Application of Optimal Homotopy Asymptotic Method to Burger Equations

Abstract

We apply optimal homotopy asymptotic method (OHAM) for finding approximate solutions of the Burger's-Huxley and Burger's-Fisher equations. The results obtained by proposed method are compared to those of Adomian decomposition method (ADM) (Ismail et al., (2004)). As a result it is concluded that the method is explicit, effective, and simple to use.

1. Introduction

Nonlinear phenomena play a vital role in applied mathematics, physics, and engineering sciences. The Burger’s equation models efficiently certain problems of a fluid flow nature, in which either shocks or viscous dissipation is a significant factor. It can be used as a model for any nonlinear wave propagation problem subject to dissipation [1]. The first steady-state solutions of Burger equation were given by Young et al. [2] However, the equation gets its name from the extensive research of Burger’s [3]. The generalized Burger’s-Huxley introduced by Satsuma shows a prototype model for describing the communication among reaction mechanisms, convection effects, and diffusion transports [4]. Burger-Fisher equation has significant applications in various fields of applied mathematics and has physical applications such as gas dynamic, traffic flow, convection effect, and diffusion transport [5–12]. Marinca and Herişanu et al. introduced a new semianalytic method OHAM for approximate solution of nonlinear problems of thin film flow of a fourth-grade fluid down a vertical cylinder. In progression of papers Marinca and Herişanu et al. have applied this method for the solution of nonlinear equations arising in the steady state flow of a fourth-grade fluid past a porous plate and for the solution of nonlinear equations arising in heat transfer [13–15]. The method has been applied by a number of researchers for solution of ordinary and partial differential equations [16–21]. The motivation of this paper is to show the effectiveness of OHAM for the solution of Burger’s-Huxley and Burger’s-Fisher equations. We consider Burger’s-Huxley equation of the form 𝜕𝑢(𝑥,𝑡)𝜕𝑡+𝛼𝑢^𝛿()()𝑥,𝑡𝜕𝑢𝑥,𝑡−𝜕𝜕𝑥²𝑢(𝑥,𝑡)𝜕𝑥²−𝛽𝑢(𝑥,𝑡)1−𝑢^𝛿(𝑥,𝑡)𝑢^𝛿(𝑥,𝑡)−𝛾=0,∀0≤𝑥≤1,𝑡≥0 (1) and Burger’s-Fisher equation of the form 𝜕𝑢(𝑥,𝑡)𝜕𝑡+𝛼𝑢^𝛿()()𝑥,𝑡𝜕𝑢𝑥,𝑡−𝜕𝜕𝑥²𝑢(𝑥,𝑡)𝜕𝑥²−𝛽𝑢(𝑥,𝑡)1−𝑢^𝛿(𝑥,𝑡)=0,∀0≤𝑥≤1,𝑡≥0, (2) where 𝛼, 𝛽, 𝛾, and 𝛿 are parameters and 𝛽≥0, 𝛿≥0, 𝛾∈(0,1).

The present paper is divided into three sections. In Section 2 fundamental mathematical theory of OHAM is presented. In Section 3 comparisons are made between the results of the proposed method and HAM for Burger’s-Huxley. In Section 4 solution of Burger’s-Fisher equation is presented, and absolute error of approximate solution of proposed method is compared with approximate solution of HAM. In all cases the proposed method yields better results than those of ADM.

2. Fundamental Theory of OHAM

Here we start by describing the basic idea of OHAM. Consider the partial differential equation of the form: ℒ(𝑢((𝑢(()ℬ𝑥,𝑡))+𝒩𝑥,𝑡))+ℊ𝑥,𝑡=0,𝑥∈Ω,𝑢,𝜕𝑢𝜕𝑡=0, (3)(4) where ℒ is a linear operator and 𝒩 is nonlinear operator. ℬ is boundary operator, 𝑢(𝑥,𝑡) is an unknown function, and 𝑥 and 𝑡 denote spatial and time variables, respectively; Ω is the problem domain and 𝑔(𝑥,𝑡) is a known function.

According to the basic idea of OHAM, one can construct the optimal homotopy 𝜓(𝑥,𝑡;𝑞)∶Ω×[0,1]→𝑅 which satisfies ℒ𝜓(𝑞ℒ𝜓𝜓(,1−𝑞𝑥,𝑡;𝑞+𝑔𝑥,𝑡)=ℋ𝑥,𝑡;𝑞+𝒩𝑥,𝑡;𝑞+ℊ𝑥,𝑡) (5) where 𝑞∈[0,1] is an embedding parameter, 𝐻(𝑞) is a nonzero auxiliary function for 𝑞≠0, 𝐻(0)=0. Equation (3) is called optimal homotopy equation. Clearly, we have 𝜓()𝜓()()𝜓()(1ℒ𝜓𝜓(𝑞=0⟹ℋ𝑥,𝑡;0,0=ℒ𝑥,𝑡;0+ℊ𝓍,𝑡=0,𝑞=1⟹ℋ𝑥,𝑡;1,1=ℋ)𝑥,𝑡;𝑞+𝒩𝑥,𝑡;𝑞+ℊ𝑥,𝑡)=0. (6)(7) Clearly, when 𝑞=0 and 𝑞=1, it holds that 𝜓(𝑥,𝑡;0)=𝑢₀(𝑥,𝑡) and 𝜓(𝑥,𝑡;1)=𝑢(𝑥,𝑡), respectively. Thus, as 𝑞 varies from 0 to 1, the solution 𝜓(𝑥,𝑡;𝑞) approaches from 𝑢₀(𝑥,𝑡) to 𝑢(𝑥,𝑡), where 𝑢₀(𝑥,𝑡) is obtained from (3) for 𝑞=0: ℒ𝑢₀(𝑥,𝑡)+𝑔(𝑥,𝑡)=0,ℬ𝑢₀,𝜕𝑢₀𝜕𝑡=0. (8) Next, we choose auxiliary function 𝐻(𝑞) in the form ℋ𝑞=𝑞𝐶₁+𝑞²𝐶₂+⋯. (9) Here 𝐶₁,𝐶₂,… are constants to be determined later.

To get an approximate solution, we expand 𝜓(𝑥,𝑡;𝑞,𝐶_𝑖) in Taylor’s series about 𝑞 in the following manner: 𝜓𝑥,𝑡;𝑞,𝐶_𝑖=𝑢₀(𝑥,𝑡)+^∞_𝑘=1𝑢_𝑘𝑥,𝑡;𝐶_𝑖𝑞^𝑘,𝑖=1,2,…. (10) Substituting (10) into (4) and equating the coefficient of like powers of 𝑞, we obtain Zeroth-order problem, given by (6), the first- and second-order problems are given by (11)-(12), respectively, and the general governing equations for 𝑢_𝑘(𝑥,𝑡) are given by (13): ℒ𝑢₁(𝑥,𝑡)=𝐶₁𝒩₀𝑢₀(𝑥,𝑡),ℬ𝑢₁,𝜕𝑢₁ℒ𝑢𝜕𝑡=0,₂(𝑥,𝑡)−ℒ𝑢₁(𝑥,𝑡)=𝐶₂𝒩₀𝑢₀(𝑥,𝑡)+𝐶₁ℒ𝑢₁(𝑥,𝑡)+𝒩₁𝑢₀(𝑥,𝑡),𝑢₁(),ℬ𝑢𝑥,𝑡₂,𝜕𝑢₂ℒ𝑢𝜕𝑡=0,_𝑘(𝑥,𝑡)−ℒ𝑢_𝑘−1(𝑥,𝑡)=𝐶_𝑘𝒩₀𝑢₀(+𝑥,𝑡)_{𝑘−1𝑖=1}𝐶_𝑖ℒ𝑢_𝑘−𝑖(𝑥,𝑡)+𝒩_𝑘−𝑖×𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),…,𝑢_𝑘−𝑖(),ℬ𝑢𝑥,𝑡_𝑘,𝜕𝑢_𝑘𝜕𝑡=0,𝑘=2,3,…, (11)(12)(13) where 𝒩_𝑘−𝑖(𝑢₀(𝑥,𝑡),𝑢₁(𝑥,𝑡),…,𝑢_𝑘−𝑖(𝑥,𝑡)) is the coefficient of 𝑞^𝑘−𝑖 in the expansion of 𝒩(𝜓(𝑥,𝑡;𝑞)) about the embedding parameter 𝑞: 𝒩𝜓𝑥,𝑡;𝑞,𝐶_𝑖=𝒩₀𝑢₀(𝑥,𝑡)+_𝑘≥1𝒩_𝑘𝑢₀,𝑢₁,𝑢₂,…,𝑢_𝑘𝑞^𝑘. (14) Here 𝑢_𝑘 for 𝑘≥0 are set of linear equations with the linear boundary conditions, which can be easily solved.

The convergence of the series in (10) depends upon the auxiliary constants 𝐶₁,𝐶₂,…. If it is convergent at 𝑞=1, one has: 𝑢𝑥,𝑡;𝐶_𝑖=𝑢₀(𝑥,𝑡)+_𝑘≥1𝑢_𝑘𝑥,𝑡;𝐶_𝑖. (15) Substituting (15) into (1) results in the following expression for residual: ℛ𝑥,𝑡;𝐶_𝑖=ℒ𝑢𝑥,𝑡;𝐶_𝑖+ℊ(𝑥,𝑡)+𝒩𝑢𝑥,𝑡;𝐶_𝑖. (16) If 𝑅(𝑥,𝑡;𝐶_𝑖)=0, then 𝑢(𝑥,𝑡;𝐶_𝑖) will be the exact solution.

For computing the auxiliary constants, 𝐶_𝑖, 𝑖=1,2,…,𝑚, there are many methods like Galerkin’s Method, Ritz Method, Least Squares Method, and Collocation Method to find the optimal values of 𝐶_𝑖, 𝑖=1,2,3,…, One can apply the Method of Least Squares as ℐ𝐶_𝑖=^𝑡_0Ωℛ²𝑥,𝑡,𝐶_𝑖𝑑𝑥𝑑𝑡, (17) where 𝑅 is the residual, 𝑅(𝑥,𝑡;𝐶_𝑖)=𝐿(𝑢(𝑥,𝑡;𝐶_𝑖))+𝑔(𝑥,𝑡)+𝑁(𝑢(𝑥,𝑡;𝐶_𝑖)), and 𝜕ℐ𝜕𝐶₁=𝜕ℐ𝜕𝐶₂=⋯=𝜕ℐ𝜕𝐶_𝑚=0. (18) The constants 𝐶_𝑖 can also be determined by another method as ℛℎ₁;𝐶_𝑖=ℛℎ₂;𝐶_𝑖=⋯=ℛℎ_𝑚;𝐶_𝑖=0,𝑖=1,2,…,𝑚, (19) at any time 𝑡, where ℎ_𝑖∈Ω. The convergence depends upon constants 𝐶₁,𝐶₂,…, can be optimally identified and minimized by (18).

3. Application of OHAM

In this section we apply OHAM for the two problems: the first is the Burger’s-Huxley equation (1) and the second is the Burger’s-Fisher equation (2).

3.1. Application of OHAM for Burger’s-Huxley Equation

Let us consider Burger’s-Huxley equation of form (1): 𝜕𝑢(𝑥,𝑡)𝜕𝑡+𝛼𝑢^𝛿()()𝑥,𝑡𝜕𝑢𝑥,𝑡−𝜕𝜕𝑥²𝑢(𝑥,𝑡)𝜕𝑥²−𝛽𝑢(𝑥,𝑡)1−𝑢^𝛿(𝑥,𝑡)𝑢^𝛿(𝑥,𝑡)−𝛾=0,∀0≤𝑥≤1,𝑡≥0, (20) Subject to constant initial condition 𝑢(𝑥,0)=0.5𝛾+0.5𝛾tanh𝐴₁𝑥^1/𝛿. (21) The exact solution of (26) with given condition is given by 𝑢(𝑥,0)=0.5𝛾+0.5𝛾tanh𝐴₁𝑥−𝐴₂𝑡^1/𝛿, (22) where 𝐴₁=−𝛼𝛿+𝛿𝛼²+4𝛽(1+𝛿)4()𝐴1+𝛿𝛾,₂=𝛼𝛾()−1+𝛿1+𝛿−𝛾−𝛼+𝛼²+4𝛽(1+𝛿)2().1+𝛿 (23) For computational work, we have taken 𝛼=1, 𝛽=1, 𝛿=1, and 𝛾=0.001 for various values of 𝑥 and 𝑡.

Following the basic idea of OHAM presented in preceding section we start with

Zeroth-Order Problem𝜕𝑢₀(𝑥,𝑡)𝑢𝜕𝑡=0,₀(𝑥,0)=(0.0005+0.0005tanh(0.00025𝑥)). (24) Its solution is 𝑢₀(𝑥,𝑡)=(0.0005+0.0005tanh(0.00025𝑥)). (25)

First-Order Problem𝜕𝑢₁(𝑥,𝑡)−𝜕𝑡1+𝐶₁𝜕𝑢₀(𝑥,𝑡)𝜕𝑡−𝐶₁𝑢₀(𝑥,𝑡)𝜕𝑢₀(𝑥,𝑡)𝜕𝑥+𝐶₁𝜕²𝑢₀(𝑥,𝑡)𝜕𝑥²−0.001𝐶₁𝑢₀(𝑥,𝑡)+0.001𝐶₁𝑢²₀(𝑥,𝑡)−𝐶₁𝑢³₀()𝑢𝑥,𝑡=0,₁(𝑥,0)=0. (26) Its solution is 𝑢₁(𝑥,𝑡)=−𝑡−2.4987500000000003×10⁻⁷𝐶₁−6.25×10⁻¹¹𝐶₁sech²(0.00025𝑥)+1.249999999999317×10⁻¹⁰𝐶₁tanh(0.00025𝑥)−1.25×10⁻¹⁰𝐶₁sech²()()0.00025𝑥×tanh0.00025𝑥+2.49875×10⁻⁷𝐶₁tanh²(0.00025𝑥)−1.25×10⁻¹⁰𝐶₁tanh³(0.00025𝑥). (27)

Second-Order Problem𝜕𝑢₂(𝑥,𝑡)−𝜕𝑡1+𝐶₁𝜕𝑢₁(𝑥,𝑡)𝜕𝑡−𝐶₂𝑢₀(𝑥,𝑡)𝜕𝑢₀(𝑥,𝑡)𝜕𝑥+𝐶₂𝜕²𝑢₀(𝑥,𝑡)𝜕𝑥²−𝐶₂𝜕𝑢₀(𝑥,𝑡)𝜕𝑡−𝐶₁𝑢₁(𝑥,𝑡)𝜕𝑢₀(𝑥,𝑡)𝜕𝑥−𝐶₁𝑢₀(𝑥,𝑡)𝜕𝑢₁(𝑥,𝑡)𝜕𝑥+𝐶₁𝜕²𝑢₁(𝑥,𝑡)𝜕𝑥²−0.001𝐶₂𝑢₀(𝑥,𝑡)+1.001𝐶₂𝑢²₀(𝑥,𝑡)−𝐶₂𝑢³₀(𝑥,𝑡)−0.001𝐶₁𝑢₁(𝑥,𝑡)+2.002𝐶₁𝑢₀(𝑥,𝑡)𝑢₁(𝑥,𝑡)−3𝐶₁𝑢²₀(𝑥,𝑡)𝑢₁()𝑢𝑥,𝑡=0,₂(𝑥,0)=0. (28) Its solution is 𝑢₂𝑥,𝑡,𝐶₁=−𝑡−2.4987500000000003×10⁻⁷𝐶₁−6.25×10⁻¹¹𝐶₁sech²(0.00025𝑥)+1.249999999999317×10⁻¹⁰𝐶₁×tanh(0.00025𝑥)−1.25×10⁻¹⁰𝐶₁×sech²(0.00025𝑥)tanh(0.00025𝑥)+2.49875×10⁻⁷𝐶₁tanh²(0.00025𝑥)−1.25×10⁻¹⁰𝐶₁tanh³(0.00025𝑥). (29)

Third-Order Problem𝜕𝑢₃(𝑥,𝑡)−𝜕𝑡1+𝐶₁𝜕𝑢₂(𝑥,𝑡)𝜕𝑡−𝐶₃𝑢₀(𝑥,𝑡)𝜕𝑢₀(𝑥,𝑡)𝜕𝑥+𝐶₃𝜕²𝑢₀(𝑥,𝑡)𝜕𝑥²−𝐶₃𝜕𝑢₀(𝑥,𝑡)𝜕𝑡−𝐶₂𝑢₁(𝑥,𝑡)𝜕𝑢₀(𝑥,𝑡)𝜕𝑥−𝐶₂𝑢₀(𝑥,𝑡)𝜕𝑢₁(𝑥,𝑡)𝜕𝑥+𝐶₂𝜕²𝑢₁(𝑥,𝑡)𝜕𝑥²−𝐶₂𝜕𝑢₁(𝑥,𝑡)𝜕𝑡−𝐶₁𝑢₂(𝑥,𝑡)𝜕𝑢₀(𝑥,𝑡)𝜕𝑥−𝐶₁𝑢₁(𝑥,𝑡)𝜕𝑢₁(𝑥,𝑡)𝜕𝑥+𝐶₁𝜕²𝑢₂(𝑥,𝑡)𝜕𝑥²−𝐶₁𝑢₀(𝑥,𝑡)𝜕𝑢₂(𝑥,𝑡)𝜕𝑥−0.001𝐶₃𝑢₀(𝑥,𝑡)+1.001𝐶₃𝑢²₀(𝑥,𝑡)−𝐶₃𝑢³₀(𝑥,𝑡)−0.001𝐶₂𝑢₁(𝑥,𝑡)−3𝐶₂𝑢²₀(𝑥,𝑡)𝑢₁(𝑥,𝑡)+1.001𝐶₁𝑢²₁(𝑥,𝑡)−3𝐶₁𝑢₀(𝑥,𝑡)𝑢²₁(𝑥,𝑡)−0.001𝐶₁𝑢₂(𝑥,𝑡)+2.002𝐶₁𝑢₀(𝑥,𝑡)𝑢₂(𝑥,𝑡)−3𝐶₁𝑢²₀(𝑥,𝑡)𝑢₂(𝑥,𝑡)+2.002𝐶₂𝑢₀(𝑥,𝑡)𝑢₁()𝑢𝑥,𝑡=0,₃(𝑥,0)=0. (30) Its solution is 𝑢₃𝑥,𝑡,𝐶₁,𝐶₂,𝐶₃=11+𝑒^0.5𝑥2𝑡sech²()×0.25𝑥−0.0625𝐶₂−0.0625𝐶₃+𝐶₁−0.0625+𝐶₁(−0.125−0.015625𝑡)+𝐶₂(−0.125−0.015625𝑡)−0.000651042𝐶²₁()×()+5.0718+𝑡18.9282+𝑡288𝐶₂+288𝐶₃+𝐶₁×288+576𝐶₂+𝐶₁(576−24𝑡)−24𝐶₂𝑡−3𝐶²₁()×()()+−6.58301+𝑡14.583+𝑡cosh0.25𝑥−96𝐶₂−96𝐶₃+𝐶₁×−96−192𝐶₂+𝐶₁(−192−24𝑡)−24𝐶₂𝑡+5𝐶²₁()×()()−7.396+𝑡2.596+𝑡sinh0.25𝑥−24𝐶₂𝑡+5𝐶²₁()()×(()().−7.396+𝑡)(2.596+𝑡×sinh0.25𝑥−0.000651042cosh0.75𝑥−0.000651042sinh0.75𝑥 (31) Adding (25), (27), (29), and (31) we obtain 𝑢𝑥,𝑡,𝐶₁,𝐶₂=𝑢₀(𝑥,𝑡)+𝑢₁𝑥,𝑡,𝐶₁+𝑢₂𝑥,𝑡,𝐶₁,𝐶₂+𝑢₃𝑥,𝑡,𝐶₁,𝐶₂,𝐶₃. (32) For the calculations of the constants 𝐶₁, 𝐶₂, and 𝐶₃ using the collocation method, we have computed that 𝐶₁=−1.0000010231545267,  𝐶₂=−9.98159444155818×10⁻⁷,  𝐶₃=−2.041939789528322×10⁻¹².

Putting the values of these constants into (32) the third order approximate solution using OHAM is 𝑢₃(𝑥,𝑡)=0.5−5.2607845247854×10⁻⁶𝑡sech²()+10.25𝑥1+𝑒^0.5𝑥2×6.360090415451543×10⁻⁹+1.3153067008227217×10⁻⁶𝑡++0.0006512058813062292𝑡1.2720180853076355×10⁻⁸−0.002604823525224917𝑡²()+cosh0.5𝑥6.36009042653818×10⁻⁹+−1.3153067008227217×10⁻⁶𝑡(𝑥)++0.0006512058813062292𝑡cosh1.2720180853076355×10⁻⁸−0.002604823525224917𝑡²()+sinh0.5𝑥6.36009042653818×10⁻⁹+−1.3153067008227217×10⁻⁹𝑡(𝑥)()+0.0006512058813062292𝑡sinh−0.5tanh0.25𝑥+0.007813813661895406𝑡²×sech²(0.25𝑥)tanh(0.25𝑥)+sech²()×.0.25𝑥0.06250525442670962𝑡 (33)

Table 1 shows a comparison between OHAM solution and ADM solution for 𝛼=1 and 𝛾=0.001. For 𝛼=0 (1) is reduced to the generalized Huxley equation which describes nerve pulse propagation in nerve fibers and wall motion in liquid crystals [22]. Tables 2 and 3 show a comparison between ADM solution and OHAM solution for 𝛼=0 and 𝛽=1 respectively. Table 4 shows absolute errors of OHAM solution for larger domain for 𝛼=0,1, 𝛽=1, and 𝛿=1,2 respectively.

Table 1 

Comparison of absolute errors of OHAM and ADM [5] for 𝛼=1, 𝛽=1, 𝛿=1, and 𝛾=0.001.

Table 2 

Comparison of absolute errors obtained by OHAM and ADM [5] for 𝛼=0, 𝛽=1, 𝛿=1, and 𝛾=0.001.

Table 3 

Comparison of absolute errors obtained by OHAM and ADM [5] for 𝛼=0, 𝛽=1, 𝛿=2, and 𝛾=0.001.

Table 4 

Absolute errors of OHAM for 𝛼=0,1, 𝛽=1, 𝛿=1,2, and 𝑥=2.

3.2. Application of OHAM for Burger’s-Fisher Equation

Consider the Burger’s-Fisher equation of form (2): 𝜕𝑢(𝑥,𝑡)𝜕𝑡+𝛼𝑢^𝛿()()𝑥,𝑡𝜕𝑢𝑥,𝑡−𝜕𝜕𝑥²𝑢(𝑥,𝑡)𝜕𝑥²−𝛽𝑢(𝑥,𝑡)1−𝑢^𝛿(𝑥,𝑡)=0,∀0≤𝑥≤1,𝑡≥0, (34) subject to constant initial condition 𝑢(𝑥,0)=0.5+0.5tanh−𝛼𝛿2()𝑥𝛿+1^1/𝛿, (35) with exact solution given by 𝑢()×𝑥,𝑡=0.5+0.5tanh−𝛼𝛿2()×𝛼𝛿+1𝑥−()+𝛽()𝛿+1𝛿+1𝛼𝑡^1/𝛿. (36) For computational work, we have taken 𝛼=0.001, 𝛽=0.001, and 𝛿=1 for various values of 𝑥 and 𝑡.

Zeroth-Order Problem𝜕𝑢₀(𝑥,𝑡)𝑢𝜕𝑡=0,₀(𝑥,0)=0.5+0.5tanh−0.0014𝑥. (37) Its solution is 𝑢₀(𝑥,𝑡)=0.5−0.5tanh0.0014𝑥. (38)

First-Order Problem𝜕𝑢₁(𝑥,𝑡)−𝜕𝑡1+𝐶₁𝜕𝑢₀(𝑥,𝑡)𝜕𝑡−0.001𝐶₁𝑢₀(𝑥,𝑡)𝜕𝑢₀(𝑥,𝑡)𝜕𝑥−0.001𝐶₁𝑢₀(𝑥,𝑡)1−𝑢₀(𝑥,𝑡)+𝐶₁𝜕²𝑢₀(𝑥,𝑡)𝜕𝑥²𝑢=0,₁(𝑥,0)=0. (39) Its solution is 𝑢₁𝑥,𝑡,𝐶₁=−𝑡0.00025𝐶₁+6.25×10⁻⁸𝐶₁×sech²(0.00025𝑥)−0.00025𝐶₁tanh²(0.00025𝑥). (40)

Second-Order Problem𝜕𝑢₂(𝑥,𝑡)−𝜕𝑡1+𝐶₁𝜕𝑢₁(𝑥,𝑡)𝜕𝑡+0.001𝐶₂𝑢₀()×𝑥,𝑡1−𝑢₀(𝑥,𝑡)+𝐶₂𝜕²𝑢₀(𝑥,𝑡)𝜕𝑥²−𝐶₂𝜕𝑢₀(𝑥,𝑡)𝜕𝑡−0.001𝐶₂𝑢₀(𝑥,𝑡)𝜕𝑢₀(𝑥,𝑡)𝜕𝑥−0.001𝐶₁𝑢₁(𝑥,𝑡)𝜕𝑢₀(𝑥,𝑡)𝜕𝑥−0.001𝐶₁𝑢₀()×𝑥,𝑡𝜕𝑢₁(𝑥,𝑡)𝜕𝑥+𝐶₁𝜕²𝑢₁(𝑥,𝑡)𝜕𝑥²−0.001𝐶₁𝑢₁(𝑥,𝑡)−0.002𝐶₁𝑢₀(𝑥,𝑡)𝑢₁()𝑢𝑥,𝑡=0,₂(𝑥,0)=0. (41) Its solution is 𝑢₂𝑥,𝑡,𝐶₁,𝐶₂=sech⁵()×0.00025𝑥−0.000187547𝐶₁𝑡−0.000187547𝐶₂𝑡+𝐶²₁(−0.000187547𝑡)cosh(0.00025𝑥)+−0.0000625156𝐶₁𝑡−0.0000625156𝐶₂𝑡+𝐶²₁(−0.0000625156𝑡)+2.71051×10⁻²⁰𝐶₁𝑡sinh(0.00025𝑥)+2.71051×10⁻²⁰𝐶²₁𝑡sinh(0.00025𝑥)+2.71051×10⁻²⁰𝐶₂𝑡sinh(0.00025𝑥)+3.12656×10⁻⁸𝐶²₁𝑡²sinh(0.00025𝑥)+3.12656×10⁻⁸𝐶²₁𝑡²sinh(0.00075𝑥). (42) The third order approximate solution using OHAM is given by 𝑢𝑥,𝑡,𝐶₁,𝐶₂=𝑢₀(𝑥,𝑡)+𝑢₁𝑥,𝑡,𝐶₁+𝑢₂𝑥,𝑡,𝐶₁,𝐶₂+𝑢₃𝑥,𝑡,𝐶₁,𝐶₂,𝐶₃, (43) where 𝑢₃(𝑥,𝑡,𝐶₁,𝐶₂,𝐶₃) is obtained in same lines as for first problem.

For the calculations of the constants 𝐶₁, 𝐶₂, and 𝐶₃ using the collocation method we have computed that 𝐶₁=−5.928318703338053×10⁻⁷,𝐶₂𝐶=−465.9630543691778,₃=1.8651679832921486. (44)

The third order OHAM solution yields very encouraging results after being compared with Fourth order approximate solution by ADM [5].

Table 5 shows a comparison between OHAM solution and ADM solution for 𝛼=0.001, 𝛽=0.001, and 𝛿=1. Table 6 compares between OHAM solution and ADM solution for 𝛼=1, 𝛽=1, and 𝛿=2. Table 7 shows the reliability of OHAM for larger domain.

Table 5 

Comparison of absolute errors obtained by OHAM and ADM [5] for 𝛼=0.001, 𝛽=0.001, and 𝛿=1.

Table 6 

Comparison of absolute errors obtained by OHAM and ADM [5] for 𝛼=1, 𝛽=1, and 𝛿=2.

Table 7 

Absolute errors of OHAM for 𝑥=2 and 𝑡∈[0.1,1].

4. Conclusion

We successfully applied OHAM for solution of Burger’s-Huxley and Burger’s-Fisher equations. The method is simple in applicability and is fast converging to the exact solution. The results obtained by OHAM are very consistent in comparison with ADM.