Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2008 / Article

Research Article | Open Access

Volume 2008 |Article ID 917407 | https://doi.org/10.1155/2008/917407

Muhammad Aslam Noor, Syed Tauseef Mohyud-Din, "Solution of Singular and Nonsingular Initial and Boundary Value Problems by Modified Variational Iteration Method", Mathematical Problems in Engineering, vol. 2008, Article ID 917407, 23 pages, 2008. https://doi.org/10.1155/2008/917407

Solution of Singular and Nonsingular Initial and Boundary Value Problems by Modified Variational Iteration Method

Academic Editor: Angelo Luongo
Received19 May 2008
Accepted05 Jun 2008
Published17 Aug 2008

Abstract

We apply the modified variational iteration method (MVIM) for solving the singular and nonsingular initial and boundary value problems in this paper. The proposed modification is made by introducing Adomian's polynomials in the correct functional. The suggested algorithm is quite efficient and is practically well suited for use in such problems. The proposed iterative scheme finds the solution without any discretization, linearization, perturbation, or restrictive assumptions. Several examples are given to verify the efficiency and reliability of the suggested algorithm.

1. Introduction

Many problems in applied sciences can be modeled by singular and nonsingular boundary value problems. The application of these problems involve astrophysics, experimental and mathematical physics, nuclear charge in heavy atoms, thermal behavior of a spherical cloud of gas, thermodynamics, population models, chemical kinetics, and fluid mechanics, see [146] and the references therein. Several techniques including decomposition, variational iteration, finite difference, polynomial spline, and homotopy perturbation have been developed for solving such problems, see [151] and the references therein. Most of these methods have their inbuilt deficiencies coupled with the major drawback of huge computational work. These facts have motivated to develop other methods for solving these problems. Adomian’s decomposition method [4244, 51] was employed for finding solution of linear and nonlinear boundary value problems. He [1218] developed the variational iteration method (VIM) for solving linear, nonlinear, initial, and boundary value problems. It is worth mentioning that the origin of variational iteration method can be traced back to Inokuti et al. [19]. In these methods, the solution is given in an infinite series usually converging to an accurate solution, see [15, 1219, 21, 23, 3144, 46, 48, 51] and the references therein. In this paper, we apply the modified variational iteration method (MVIM), which is obtained by the elegant coupling of variational iteration method and the Adomian’s polynomials for solving singular and nonsingular initial and boundary value problems. This idea has been used by Abbasbandy [1, 2] implicitly, and by Noor and Mohyud-Din [36, 38, 40] for the solution of nonlinear boundary value problems. The basic motivation of this paper is to apply this modified variational iteration method (MVIM) for finding the solution of singular and nonsingular initial and boundary value problems. It is shown that the MVIM provides the solution in a rapid convergent series with easily computable components. We write the correct functional for the boundary value problem and calculate the Lagrange multiplier optimally. The Adomian’s polynomials are introduced in the correct functional and evaluated by using the specific algorithm [4244] and the references therein. Finally, the approximants are calculated by employing the Lagrange multipliers and the Adomian’s polynomial scheme simultaneously. The use of Lagrange multiplier reduces the successive application of the integral operator and minimizes the computational work. Moreover, the selection of the initial value is done by exploiting the concept of modified decomposition method. In the present study, we apply this technique to solve boundary layer problem, unsteady flow of gas, singularly perturbed sixth-order Boussinesq, third-order dispersive, and fourth-order parabolic equations. To make the work more concise and to get a better understanding of the solution behavior, in case of boundary layer problem and the unsteady flow of gas, we replace the series solutions by the powerful Pade approximants [22, 28, 34, 43, 44, 47]. The use of Pade approximants shows real promise in solving boundary value problems in an infinite domain. The proposed MVIM solves effectively, easily, and accurately a large class of linear, nonlinear, partial, deterministic, or stochastic differential equations with approximate solutions which converge very rapidly to accurate solutions. Our results can be viewed as important and significant improvement of the previously known results.

2. Variational Iteration Method

To illustrate the basic concept of the technique, we consider the following general differential equation: 𝐿𝑢+𝑁𝑢=𝑔(𝑥),(2.1)where 𝐿 is a linear operator, 𝑁 a nonlinear operator, and 𝑔(𝑥) is the inhomogeneous term. According to variational iteration method [15, 1319, 21, 23, 3141, 46, 48], we can construct a correct functional as follows: 𝑢𝑛+1(𝑥)=𝑢𝑛(𝑥)+𝑥0𝜆𝑠𝐿𝑢𝑛(𝑠)+𝑁𝑢𝑛(𝑠)𝑔(𝑠)𝑑𝑠,(2.2)where 𝜆(𝑠) is a Lagrange multiplier [1318], which can be identified optimally via variational iteration method. The subscripts 𝑛 denote the 𝑛th approximation, 𝑢𝑛 is considered as a restricted variation, that is, 𝛿𝑢𝑛=0. Relational (2.2) is called as a correct functional. The solution of the linear problems can be solved in a single iteration step due to the exact identification of the Lagrange multiplier. The principles of variational iteration method and its applicability for various kinds of differential equations are given in [1318]. In this method, it is required first to determine the Lagrange multiplier 𝜆 optimally. The successive approximation 𝑢𝑛+1,𝑛0 of the solution 𝑢 will be readily obtained upon using the determined Lagrange multiplier and any selective function. Consequently, the solution is given by 𝑢=lim𝑛𝑢𝑛. For the convergence criteria and error estimates of variational iteration method, see Ramos [41].

3. Adomian’s Decomposition Method

To convey an idea of the technique, we consider the differential equation [4244] of the form 𝐿𝑢+𝑅𝑢+𝑁𝑢=𝑔,(3.1) where 𝐿 is the highest-order derivative which is assumed to be invertible, 𝑅 is a linear differential operator of order lesser than 𝐿, 𝑁𝑢 represents the nonlinear terms, and 𝑔 is the source term. Applying the inverse operator 𝐿1 to both sides of (3.1) and using the given conditions, we obtain 𝑢=𝑓𝐿1(𝑅𝑢)𝐿1(𝑁𝑢),(3.2)where the function 𝑓 represents the terms arising from integrating the source term 𝑔 and by using the given conditions. Adomian’s decomposition method [4244] defines the solution by the series 𝑢(𝑥)=𝑛=0𝑢𝑛(𝑥),(3.3)where the components 𝑢𝑛(𝑥) are usually determined recurrently by using the relation 𝑢0𝑢=𝑓,𝑘+1=𝐿1𝑅𝑢𝑘𝐿1𝑁𝑢𝑘,𝑘0.(3.4) The nonlinear operator 𝑁(𝑢) can be decomposed into an infinite series of polynomials given by 𝑁(𝑢)=𝑛=0𝐴𝑛,(3.5)where 𝐴𝑛 are the so-called Adomian’s polynomials that can be generated for various classes of nonlinearities according to the specific algorithm developed in [4244] which yields 𝐴𝑛=1𝑑𝑛!𝑛𝑑𝜆𝑛𝑁(𝑛𝑖=0𝜆𝑖𝑢𝑖)𝜆=0,𝑛=0,1,2,.(3.6)

4. Modified Variational Iteration Method (MVIM)

To illustrate the basic concept of the variational decomposition method, we consider the following general differential (4.1), we have 𝐿𝑢+𝑁𝑢=𝑔(𝑥),(4.1)where 𝐿 is a linear operator, 𝑁 is a nonlinear operator, and 𝑔(𝑥) is the forcing term.

According to variational iteration method [15, 1319, 21, 23, 3141, 46, 48], we can construct a correct functional as follows: 𝑢𝑛+1(𝑥)=𝑢𝑛(𝑥)+𝑥0𝜆𝐿𝑢𝑛(𝑠)+𝑁𝑢𝑛(𝑠)𝑔(𝑠)𝑑𝑠,(4.2)where 𝜆 is a Lagrange multiplier [1318], which can be identified optimally via variational iteration method. The subscripts 𝑛 denote the 𝑛th approximation, 𝑢𝑛 is considered as a restricted variation, that is, 𝛿𝑢𝑛=0 (4.2) is called as a correct functional. We define the solution 𝑢(𝑥) by the series 𝑢(𝑥)=𝑖=0𝑢𝑖(𝑥),(4.3)and the nonlinear term 𝑁(𝑢)=𝑛=0𝐴𝑛(𝑢0,𝑢1,𝑢2,,𝑢𝑖),(4.4)where 𝐴𝑛 are the so-called Adomian’s polynomials and can be generated for all types of nonlinearities according to the algorithm developed, in [4244] which yields the following: 𝐴𝑛=1𝑑𝑛!𝑛𝑑𝜆𝑛𝑁𝑢𝜆.(4.5)Hence, we obtain the following iterative scheme for finding the approximate solution 𝑢𝑛+1(𝑥)=𝑢𝑛(𝑥)+𝑡0𝜆(𝐿𝑢𝑛(𝑥)+𝑛=0𝐴𝑛𝑔(𝑥))𝑑𝑥,(4.6)which is called the modified variational iteration method (MVIM) and is formulated by the elegant coupling of variational iteration method and the Adomian’s polynomials.

5. Pade Approximants

A Pade approximant is the ratio of two polynomials constructed from the coefficients of the Taylor series expansion of a function 𝑢(𝑥). The [𝐿/𝑀] Pade approximants to a function 𝑦(𝑥) are given by [22, 28, 34, 43, 44, 47] 𝐿𝑀=𝑃𝐿(𝑥)𝑄𝑀(𝑥),(5.1)where 𝑃𝐿(𝑥) is polynomial of degree at most 𝐿 and 𝑄𝑀(𝑥) is a polynomial of degree at most 𝑀. The formal power series 𝑦(𝑥)=𝑖=1𝑎𝑖𝑥𝑖,𝑃𝑦(𝑥)𝐿(𝑥)𝑄𝑀𝑥(𝑥)=𝑂𝐿+𝑀+1,(5.2) determine the coefficients of 𝑃𝐿(𝑥)and𝑄𝑀(𝑥) by the equation. Since we can clearly multiply the numerator and denominator by a constant and leave [𝐿/𝑀] unchanged, we imposed the normalization condition 𝑄𝑀(0)=1.0.(5.3) Finally, we require that 𝑃𝐿(𝑥)and𝑄𝑀(𝑥) have non common factors. If we write the coefficient of 𝑃𝐿(𝑥)and𝑄𝑀(𝑥) as 𝑃𝐿(𝑥)=𝑝0+𝑝1𝑥+𝑝2𝑥2++𝑝𝐿𝑥𝐿,𝑄𝑀(𝑥)=𝑞0+𝑞1𝑥+𝑞2𝑥2++𝑞𝑀𝑥𝑀.(5.4)Then by (5.3) and (5.4), we may multiply (5.5) by 𝑄𝑀(𝑥), which linearizes the coefficient equations. We can write out (5.5) in more details as 𝑎𝐿+1+𝑎𝐿𝑞1++𝑎𝐿𝑀𝑞𝑀𝑞=0,𝐿+2+𝑞𝐿+1𝑞1++𝑎𝐿𝑀+2𝑞𝑀𝑎=0,𝐿+𝑀+𝑎𝐿+𝑀1𝑞1++𝑎𝐿𝑞𝑀=0,(5.5)𝑞To solve these equations, we start with (5.5), which is a set of linear equations for all the unknown 𝑞's. Once the 𝑝's are known, then (5.6) gives an explicit formula for the unknown 𝐿𝑀=𝑎det𝐿𝑀+1𝑎𝐿𝑀+2𝑎𝐿+1𝑎𝐿𝑎𝐿+1𝑎𝐿+𝑀𝐿𝑗=𝑀𝑎𝑗𝑀𝑥𝑗𝐿𝑗=𝑀1𝑎𝑗𝑀+1𝑥𝑗𝐿𝑗=0𝑎𝑗𝑥𝑗𝑎det𝐿𝑀+1𝑎𝐿𝑀+2𝑎𝐿+1𝑎𝐿𝑎𝐿+1𝑎𝐿+𝑀𝑥𝑀𝑥𝑀11.(5.7)'s, which complete the solution. If (5.5) and (5.6) are nonsingular, then we can solve them directly and obtain (5.7) [22], where (5.7) holds, and if the lower index on a sum exceeds the upper, the sum is replaced by zero: 𝑓(𝑥)+(𝑘1)𝑓(𝑥)𝑓𝑓(𝑥)2𝑛(𝑥)2=0,𝑛>0,(6.1)To obtain diagonal Pade approximants of different order such as [2/2], [4/4], or [6/6], we can use the symbolic calculus software Maple.

6. Numerical Applications

In this section, we apply the modified variational iteration method (MVIM) for solving the singular and nonsingular boundary value problems. We write the correct functional for the boundary value problem and carefully select the initial value because the approximants are heavily dependant on the initial value. The Adomian’s polynomials are introduced in the correct functional for the nonlinear terms. The results are very encouraging indicating the reliability and efficiency of the proposed method. We apply the MVIM for solving the boundary layer problem; unsteady flow of gas through a porous medium; Boussinesq equations, third-order dispersive, and fourth-order parabolic singular partial differential equations. The powerful Pade approximants are applied in case of boundary-layer problem and unsteady flow in order to make the work more concise and for better understanding of the solution behavior.

Example 6.1 (see [43]). Consider the following nonlinear third-order boundary layer problem which appears mostly in the mathematical modeling of physical phenomena in fluid mechanics [43, 45]: 𝑓(0)=0,𝑓(0)=1,𝑓()=0,𝑘>0.(6.2)with boundary conditions 𝑓𝑛+1(𝑥)=𝑓𝑛(𝑥)+𝑥0𝜆(𝑠)(𝑓𝑛𝑓(𝑠)+(𝑘1)𝑛𝑓(𝑥)𝑛𝑓(𝑠)2𝑛𝑛(𝑠)2)𝑑𝑠=0,𝑘>0.(6.3) The correct functional is given as 𝜆(𝑠)=(1/2!)(𝑠𝑥)2,Making the correct functional stationary, the Lagrange multipliers can be identified as 𝑓𝑛+1(𝑥)=𝑓𝑛(𝑥)𝑥012!𝑠𝑥2(𝑓𝑛𝑓(𝑠)+(𝑘1)𝑛𝑓(𝑠)𝑛𝑓(𝑥)2𝑛𝑛(𝑠)2)𝑑𝑠=0,𝑘>0,(6.4) consequently, we have 𝑓(0)=𝛼<0.where 𝑓𝑛+1(𝑥)=𝑓𝑛(𝑥)+𝑥012!𝑠𝑥2(𝑓𝑛(𝑠)+(𝑘1)𝑛=0𝐴𝑛2𝑛𝑛=0𝐵𝑛)𝑑𝑠=0,(6.5) Applying the modified variational iteration method, we have 𝐴𝑛where 𝐵𝑛 and 𝑓0𝑓(𝑥)=𝑥,11(𝑥)=𝑥+2𝛼𝑥2+13𝑥3,𝑓21(𝑥)=𝑥+2𝛼𝑥2+13𝑥3+124𝛼(3𝑛+1)𝑥4+130𝑛(𝑛+1)𝑥5,𝑓31(𝑥)=𝑥+2𝛼𝑥2+13𝑥3+124𝛼(3𝑛+1)𝑥4+130𝑛(𝑛+1)𝑥5+1𝛼1202(3𝑛+1)𝑥5+1𝛼72019𝑛2𝑥+18𝑛+36+1𝑛3152𝑛2𝑥+2𝑛+17,𝑓41(𝑥)=𝑥+2𝛼𝑥2+13𝑥3+124𝛼(3𝑛+1)𝑥4+130𝑛(𝑛+1)𝑥5+1𝛼1202(3𝑛+1)𝑥5+1𝛼72019𝑛2𝑥+18𝑛+36+1𝑛3152𝑛2𝑥+2𝑛+17,1𝛼5040227𝑛2𝑥+42𝑛+117+1𝛼40320167𝑛3+297𝑛2𝑥+161𝑛+158+1𝑛2268013𝑛3+38𝑛2𝑥+23𝑛+69,(6.6) are the so-called Adomian’s polynomials and can be generated for all types of nonlinearities according to the algorithm defined in [4244]. Consequently the following approximants are made: 𝑓(𝑥)=𝑥+𝛼𝑥22+𝑛𝑥33+181𝑛𝛼+𝛼𝑥244+1𝑛302+140𝑛𝛼2+1𝛼1202+1𝑛𝑥305+19𝑛72021𝛼+1240𝛼+𝑥40𝑛𝛼6+1120𝑛𝛼2+12315𝑛+𝑛3153+11𝛼50402+3𝑛5602𝛼2+2𝑛3152𝑥7+11𝛼403203+33𝑛448023𝛼+𝛼44803𝑛2+2315760𝑛𝛼+2688𝛼+167𝑛4032031𝛼+𝛼9603𝑛𝑥8+13780𝑛+527𝑛3628803𝛼2+19𝑛113403+709362880𝑛𝛼2+23𝑛80642𝛼2+23𝑛226802+13𝑛226804+43𝛼1209602𝑥9+.(6.7)The series solution is given as 𝑦(𝑥)+2𝑥𝑦1𝛼𝑦(𝑥)=0,0<𝛼<1.(6.8)

Example 6.2 (see [34, 44]). Consider the following nonlinear differential equation which governs the unsteady flow of gas through a porous medium: 𝑦(0)=1,lim𝑥𝑦(𝑥)=0.(6.9)With the following typical boundary conditions imposed by the physical properties [34, 44], 𝑦𝑛+1(𝑥)=𝑦𝑛(𝑥)+𝑥0𝑦𝜆(𝑠)(𝑠)+2𝑥𝑦1𝛼𝑦(𝑠)𝑑𝑠,0<𝛼<1.(6.10)The correct functional is given as 𝜆=𝑥𝑠,Making the correct functional stationary, using 𝑦𝑛+1(𝑥)=𝑦𝑛(𝑥)+𝑥0𝑦(𝑠𝑥)(𝑠)+2𝑥𝑦1𝛼𝑦(𝑠)𝑑𝑠,0<𝛼<1,(6.11) as the Lagrange multiplier, we get the following iterative formula: 𝐴=𝑦(0).(6.12)where 𝑦𝑛+1(𝑥)=𝑦𝑛(𝑥)+𝑥0(𝑠𝑥)(𝑦(𝑠)+2𝑥𝑛=0𝐴𝑛)𝑑𝑠,0<𝛼<1,(6.13)Applying the modified variational iteration method, we have 𝐴𝑛where 𝐴0=1𝛼𝑦01/2𝑦0,𝐴1=1𝛼𝑦01/2𝑦1+𝛼21𝛼𝑦03/2𝑦0𝑦1,𝐴2=1𝛼𝑦01/2𝑦2+𝛼21𝛼𝑦03/2𝑦1𝑦1+𝛼21𝛼𝑦03/2𝑦0𝑦2+38𝛼21𝛼𝑦05/2𝑦0𝑦21,𝐴3=1𝛼𝑦01/2𝑦3+𝛼21𝛼𝑦03/2𝑦2𝑦1+𝛼21𝛼𝑦03/2𝑦1𝑦2+𝛼21𝛼𝑦03/2𝑦0𝑦3+38𝛼21𝛼𝑦05/2𝑦1𝑦21+34𝛼21𝛼𝑦05/2𝑦0𝑦1𝑦2+5𝛼1631𝛼𝑦07/2𝑦0𝑦31,(6.14) are the so-called Adomian’s polynomials and can be generated for all types of nonlinearities according to the algorithm defined in [4244]. First few Adomian’s polynomials are as under: 𝑦0𝑦(𝑥)=1,1𝑦(𝑥)=1+𝐴𝑥,2𝐴(𝑥)=1+𝐴𝑥3𝑥1𝛼3,𝑦3𝐴(𝑥)=1+𝐴𝑥3𝑥1𝛼3𝛼𝐴212(1𝛼)3/2𝑥4+𝐴𝑥10(1𝛼)5,𝑦4𝐴(𝑥)=1+𝐴𝑥3𝑥1𝛼3𝛼𝐴212(1𝛼)3/2𝑥4+𝐴𝑥10(1𝛼)53𝛼2𝐴380(1𝛼)5/2𝑥5+𝛼𝐴215(1𝛼)2𝑥6𝑥+𝑂7,𝑦5𝐴(𝑥)=1+𝐴𝑥3𝑥1𝛼3𝛼𝐴212(1𝛼)3/2𝑥4+𝐴𝑥10(1𝛼)53𝛼2𝐴380(1𝛼)5/2𝑥5+𝛼𝐴215(1𝛼)2𝑥6𝛼3𝐴448(1𝛼)7/2𝑥6𝑥+𝑂7,(6.15)Consequently, the following approximants are obtained: 𝐴𝑦(𝑥)=1+𝐴𝑥3𝑥1𝛼3𝛼𝐴212(1𝛼)3/2𝑥4+𝐴10(1𝛼)3𝛼2𝐴380(1𝛼)5/2𝑥5+𝛼𝐴215(1𝛼)2𝛼3𝐴448(1𝛼)7/2𝑥6𝑥+𝑂7.(6.16) The series solution is given as 𝑦(𝑥)Now, we investigate the mathematical behavior of the solution 𝑦(0). in order to determine the initial slope 𝑦(𝑥) This goal can be achieved by forming diagonal Pade approximants [34, 44, 47] which have the advantage of manipulating the polynomial approximation into a rational function to gain more information about 𝑦(𝑥). It is well-known that Pade approximants will converge on the entire real axis [20, 22, 28, 34, 43, 44, 47], if 𝑦()=0 is free of singularities on the real axis. It is of interest to note that Pade approximants give results with no greater error bounds than approximation by polynomials. More importantly, the diagonal approximant is the most accurate approximant; therefore we will construct only the diagonal approximants in the following discussions. Using the boundary condition [𝑀/𝑀], the diagonals approximant 𝑥 vanishes if the coefficient of 𝑦(𝑥) with the highest power in the numerator vanishes. The computational work can be performed by using the mathematical software MAPLE. The physical behavior indicates that 𝑦(0)<0 is a decreasing function, hence 𝐴. Using this fact, and following [20, 34, 44], complex roots and nonphysical positive roots should be excluded. Based on this, the [2/2] Pade approximant produced the slope 𝐴=2(1𝛼)1/43𝛼,(6.17) to be 𝐴=(4674𝛼+8664)1𝛼144𝛾57𝛼,(6.18)and using [3/3] Pade approximants we find 𝛾=5(1𝛼)1309𝛼22280𝛼+1216.(6.19)where 𝐴=𝑦(0)Using (6.17)–(6.19) gives the values of the initial slope 𝐴=𝑦(0) listed in Table 1. The formulas (6.17) and (6.19) suggest that the initial slope 𝛼 depends mainly on the parameter 0<𝛼<1, where 𝐴=𝑦(0). Table 3 shows that the initial slope 𝛼 increases with the increase of 𝑦(𝑥). The mathematical structure of 𝑦(𝑥) was successfully enhanced by using the Pade approximants. Table 4 indicates the values of 𝛼=0.5 [34, 44] and by using the [2/2] and [3/3] approximants for specific value of 𝛼=𝑓(0).


𝛼 = 𝑓 ( 0 ) [2/2][3/3][4/4][5/5][6/6]

0.2−0.3872983347−0.3821533832−0.3819153845−0.3819148088−0.3819121854
1/3−0.5773502692−0.5615999244−0.5614066588−0.5614481405−0.561441934
0.4−0.6451506398−0.6397000575−0.6389732578−0.6389892681−0.6389734794
0.6−0.8407967591−0.8393603021−0.8396060478−0.8395875381−0.8396056769
0.8−1.007983207−1.007796981−1.007646828−1.007646828−1.007792100


𝛼 𝐴 = 𝑦 ( 0 )

4−2.483954032
10−4.026385103
100−12.84334315
1000−40.65538218
5000−104.8420672


𝐵 [ 2 / 2 ] = 𝑦 ( 0 ) 𝐵 [ 3 / 3 ] = 𝑦 ( 0 ) 𝑦 ( 𝑥 )

0.1−3.556558821−1.957208953
0.2−2.441894334−1.786475516
0.3−1.928338405−1.478270843
0.4−1.606856838−1.231801809
0.5−1.373178096−1.025529704
0.6−1.185519607−0.8400346085
0.7−1.021411309−0.6612047893
0.8−0.8633400217−0.4776697286
0.9−0.6844600642−0.2772628386


𝑦 𝑦 [ 2 / 2 ] kidder 𝑦 [ 3 / 3 ] 𝑥 𝑖

0.10.88165882830.86330606410.8979167028
0.20.76630767810.73012622610.7985228199
0.30.65653799950.60330541400.7041129703
0.40.55440240320.48488987170.6165037901
0.50.46136502950.37616038690.5370533796
0.60.37831093150.27773116280.4665625669
0.70.30559765460.18968433710.4062426033
0.80.24313254730.11171051650.3560801699
0.90.19046236810.043236732360.3179966614
1.00.15876898260.016467508470.2900255005


𝑡 𝑗 𝐸 1 4
0.010.020.040.10.20.5

−12.80886 𝐸 1 2 1.79667 𝐸 1 0 1.15235 𝐸 8 2.83355 𝐸 6 1.83899 𝐸 4 4.74681 𝐸 1 4
−0.86.27276 𝐸 1 2 4.01362 𝐸 1 0 2.57471 𝐸 8 6.33178 𝐸 6 4.10454 𝐸 3 1.04489 𝐸 1 4
−0.66.08402 𝐸 1 2 3.90188 𝐸 1 0 2.25663 𝐸 8 6.18024 𝐸 6 4.02299 𝐸 3 1.03093 𝐸 1 4
−0.41.16573 𝐸 1 3 7.41129 𝐸 1 1 4.82756 𝐸 8 1.23843 𝐸 6 8.53800 𝐸 4 2.46302 𝐸 1 4
−0.25.53446 𝐸 1 2 3.53395 𝐸 1 0 2.25663 𝐸 8 5.47485 𝐸 6 3.47264 𝐸 4 8.35783 𝐸 1 4
08.63198 𝐸 1 2 5.53357 𝐸 1 0 2.54174 𝐸 8 8.65197 𝐸 6 5.54893 𝐸 3 1.37353 𝐸 1 4
0.25.56222 𝐸 1 2 3.55044 𝐸 1 0 2.27779 𝐸 8 5.60362 𝐸 6 3.63600 𝐸 4 9.29612 𝐸 1 4
0.41.14353 𝐸 1 3 7.14928 𝐸 1 1 4.49107 𝐸 8 1.03370 𝐸 7 5.93842 𝐸 5 9.61260 𝐸 1 4
0.66.06182 𝐸 1 2 3.87551 𝐸 1 0 2.47218 𝐸 8 5.97562 𝐸 6 3.76275 𝐸 4 8.79002 𝐸 1 4
0.86.23945 𝐸 1 2 3.99519 𝐸 1 0 2.55127 𝐸 8 6.18881 𝐸 6 3.92220 𝐸 4 9.36404 𝐸 1 4
12.79776 𝐸 1 2 1.78946 𝐸 1 0 1.14307 𝐸 8 2.77684 𝐸 6 1.76607 𝐸 4 4.28986 𝑥 𝑖


𝑡 𝑗 𝐸 1 6
0.010.020.040.10.20.5

−17.77156 𝐸 1 4 1.36557 𝐸 1 3 8.57869 𝐸 1 0 2.09264 𝐸 8 1.33823 𝐸 6 3.25944 𝐸 1 6
−0.81.11022 𝐸 1 5 1.99840 𝐸 1 3 1.12688 𝐸 1 1 2.73880 𝐸 9 1.74288 𝐸 7 4.14094 𝐸 1 6
−0.62.22045 𝐸 1 4 1.09912 𝐸 1 3 7.28861 𝐸 1 0 1.78030 𝐸 8 1.14025 𝐸 6 2.79028 𝐸 1 6
−0.41.11022 𝐸 1 4 2.32037 𝐸 1 2 1.50302 𝐸 1 0 3.67002 𝐸 8 2.34944 𝐸 6 5.74091 𝐸 1 6
−0.26.66134 𝐸 1 4 3.23075 𝐸 1 2 2.04747 𝐸 1 0 4.99918 𝐸 9 3.19983 𝐸 6 7.81509 𝐸 1 6
04.44089 𝐸 1 4 3.49720 𝐸 1 2 2.24365 𝐸 1 0 5.47741 𝐸 8 3.50559 𝐸 6 8.55935 𝐸 1 6
0.25.55112 𝐸 1 4 3.19744 𝐸 1 2 2.04714 𝐸 1 0 4.99820 𝐸 8 3.19858 𝐸 6 7.80749 𝐸 1 6
0.43.33067 𝐸 1 4 2.32037 𝐸 1 2 1.50324 𝐸 1 0 3.66815 𝐸 8 2.34706 𝐸 6 5.72641 𝐸 1 6
0.63.33067 𝐸 1 4 1.12133 𝐸 1 2 7.28528 𝐸 1 0 1.77772 𝐸 8 1.13695 𝐸 6 2.77022 𝐸 1 6
0.83.33067 𝐸 1 5 1.99840 𝐸 1 3 1.13132 𝐸 1 1 2.76944 𝐸 9 1.78208 𝐸 7 4.41936 𝐸 1 6
17.77156 𝐸 1 4 1.38778 𝐸 1 3 8.58313 𝐸 1 0 2.09593 𝐸 8 1.34244 𝐸 6 3.28504 𝑢 𝑡 𝑡 = 𝑢 𝑥 𝑥 + 𝑝 ( 𝑢 ) 𝑥 𝑥 + 𝛼 𝑢 𝑥 𝑥 𝑥 𝑥 + 𝛽 𝑢 𝑥 𝑥 𝑥 𝑥 𝑥 𝑥 , ( 6 . 2 0 )

Example 6.3 (see [32, 51]). Consider the following singularly perturbed sixth-order Boussinesq equation: 𝛼=1,𝛽=0,and𝑝(𝑢)=3𝑢2taking 𝑢𝑡𝑡=𝑢𝑥𝑥𝑢+32𝑥𝑥+𝑢𝑥𝑥𝑥𝑥,(6.21), the model equation is given as 𝑢(𝑥,0)=2𝑎𝑘2𝑒𝑘𝑥1+𝑎𝑒𝑘𝑥2,𝑢𝑡(𝑥,0)=2𝑎𝑘31+𝑘21𝑎𝑒𝑘𝑥𝑒𝑘𝑥1+𝑎𝑒𝑘𝑥3,(6.22)with initial conditions 𝑎where 𝑘 and 𝑢(𝑥,𝑡) are arbitrary constants. The exact solution 𝑢(𝑥,𝑡)=2𝑎𝑘2exp(𝑘𝑥+𝑘1+𝑘2𝑡)1+𝑎exp𝑘𝑥+𝑘1+𝑘2𝑡2.(6.23) of the problem is given as [32, 51] 𝑢𝑛+1(𝑥,𝑡)=2𝑎𝑘2𝑒𝑘𝑥1+𝑎𝑒𝑘𝑥2+2𝑎𝑘31+𝑘21𝑎𝑒𝑘𝑥𝑒𝑘𝑥1+𝑎𝑒𝑘𝑥3𝑡+𝑡0𝜕𝜆(2𝑢𝑛𝜕𝑡2𝑢𝑛𝑥𝑥𝑢𝑛𝑥𝑥𝑥𝑥3𝑛=0𝐵𝑛)𝑑𝑡,(6.24)The correct functional is given as 𝐵𝑛where 𝐹(𝑢)=𝑢2(𝑥) are Adomian’s polynomials for nonlinear operator 𝐵0=𝑢20𝑥𝑥,𝐵1=2𝑢0𝑢1𝑥𝑥+4𝑢0𝑥𝑢1𝑥+2𝑢0𝑥𝑥𝑢1,𝐵2=2𝑢0𝑢2𝑥𝑥+4𝑢0𝑥𝑢2𝑥+2𝑢0𝑥𝑥𝑢2+2𝑢1𝑥𝑢1𝑥𝑥𝑢+21𝑥2𝐵3=2𝑢0𝑢3+2𝑢1𝑢2𝑥𝑥,(6.25) and can be generated for all types of nonlinearities according to the algorithm developed in [4244, 51] which yields the following: 𝜆=𝑥𝑡 Making the correct functional stationary, the Lagrange multiplier can be identified as 𝑢𝑛+1(𝑥,𝑡)=2𝑎𝑘2𝑒𝑘𝑥1+𝑎𝑒𝑘𝑥2+2𝑎𝑘31+𝑘21𝑎𝑒𝑘𝑥𝑒𝑘𝑥1+𝑎𝑒𝑘𝑥3𝑡+𝑡0(𝜕𝑥𝑡2𝑢𝑛𝜕𝑡2𝑢𝑛𝑥𝑥𝑢𝑛𝑥𝑥𝑥𝑥3𝑛=0𝐵𝑛)𝑑𝑡.(6.26), consequently, 𝑢0(𝑥,𝑡)=2𝑒𝑥1+𝑒𝑥2,𝑢1(𝑥,𝑡)=2𝑒𝑥1+𝑒𝑥2+2𝑎𝑘31+𝑘21𝑎𝑒𝑘𝑥𝑒𝑘𝑥1+𝑎𝑒𝑘𝑥3𝑡+2𝑒𝑥14𝑒𝑥+𝑒2𝑥1+𝑒𝑥4𝑡2,𝑢2(𝑥,𝑡)=2𝑒𝑥1+𝑒𝑥2+2𝑎𝑘31+𝑘21𝑎𝑒𝑘𝑥𝑒𝑘𝑥1+𝑎𝑒𝑘𝑥3𝑡+2𝑒𝑥14𝑒𝑥+𝑒2𝑥1+𝑒𝑥4𝑡222𝑒𝑥1+𝑒𝑥110𝑒𝑥+𝑒2𝑥31+𝑒𝑥5𝑡3+𝑒𝑥14𝑒𝑥+𝑒2𝑥144𝑒𝑥+78𝑒2𝑥44𝑒3𝑥+𝑒4𝑥31+𝑒𝑥8𝑡4,𝑢3(𝑥,𝑡)=2𝑒𝑥1+𝑒𝑥2+2𝑎𝑘31+𝑘21𝑎𝑒𝑘𝑥𝑒𝑘𝑥1+𝑎𝑒𝑘𝑥3𝑡+2𝑒𝑥14𝑒𝑥+𝑒2𝑥1+𝑒𝑥4𝑡222𝑒𝑥1+𝑒𝑥110𝑒𝑥+𝑒2𝑥31+𝑒𝑥5𝑡3+𝑒𝑥14𝑒𝑥+𝑒2𝑥144𝑒𝑥+78𝑒2𝑥44𝑒3𝑥+𝑒4𝑥31+𝑒𝑥8𝑡42𝑒𝑥1+𝑒𝑥156𝑒𝑥+246𝑒2𝑥56𝑒3𝑥+𝑒4𝑥151+𝑒𝑥7𝑡5+𝑒𝑥1452𝑒𝑥+19149𝑒2𝑥207936𝑒3𝑥+807378𝑒4𝑥1256568𝑒5𝑥451+𝑒𝑥12𝑡6+𝑒𝑥807378𝑒6𝑥207936𝑒7𝑥+19149𝑒8𝑥452𝑒9𝑥+𝑒10𝑥451+𝑒𝑥12𝑡6,(6.27)The following approximants are obtained: 𝑢(𝑥,𝑡)=2𝑒𝑥1+𝑒𝑥2+2𝑎𝑘31+𝑘21𝑎𝑒𝑘𝑥𝑒𝑘𝑥1+𝑎𝑒𝑘𝑥3𝑡+2𝑒𝑥14𝑒𝑥+𝑒2𝑥1+𝑒𝑥4𝑡222𝑒𝑥1+𝑒𝑥110𝑒𝑥+𝑒2𝑥31+𝑒𝑥5𝑡3+𝑒𝑥14𝑒𝑥+𝑒2𝑥144𝑒𝑥+78𝑒2𝑥44𝑒3𝑥+𝑒4𝑥31+𝑒𝑥8𝑡4+8𝑒2𝑥110𝑒𝑥+20𝑒2𝑥10𝑒3𝑥+𝑒4𝑥1+𝑒𝑥8𝑡42𝑒𝑥1+𝑒𝑥156𝑒𝑥+246𝑒2𝑥56𝑒3𝑥+𝑒4𝑥151+𝑒𝑥7𝑡5+𝑒𝑥1452𝑒𝑥+19149𝑒2𝑥207936𝑒3𝑥+807378𝑒4𝑥1256568𝑒5𝑥451+𝑒𝑥12𝑡6+𝑒𝑥807378𝑒6𝑥207936𝑒7𝑥+19149𝑒8𝑥452𝑒9𝑥+𝑒10𝑥451+𝑒𝑥12𝑡6+.(6.28)The series solution is given as 𝑢𝑡𝑡=𝑢𝑥𝑥+𝑢2𝑥𝑥𝑢𝑥𝑥𝑥𝑥+12𝑢𝑥𝑥𝑥𝑥𝑥𝑥,(6.29)

Example 6.4 (see [32, 51]). Consider the following singularly perturbed sixth-order Boussinesq equation: 𝑢(𝑥,0)=105169sec4𝑥26,𝑢𝑡(𝑥,0)=210194/13sec4𝑥/26tanh𝑥/262197.(6.30)with initial conditions 𝑢(𝑥,𝑡)=105169sec4[126(𝑥97169𝑡)].(6.31)The exact solution of the problem is given as 𝑢𝑛+1(𝑥,𝑡)=105169sec4𝑥+26210194/13sec4𝑥/26tanh𝑥/26𝑡+2197𝑡0𝜕𝜆(2𝑢𝑛𝜕𝑡2𝑢𝑛𝑥𝑥+𝑢𝑛𝑥𝑥𝑥𝑥12𝑢𝑛𝑥𝑥𝑥𝑥𝑥𝑥+𝑛=0𝑏𝑛)𝑑𝑡.(6.32)Applying the modified variational iteration method, we obtain 𝜆=𝑥𝑡Making the correct functional stationary, the Lagrange multiplier can be identified as 𝑢𝑛+1(𝑥,𝑡)=105169sec4𝑥+26210194/13sec4𝑥/26tanh𝑥/26𝑡+2197𝑡0(𝜕𝑥𝑡2𝑢𝑛𝜕𝑡2𝑢𝑛𝑥𝑥+𝑢𝑛𝑥𝑥𝑥𝑥12𝑢𝑛𝑥𝑥𝑥𝑥𝑥𝑥+𝑛=0𝐵𝑛)𝑑𝑡,(6.33), consequently 𝐵𝑛where 𝐹(𝑢)=𝑢2(𝑥) are Adomian’s polynomials for nonlinear operator 𝑢0(𝑥,𝑡)=105169sec4𝑥,𝑢261(𝑥,𝑡)=105169sec4𝑥26105194/13sec6𝑥/26sinh2𝑥/13𝑡2197105371293291+194cosh2𝑥13sec6𝑥𝑡262,𝑢2(𝑥,𝑡)=105169sec4𝑥26105194/13sec6𝑥/26sinh2𝑥/13𝑡2197105371293291+194cosh2𝑥13sec6𝑥𝑡262+395sec7𝑥/265220676614410816𝑥2522sinh2616642522sinh3𝑥𝑡263+334200sec5𝑥226+354247cosh𝑥13sec5𝑥22647164cosh2𝑥13sec5𝑥𝑡264+3201cosh332𝑥13sec5𝑥426388cosh2𝑥13sec5𝑥𝑡264+,(6.34) and can be generated for all types of nonlinearities according to the algorithm developed in [4244, 51]. Consequently, the following approximants are obtained: 𝑢(𝑥,𝑡)=105169sec4𝑥26105194/13sec6𝑥/26sinh2𝑥/13𝑡2197105371293291+194cosh2𝑥13sec6𝑥𝑡262+395sec7(𝑥/26)5220676614410816𝑥2522sinh2616642522sinh3𝑥𝑡263+334200sec5𝑥226+354247cosh𝑥13sec5𝑥22647164cosh2𝑥13sec5𝑥𝑡264+3201cosh332𝑥13sec5𝑥426388cosh2𝑥13sec5𝑥𝑡264+.(6.35)The series solution is obtained as 𝑢𝑡+2𝑢𝑥+𝑢𝑥𝑥𝑥=0,𝑡>0,(6.36)

Example 6.5. Consider the following linear third-order dispersive KdV equation: 𝑢(𝑥,0)=sin𝑥.(6.37)with initial condition 𝑢𝑛+1(𝑥,𝑡)=𝑢𝑛(𝑥,𝑡)+𝑥0𝜆(𝑠)𝜕𝑢𝑛𝜕𝑡+2𝜕𝑢𝑛+𝜕𝜕𝑥3𝑢𝑛𝜕𝑥3𝑑𝑠.(6.38)The correct functional is given as 𝜆=1,Making the correct functional stationary, the Lagrange multiplier can be identified as 𝑢𝑛+1(𝑥,𝑡)=𝑢𝑛(𝑥,𝑡)𝑥0𝜕𝑢𝑛𝜕𝑡+2𝜕𝑢𝑛+𝜕𝜕𝑥3𝑢𝑛𝜕𝑥3𝑑𝑠.(6.39) consequently, 𝑢𝑛+1(𝑥,𝑡)=𝑢𝑛(𝑥,𝑡)𝑥0(𝜕𝑢𝑛𝜕𝑡+2𝑛=0𝜕𝑢𝑛+𝜕𝑥𝑛=0𝜕3𝑢𝑛𝜕𝑥3)𝑑𝑠.(6.40)Applying the modified variational iteration method, 𝑢0𝑢(𝑥,𝑡)=sin𝑥,1𝑢(𝑥,𝑡)=sin𝑥𝑡cos𝑥,2𝑡(𝑥,𝑡)=sin𝑥12𝑢2!𝑡cos𝑥,3𝑡(𝑥,𝑡)=sin𝑥12𝑡2!cos𝑥𝑡3,3!(6.41)Consequently, the following approximants are obtained: 𝑡𝑢(𝑥,𝑡)=sin𝑥12+𝑡2!4𝑡4!cos𝑥𝑡3+𝑡3!55!+,(6.42)The series solution is given by 𝑢(𝑥,𝑡)=sin(𝑥𝑡).(6.43)and in a closed form by 𝑢(𝑥,0)=cos𝑥,If we change the initial condition as 𝑢(𝑥,𝑡)=cos(𝑥𝑡).(6.44) than the following closed-form solution will be obtained: 𝑢𝑡+𝑢𝑥𝑥𝑥+𝑢𝑦𝑦𝑦=0,𝑡>0,(6.45)

Example 6.6. Consider the following linear third-order dispersive KdV equation in a two-dimensional space: 𝑢(𝑥,𝑦,0)=cos(𝑥+𝑦).(6.46)with initial condition 𝑢𝑛+1(𝑥,𝑦,𝑡)=𝑢𝑛(𝑥,𝑦,𝑡)+𝑥0𝜆(𝑠)𝜕𝑢𝑛𝜕𝜕𝑡+23𝑢𝑛𝜕𝑥3+𝜕3𝑢𝑛𝜕𝑦3𝑑𝑠.(6.47)The correct functional is given as 𝜆=1,Making the correct functional stationary, the Lagrange multiplier can be identified as 𝑢𝑛+1(𝑥,𝑦,𝑡)=𝑢𝑛(𝑥,𝑦,𝑡)𝑥0𝜕𝑢𝑛𝜕𝜕𝑡+23𝑢𝑛𝜕𝑥3+𝜕3𝑢𝑛𝜕𝑦3𝑑𝑠.(6.48) consequently, 𝑢𝑛+1(𝑥,𝑦,𝑡)=𝑢𝑛(𝑥,𝑦,𝑡)𝑥0(𝜕𝑢𝑛𝜕𝑡+2𝑛=0𝜕3𝑢𝑛𝜕𝑥3+𝑛=0𝜕3𝑢𝑛𝜕𝑦3)𝑑𝑠.(6.49)Applying the modified variational iteration method, 𝑢0𝑢(𝑥,𝑦,𝑡)=cos(𝑥+𝑦),1𝑢(𝑥,𝑦,𝑡)=cos(𝑥+𝑦)2𝑡sin(𝑥+𝑦),2(𝑥,𝑦,𝑡)=cos(𝑥+𝑦)12𝑡2𝑢2!2𝑡sin(𝑥+𝑦),3(𝑥,𝑦,𝑡)=cos(𝑥+𝑦)12𝑡22!sin(𝑥+𝑦)2𝑡2𝑡3,3!(6.50)Consequently, the following approximants are obtained: 𝑢(𝑥,𝑦,𝑡)=cos(𝑥+𝑦)1(2𝑡)22!+sin(𝑥+𝑦)(2𝑡)(2𝑡)33!+,(6.51)The series solution is given by 𝑢(𝑥,𝑦,𝑡)=cos(𝑥+𝑦+2𝑡).(6.52)and in a closed form by 𝑢(𝑥,𝑦,0)=sin(𝑥+𝑦),If we change the initial condition as 𝑢(𝑥,𝑦,𝑡)=sin(𝑥+𝑦+2𝑡).(6.53) than the following closed-form solution will be obtained: 𝜕2𝑢𝜕𝑡21+2𝑥2+𝑥4𝜕6!4𝑢𝜕𝑥41+2𝑦2+𝑦4𝜕6!4𝑢𝜕𝑦4=0,(6.54)

Example 6.7 ([42]). Consider the following singular fourth-order parabolic partial differential equation in two space variables: 𝑢(𝑥,𝑦,0)=0,𝜕𝑢𝑥𝜕𝑡(𝑥,𝑦,0)=2+6+𝑦6!66!,(6.55)with initial conditions 𝑢12=,𝑦,𝑡2+(0.5)6+𝑦6!616!sin𝑡,𝑢(1,𝑦,𝑡)=2++𝑦6!6𝜕6!sin𝑡,2𝑢𝜕𝑥212=,𝑦,𝑡(0.5)4𝜕24sin𝑡,2𝑢𝜕𝑥21(1,𝑦,𝑡)=𝜕24sin𝑡,2𝑢𝜕𝑦21𝑥,2=,𝑡(0.5)4𝜕24sin𝑡,2𝑢𝜕𝑦21(𝑥,1,𝑡)=24sin𝑡.(6.56)and the boundary conditions 𝑢𝑛+1(𝑥,𝑡)=𝑢0(𝑥,𝑡)+𝑡0𝜕𝜆(𝜉)2𝑢𝑛𝜕𝑡21+2𝑥2+𝑥4𝜕6!4𝑢𝑛𝜕𝑥4+21𝑦2+𝑦4𝜕6!4𝑢𝑛𝜕𝑦4𝑑𝜉,(6.57)The correct functional is given as 𝑢𝑛where 𝜆=𝜉𝑡, is considered as a restricted variation. Making the above functional stationary, the Lagrange multiplier can be identified as 𝑢𝑛+1(𝑥,𝑡)=𝑢0(𝑥,𝑡)+𝑡0𝜕(𝜉𝑡)2𝑢𝑛𝜕𝑡21+2𝑥2+𝑥4𝜕6!4𝑢𝑛𝜕𝑥4+21𝑦2+𝑦4𝜕6!4𝑢𝑛𝜕𝑦4𝑑𝜉.(6.58) consequently, 𝑢𝑛+1(𝑥,𝑡)=𝑢0(𝑥,𝑡)+𝑡0𝜕(𝜉𝑡)(2𝑢𝑛𝜕𝑡21+2𝑥2+𝑥46!𝑛=0𝜕4𝑢𝑛𝜕𝑥41+(2𝑦2+𝑦46!𝑛=0𝜕4𝑢𝑛𝜕𝑦4))𝑑𝜉.(6.59)Applying the modified variational iteration method, we have 𝑢0𝑥(𝑥,𝑡)=2+6+𝑦6!6𝑡𝑢6!1𝑥(𝑥,𝑡)=2+6+𝑦6!6𝑡6!𝑡3𝑢3!2𝑥(𝑥,𝑡)=2+6+𝑦6!6𝑡6!𝑡3+𝑡3!5,𝑢5!3𝑥(𝑥,𝑡)=2+6+𝑦6!6𝑡6!𝑡3+𝑡3!5𝑡5!7,𝑢7!4𝑥(𝑥,𝑡)=2+6+𝑦6!6𝑡6!𝑡3+𝑡3!5𝑡5!7+𝑡7!9,9!(6.60)Consequently, the following approximants are obtained as: 𝑥𝑢(𝑥,𝑦,𝑡)=2+6+𝑦6!6𝑡6!𝑡3+𝑡3!5𝑡5!7+𝑡7!9=𝑥9!+2+6+𝑦6!66!sin𝑡.(6.61)The solution is given by 𝛼=2.483954032

Remark 6.8. It is worth mentioning that Ghorbani and Saberi-Nadjafi [49] and Ghorbani [50] introduced He polynomials which are compatible to Adomian’s polynomials, are easier to calculate, and hence make the solution procedure simpler.

7. Conclusion

In this paper, we applied the modified variational iteration method (MVIM) for solving singular and nonsingular initial and boundary value problems. The proposed technique is applied on boundary layer problem, unsteady flow of gas, Boussinesq equations, third-order dispersive and fourth-order parabolic partial differential equations. The Pade approximants were employed in order to make the work more concise and for better understanding of the solution behavior. It may be concluded that the proposed frame work is very powerful and efficient in finding the analytical solutions for singular and nonsingular boundary value problems. The method gives more realistic series solutions that converge very rapidly in physical problems.

917407.fig.001
Figure 1: (𝑡=1).
917407.fig.002
Figure 2
917407.fig.003
Figure 3
917407.fig.004
Figure 4
917407.fig.005
Figure 5: (𝑡=1).
917407.fig.006
Figure 6: ().

Acknowledgments

The authors are highly grateful to the referee for his/her very constructive comments. The authors also would like to thank Dr. S. M. Junaid Zaidi, Rector CIIT for providing the excellent research facilities and environment. The authors are grateful to Asif Waheed for drawing the figures and useful comments. This research is supported by HEC research Grant.

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Copyright © 2008 Muhammad Aslam Noor and Syed Tauseef Mohyud-Din. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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