Research Article  Open Access
Muhammad Aslam Noor, Syed Tauseef MohyudDin, "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
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 [1–46] 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 [1–51] 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 [42–44, 51] was employed for finding solution of linear and nonlinear boundary value problems. He [12–18] 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 [1–5, 12–19, 21, 23, 31–44, 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 MohyudDin [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 [42–44] 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 sixthorder Boussinesq, thirdorder dispersive, and fourthorder 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: where is a linear operator, a nonlinear operator, and is the inhomogeneous term. According to variational iteration method [1–5, 13–19, 21, 23, 31–41, 46, 48], we can construct a correct functional as follows: where is a Lagrange multiplier [13–18], which can be identified optimally via variational iteration method. The subscripts denote the th approximation, is considered as a restricted variation, that is, . 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 [13–18]. In this method, it is required first to determine the Lagrange multiplier optimally. The successive approximation of the solution will be readily obtained upon using the determined Lagrange multiplier and any selective function. Consequently, the solution is given by 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 [42–44] of the form where is the highestorder derivative which is assumed to be invertible, is a linear differential operator of order lesser than , represents the nonlinear terms, and is the source term. Applying the inverse operator to both sides of (3.1) and using the given conditions, we obtain where the function represents the terms arising from integrating the source term and by using the given conditions. Adomian’s decomposition method [42–44] defines the solution by the series where the components are usually determined recurrently by using the relation The nonlinear operator can be decomposed into an infinite series of polynomials given by where are the socalled Adomian’s polynomials that can be generated for various classes of nonlinearities according to the specific algorithm developed in [42–44] which yields
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 where is a linear operator, is a nonlinear operator, and is the forcing term.
According to variational iteration method [1–5, 13–19, 21, 23, 31–41, 46, 48], we can construct a correct functional as follows: where is a Lagrange multiplier [13–18], which can be identified optimally via variational iteration method. The subscripts denote the th approximation, is considered as a restricted variation, that is, (4.2) is called as a correct functional. We define the solution by the series and the nonlinear term where are the socalled Adomian’s polynomials and can be generated for all types of nonlinearities according to the algorithm developed, in [42–44] which yields the following: Hence, we obtain the following iterative scheme for finding the approximate solution 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 . The Pade approximants to a function are given by [22, 28, 34, 43, 44, 47] where is polynomial of degree at most and is a polynomial of degree at most . The formal power series determine the coefficients of by the equation. Since we can clearly multiply the numerator and denominator by a constant and leave unchanged, we imposed the normalization condition Finally, we require that have non common factors. If we write the coefficient of as Then by (5.3) and (5.4), we may multiply (5.5) by , which linearizes the coefficient equations. We can write out (5.5) in more details as To solve these equations, we start with (5.5), which is a set of linear equations for all the unknown 's. Once the 's are known, then (5.6) gives an explicit formula for the unknown '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: 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, thirdorder dispersive, and fourthorder parabolic singular partial differential equations. The powerful Pade approximants are applied in case of boundarylayer 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 thirdorder boundary layer problem which appears mostly in the mathematical modeling of physical phenomena in fluid mechanics [43, 45]: with boundary conditions The correct functional is given as Making the correct functional stationary, the Lagrange multipliers can be identified as consequently, we have where Applying the modified variational iteration method, we have where and are the socalled Adomian’s polynomials and can be generated for all types of nonlinearities according to the algorithm defined in [42–44]. Consequently the following approximants are made: The series solution is given as
Example 6.2 (see [34, 44]). Consider the following nonlinear differential equation which governs the unsteady flow of gas through a porous medium: With the following typical boundary conditions imposed by the physical properties [34, 44], The correct functional is given as Making the correct functional stationary, using as the Lagrange multiplier, we get the following iterative formula: where Applying the modified variational iteration method, we have where are the socalled Adomian’s polynomials and can be generated for all types of nonlinearities according to the algorithm defined in [42–44]. First few Adomian’s polynomials are as under: Consequently, the following approximants are obtained: The series solution is given as Now, we investigate the mathematical behavior of the solution in order to determine the initial slope 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 . It is wellknown that Pade approximants will converge on the entire real axis [20, 22, 28, 34, 43, 44, 47], if 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 , the diagonals approximant vanishes if the coefficient of 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 is a decreasing function, hence . 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 to be and using [3/3] Pade approximants we find where Using (6.17)–(6.19) gives the values of the initial slope listed in Table 1. The formulas (6.17) and (6.19) suggest that the initial slope depends mainly on the parameter , where . Table 3 shows that the initial slope increases with the increase of . The mathematical structure of was successfully enhanced by using the Pade approximants. Table 4 indicates the values of [34, 44] and by using the [2/2] and [3/3] approximants for specific value of .






Example 6.3 (see [32, 51]). Consider the following singularly perturbed sixthorder Boussinesq equation: taking , the model equation is given as with initial conditions where and are arbitrary constants. The exact solution of the problem is given as [32, 51] The correct functional is given as where are Adomian’s polynomials for nonlinear operator and can be generated for all types of nonlinearities according to the algorithm developed in [42–44, 51] which yields the following: Making the correct functional stationary, the Lagrange multiplier can be identified as , consequently, The following approximants are obtained: The series solution is given as
Example 6.4 (see [32, 51]). Consider the following singularly perturbed sixthorder Boussinesq equation: with initial conditions The exact solution of the problem is given as Applying the modified variational iteration method, we obtain Making the correct functional stationary, the Lagrange multiplier can be identified as , consequently where are Adomian’s polynomials for nonlinear operator and can be generated for all types of nonlinearities according to the algorithm developed in [42–44, 51]. Consequently, the following approximants are obtained: The series solution is obtained as
Example 6.5. Consider the following linear thirdorder dispersive KdV equation: with initial condition The correct functional is given as Making the correct functional stationary, the Lagrange multiplier can be identified as consequently, Applying the modified variational iteration method, Consequently, the following approximants are obtained: The series solution is given by and in a closed form by If we change the initial condition as than the following closedform solution will be obtained:
Example 6.6. Consider the following linear thirdorder dispersive KdV equation in a twodimensional space: with initial condition The correct functional is given as Making the correct functional stationary, the Lagrange multiplier can be identified as consequently, Applying the modified variational iteration method, Consequently, the following approximants are obtained: The series solution is given by and in a closed form by If we change the initial condition as than the following closedform solution will be obtained:
Example 6.7 ([42]). Consider the following singular fourthorder parabolic partial differential equation in two space variables: with initial conditions and the boundary conditions The correct functional is given as where is considered as a restricted variation. Making the above functional stationary, the Lagrange multiplier can be identified as consequently, Applying the modified variational iteration method, we have Consequently, the following approximants are obtained as: The solution is given by
Remark 6.8. It is worth mentioning that Ghorbani and SaberiNadjafi [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, thirdorder dispersive and fourthorder 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.
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
Copyright © 2008 Muhammad Aslam Noor and Syed Tauseef MohyudDin. 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.