem 𝑢_𝑡𝑡−𝑢_𝑥𝑥=𝑢_𝑥𝑢−𝑒^2𝑡𝑥+𝑒^𝑡𝑢()𝑥,𝑥,0=𝑥,𝑢_𝑡(𝑥,0)=𝑥,𝑢(0,𝑡)=0. (2.9) The exact solution is 𝑢=𝑒^𝑡𝑥. We take again 𝐿=𝑢_𝑡𝑡−𝑢_𝑥𝑥, 𝑣₀=0, and construct the homotopy 𝐿𝑢−𝐿𝑣₀+𝑝𝐿𝑣₀−𝑢_𝑥𝑢+𝑒^2𝑡𝑥−𝑒^𝑡𝑥=0. (2.10) Substituting (1.7) into (2.10) we obtain 𝐿𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯−𝐿𝑣₀+𝑝𝐿𝑣₀−𝑢_0𝑥+𝑝𝑢_1𝑥+𝑝²𝑢_2𝑥+⋯𝑢₀+𝑝𝑢₁+⋯+𝑒^2𝑡𝑥−𝑒^𝑡𝑥=0, (2.11) and equating terms of the coefficients of like powers of 𝑝 gives 𝐿𝑢₀=𝐿𝑣₀,𝑢₀(𝑥,0)=𝑥,𝑢_0𝑡(𝑥,0)=𝑥,𝑢₀(0,𝑡)=0,𝐿𝑢₁+𝐿𝑣₀−𝑢_0𝑥𝑢₀+𝑒^2𝑡𝑥−𝑒^𝑡𝑥=0,𝑢₁(𝑥,0)=0,𝑢_1𝑡(𝑥,0)=0,𝑢₁(0,𝑡)=0,𝐿𝑢₂−𝑢_0𝑥𝑢₁−𝑢_1𝑥𝑢₀=0,𝑢₂(𝑥,0)=0,𝑢_2𝑡(𝑥,0)=0,𝑢₂(0,𝑡)=0,𝐿𝑢₃−𝑢_2𝑥𝑢₀−𝑢_1𝑥𝑢₁−𝑢_0𝑥𝑢₂=0,𝑢₃(𝑥,0)=0,𝑢_3𝑡(𝑥,0)=0,𝑢₃(0,𝑡)=0,…. (2.12)

Solving these equations we obtain (see [21, Chapter 3.4])𝑢₀=12)+(+1[(𝑥+𝑡𝑥−𝑡)]2^𝑥+𝑡_𝑥−𝑡()𝑦𝑑𝑦=𝑥+𝑥+𝑡²4−(𝑥−𝑡)²4(),𝑢=𝑥𝑡+1₁=12^𝑡₀^{𝑥+𝑡−𝑠}_{𝑥−𝑡+𝑠}𝑦(𝑠+1)²−𝑒^2𝑠𝑦+𝑒^𝑠𝑦𝑥𝑑𝑦𝑑𝑠=𝑡12⁴+4𝑡³+6𝑡²−6𝑡−3𝑒^2𝑡+12𝑒^𝑡−9, (2.13)for 𝑥>𝑡 and 𝑢₁=12₀^{𝑡−𝑥𝑥+𝑡−𝑠}_{𝑡−𝑥−𝑠}𝑔1𝑦,𝑠𝑑𝑦𝑑𝑠+2^𝑡_𝑡−𝑥^{𝑥+𝑡−𝑠}_{𝑥−𝑡+𝑠}𝑔𝑦,𝑠𝑑𝑦𝑑𝑠=𝑥²𝑒^𝑡−𝑥−14𝑥𝑒^{2𝑡−2𝑥}−341𝑥+2𝑡²𝑥−𝑡𝑥³+13𝑡³2𝑥+3𝑡𝑥⁴+1𝑡12⁴1𝑥−2𝑥²𝑒^{2𝑡−2𝑥}+𝑥𝑒^𝑡−𝑥−12𝑡²𝑥³−121𝑡𝑥−2𝑥³+23𝑥⁴−14𝑥⁵+𝑡𝑥³−23𝑡𝑥⁴−14𝑥𝑒^2𝑡+12𝑡²𝑥³+𝑥𝑒^𝑡+12𝑥³−23𝑥⁴+14𝑥⁵+12𝑥²𝑒^2𝑡𝑒^2𝑥𝑒−𝑥^𝑡𝑒^𝑥−𝑥²𝑒^𝑡𝑒^𝑥+14𝑥𝑒^2𝑡𝑒^2𝑥=112𝑥𝑡⁴+13𝑥𝑡³+12𝑥𝑡²−121𝑥𝑡−4𝑥𝑒^2𝑡+𝑥𝑒^𝑡−34𝑥, (2.14)for 𝑥<𝑡. In a similar manner we have 𝐿𝑢₂=𝑢_0𝑥𝑢₁+𝑢_1𝑥𝑢₀=()1𝑡+1𝑥𝑡12⁴+4𝑡³+6𝑡²−6𝑡−3𝑒^2𝑡+12𝑒^𝑡+1−9𝑡12⁴+4𝑡³+6𝑡²−6𝑡−3𝑒^2𝑡+12𝑒^𝑡()𝑥=1−9𝑡+16𝑥(𝑡+1)𝑡⁴+4𝑡³+6𝑡²−6𝑡−3𝑒^2𝑡+12𝑒^𝑡−9, (2.15)and therefore 𝑢₂=𝐿⁻¹16𝑥(𝑡+1)𝑡⁴+4𝑡³+6𝑡²−6𝑡−3𝑒^2𝑡+12𝑒^𝑡=1−92^𝑡₀^{𝑥+𝑡−𝑠}_{𝑥−𝑡+𝑠}16𝑦(𝑠+1)𝑠⁴+4𝑠³+6𝑠²−6𝑠−3𝑒^2𝑠+12𝑒^𝑠=𝑥−9𝑑𝑦𝑑𝑠5042𝑡⁷+14𝑡⁶+42𝑡⁵−210𝑡³−378𝑡²−63𝑡𝑒^2𝑡+1008𝑡𝑒^𝑡+63𝑡−1008𝑒^𝑡+1008, (2.16)

for 𝑥>𝑡. In a similar manner we obtain 𝐿𝑢₂=𝑢_0𝑥𝑢₁+𝑢_1𝑥𝑢₀=𝑥𝑡⁵6+56𝑥𝑡⁴+53𝑥𝑡³−𝑥𝑡𝑒^2𝑡2+2𝑥𝑡𝑒^𝑡−5𝑥𝑡2−𝑥𝑒^2𝑡2+2𝑥𝑒^𝑡−3𝑥2,𝑢₂=12₀^{𝑡−𝑥𝑥+𝑡−𝑠}_{𝑡−𝑥−𝑠}𝑔1𝑦,𝑠𝑑𝑦𝑑𝑠+2^𝑡_𝑡−𝑥^{𝑥+𝑡−𝑠}_{𝑥−𝑡+𝑠}𝑔𝑦,𝑠𝑑𝑦𝑑𝑠, (2.17)for 𝑥<𝑡, where 𝑔(𝑦,𝑠)=(1/6)𝑦𝑠⁵+(5/6)𝑦𝑠⁴+(5/3)𝑦𝑠³−(1/2)𝑦𝑠𝑒^2𝑠+2𝑦𝑠𝑒^𝑠−(5/2)𝑦𝑠−(1/2)𝑦𝑒^2𝑠+2𝑦𝑒^s−(3/2)𝑦 and so 𝑢₂=1252𝑥𝑡⁷+136𝑥𝑡⁶+112𝑥𝑡⁵−512𝑥𝑡³−34𝑥𝑡²−18𝑥𝑡𝑒^2𝑡+2𝑥𝑡𝑒^𝑡+18𝑥𝑡−2𝑥𝑒^𝑡+2𝑥. (2.18) The absolute errors between the exact and three term approximation of the series solution for some values of (𝑥,𝑡)∈[0,1]×[0,1] are shown in Table 1.

Table 1 

Maximum errors for Example 2.2.

Example 2.3. Now we consider the problem 𝑢_𝑡𝑡−𝑢_𝑥𝑥=𝑢²_𝑥𝑡−16𝑡²𝑥²+16𝑡²𝑥−2𝑡²−2𝑥²𝑢()+2𝑥,𝑥,0=𝑥,𝑢_𝑡(𝑥,0)=0,𝑢(0,𝑡)=0. (2.19)

The exact solution is 𝑢=𝑥+𝑡²𝑥−𝑡²𝑥². We take 𝑣₀=0, and construct the homotopy 𝐿𝑢−𝐿𝑣₀+𝑝𝐿𝑣₀−𝑢²_𝑥𝑡+16𝑡²𝑥²−16𝑡²𝑥+2𝑡²+2𝑥²−2𝑥=0. (2.20) Now substituting (1.7) into (2.20) and equating terms of the coefficients of like powers of 𝑝, we obtain𝐿𝑢₀=0,𝑢₀(𝑥,0)=𝑥,𝑢_0𝑡(𝑥,0)=𝑢₀(0,𝑡)=0,𝐿𝑢₁+𝐿𝑣₀−𝑢²_0𝑥𝑡+16𝑡²𝑥²−16𝑡²𝑥+2𝑡²+2𝑥²−2𝑥=0,𝑢₁(𝑥,0)=𝑢_1𝑡(𝑥,0)=𝑢₁(0,𝑡)=0,𝐿𝑢₂−2𝑢_0𝑥𝑡𝑢_1𝑥𝑡=0,𝑢₂(𝑥,0)=𝑢_2𝑡(𝑥,0)=𝑢₂(0,𝑡)=0,𝐿𝑢₃−𝑢_1𝑥𝑡²−2𝑢_0𝑥𝑡𝑢_2𝑥𝑡=0,𝑢₃(𝑥,0)=𝑢_3𝑡(𝑥,0)=𝑢₃(0,𝑡)=0. (2.21)

Solving equations yields 𝑢₀=12()𝑢𝑥+𝑡+𝑥−𝑡=𝑥,₁=12^𝑡₀^{𝑥+𝑡−𝑠}_{𝑥−𝑡+𝑠}−16𝑠²𝑦²+16𝑠²𝑦−2𝑠²−2𝑦²4+2𝑦𝑑𝑦𝑑𝑠=−𝑡45⁶−43𝑡⁴𝑥²+43𝑡⁴1𝑥−3𝑡⁴−𝑡²𝑥²+𝑡²𝑥for𝑥>𝑡. (2.22) In a similar manner we get 𝑢₁=12₀^{𝑡−𝑥𝑥+𝑡−𝑠}_{𝑡−𝑥−𝑠}−16𝑠²𝑦²+16𝑠²𝑦−2𝑠²−2𝑦²+1+2𝑦𝑑𝑦𝑑𝑠2^𝑡_𝑡−𝑥^{𝑥+𝑡−𝑠}_{𝑥−𝑡+𝑠}−16𝑠²𝑦²+16𝑠²𝑦−2𝑠²−2𝑦²8+2𝑦𝑑𝑦𝑑𝑠=−𝑡15⁵4𝑥−3𝑡⁴𝑥−169𝑡³𝑥³−43𝑡³4𝑥+3𝑡²𝑥⁴+𝑡²𝑥²+𝑡²8𝑥−15𝑡𝑥⁵−43𝑡𝑥³+4𝑥⁶+𝑥453⁴, (2.23)

for 𝑥<𝑡. Taking only two-term approximation for 𝑢 we have 𝑢=𝑢₀+𝑢₁4=𝑥−𝑡45⁶−43𝑡⁴𝑥²+43𝑡⁴1𝑥−3𝑡⁴−𝑡²𝑥²+𝑡²8𝑥for𝑥>𝑡,𝑢=𝑥−𝑡15⁵4𝑥+3𝑡⁴𝑥−169𝑡³𝑥³−43𝑡³4𝑥+3𝑡²𝑥⁴+𝑡²𝑥²+𝑡²8𝑥−15𝑡𝑥⁵−43𝑡𝑥³+4𝑥45⁶+13𝑥⁴, (2.24)for 𝑥<𝑡. The absolute errors between the exact and two-term approximation of the series solution for some values of (𝑥,𝑡)∈[0,1]×[0,1] are shown in Table 2. A higher accuracy level can be attained by evaluating some more terms.

Table 2 

Maximum errors for Example 2.3.

Example 2.4. Now we consider the problem 𝑢_𝑡𝑡−𝑢_𝑥𝑥=𝑢_𝑥+𝑢_𝑡𝑢()−2𝑥−2,𝑥,0=𝑥²,𝑢_𝑡(𝑥,0)=0,𝑢(0,𝑡)=sin²𝑡. (2.25) It is easy to show that the HPM can not be applied if 𝐿 is taken as 𝑢_𝑡𝑡. The exact solution is 𝑢=𝑥²+sin²(𝑡−𝑥)for𝑥<𝑡=𝑥²for𝑥≥𝑡. (2.26) We construct the homotopy 𝐿𝑢−𝐿𝑣₀+𝑝𝐿𝑣₀−𝑢_𝑥−𝑢_𝑡+2𝑥+2=0, (2.27) with 𝐿=𝑢_𝑡𝑡−𝑢_𝑥𝑥. Substituting (1.7) into 𝑢 and equating the coefficients of like powers of 𝑝, we get a system of linear equations 𝐿𝑢₀=0,𝑢₀(𝑥,0)=𝑥²,𝑢_0𝑡(𝑥,0)=0,𝑢₀(0,𝑡)=sin²𝑡,𝐿𝑢₁=𝑢_0𝑥+𝑢_0𝑡−2𝑥−2,𝑢₁(𝑥,0)=0,𝑢_1𝑡(𝑥,0)=0,𝑢₁(0,𝑡)=0,𝐿𝑢₂=𝑢_1𝑥+𝑢_1𝑡,𝑢₂(𝑥,0)=0,𝑢_2𝑡(𝑥,0)=0,𝑢₂(0,𝑡)=0,…. (2.28) Solving we obtain 𝑢₀=12(𝑥+𝑡)²−(𝑡−𝑥)²+sin²(𝑡−𝑥)=2𝑡𝑥+sin²()𝑢𝑡−𝑥for𝑥<𝑡,₀=12(𝑥+𝑡)²+(𝑥−𝑡)²=𝑥²+𝑡²for𝑥>𝑡,𝐿𝑢₁=𝑢_0𝑥+𝑢_0𝑡𝑢−2𝑥−2=2𝑡+2𝑥−2𝑥−2=2𝑡−2,₁=12₀^{𝑡−𝑥𝑥+𝑡−𝑠}_{𝑡−𝑥−𝑠}()12𝑠−2𝑑𝑦𝑑𝑠+2^𝑡_𝑡−𝑥^{𝑥+𝑡−𝑠}_{𝑥−𝑡+𝑠}(2𝑠−2)𝑑𝑦𝑑𝑠=𝑡²𝑥−𝑡𝑥²1−2𝑡𝑥+3𝑥³+𝑥²for𝑥<𝑡,𝐿𝑢₁=𝑢_0𝑥+𝑢_0𝑡𝑢−2𝑥−2=2𝑡−2,₁=12^𝑡₀^{𝑥+𝑡−𝑠}_{𝑥−𝑡+𝑠}()12𝑠−2𝑑𝑦𝑑𝑠=3𝑡²(𝑡−3)for𝑥>𝑡. (2.29) Continuing we obtain 𝐿𝑢₂=𝑢_1𝑥+𝑢_1𝑡=𝑡²−2𝑡𝑥−2𝑡+𝑥²+2𝑥+2𝑡𝑥−𝑥²−2𝑥=𝑡²𝑢−2𝑡,₂=12₀^{𝑡−𝑥𝑥+𝑡−𝑠}_{𝑡−𝑥−𝑠}𝑠²1−2𝑠𝑑𝑦𝑑𝑠+2^𝑡_𝑡−𝑥^{𝑥+𝑡−𝑠}_{𝑥−𝑡+𝑠}𝑠²=7−2𝑠𝑑𝑦𝑑𝑠3𝑡³5𝑥−2𝑡²𝑥²−5𝑡²4𝑥+3𝑡𝑥³+3𝑡𝑥²−14𝑥⁴−23𝑥³𝑢for𝑥<𝑡,₂=12^𝑡₀^{𝑥+𝑡−𝑠}_{𝑥−𝑡+𝑠}𝑠²1−2𝑠𝑑𝑦𝑑𝑠=𝑡12⁴−13𝑡³for𝑥>𝑡. (2.30) Thus the three-term approximation is 𝑢=sin²(𝑡−𝑥)+𝑥²+73𝑡³5𝑥−2𝑡²𝑥²−4𝑡²4𝑥+3𝑡𝑥³+2𝑡𝑥²−𝑥⁴4−𝑥³3for𝑥<𝑡,𝑢=𝑥²+1𝑡12⁴for𝑥>𝑡, (2.31) and the error is less than (1/12)𝑡⁴ for 𝑥>𝑡.

3. Conclusion

Homotopy perturbation method has been successful for solving many linear and nonlinear wave-type problems. However, it has difficulties in dealing with initial boundary problems, namely, in including all initial and boundary conditions together into the process of homotopy perturbation and computation. Our main goal is to construct the homotopy perturbation scheme containing all initial and boundary conditions together. The goal is achieved by involving an auxiliary operator which includes both variables 𝑥 and 𝑡.