Nonlinear Singular Two-point Boundary Value Research Articles

A highly accurate and computationally efficient technique for solving the electrohydrodynamic flow in a circular cylindrical conduit

The present work proposes a new, highly accurate and efficient computational technique for numerical solution of a strongly nonlinear singular two-point boundary value problem which governs electrohydrodynamic flow (EHF) of a fluid in an ion drag configuration in a circular cylindrical conduit. The nonlinearity confronted in the underlying problem is in the form of rational function and the governing equation has a singularity at the point z=0. The proposed method tackles the singularity behavior at the origin. Analysis of the proposed method is carried out. Numerical experiments were performed to demonstrate the applicability and robustness of the new method. The velocity field of EHF of a fluid in radial direction is computed. In order to justify the advantage of the method, the computed results are compared with the results obtained from some recently developed computational techniques in the literature. The effects of two relevant parameters, namely the strength of nonlinearity (σ) and the Hartmann electric number (H) on the velocity profile are investigated.

A Novel Neuroevolutionary Paradigm for Solving Strongly Nonlinear Singular Boundary Value Problems in Physiology

In the present research work, we designed a hybrid stochastic numerical solver to investigate nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions arising in various physical models. In this method, we hybridized Harris Hawks Optimizer with Interior Point Algorithm named HHO-IPA. We construct artificial neural networks (ANNs) model for the problem, and this model is tuned with the proposed scheme. This scheme overcomes the singular behavior of problems. The accuracy and applicability of the method are illustrated by finding absolute errors in the solution. The outcomes are compared with the results present in the literature to demonstrate the effectiveness and robustness of the scheme by considering four different nonlinear singular boundary value problems. Further, the convergence of the scheme is proved by performing computational complexity analysis. Moreover, the graphical overview of statistical analysis is added to our investigation to elaborate further on the scheme’s stability, accuracy, and consistency.

The series expansion and Chebyshev collocation method for nonlinear singular two-point boundary value problems

The solution of singular two-point boundary value problem is usually not sufficiently smooth at one or two endpoints of the interval, which leads to a great difficulty when the problem is solved numerically. In this paper, an algorithm is designed to recognize the singular behavior of the solution and then solve the equation efficiently. First, the singular problem is transformed to a Fredholm integral equation of the second kind via Green’s function. Second, the truncated fractional series of the solution about the singularity is formulated by using Picard iteration and implementing series expansion for the nonlinear function. Third, a suitable variable transformation is performed by using the known singular information of the solution such that the solution of the transformed equation is sufficiently smooth. Fourth, the Chebyshev collocation method is used to solve the deduced equation to obtain approximate solution with high precision. Fifth, the convergence analysis of the collocation method is conducted in weighted Sobolev spaces for linear singular equations. Sixth, numerical examples confirm the effectiveness of the algorithm. Finally, the Thomas–Fermi equation and the Emden–Fowler equation as some applications are accurately solved by the method.

An improved iterative technique for solving nonlinear doubly singular two-point boundary value problems

This paper presents a new iterative technique for solving nonlinear singular two-point boundary value problems with Neumann and Robin boundary conditions. The method is based on the homotopy perturbation method and the integral equation formalism in which a recursive scheme is established for the components of the approximate series solution. This method does not involve solution of a sequence of nonlinear algebraic or transcendental equations for the unknown coefficients as in some other iterative techniques developed for singular boundary value problems. The convergence result for the proposed method is established in the paper. The method is illustrated by four numerical examples, two of which have physical significance: The first problem is an application of the reaction-diffusion process in a porous spherical catalyst and the second problem arises in the study of steady-state oxygen-diffusion in a spherical cell with Michaelis-Menten uptake kinetics.

Optimal L ∞ estimates for Galerkin methods for nonlinear singular two-point boundary value problems

In this paper, we apply the symmetric Galerkin methods to the numerical solutions of a kind of singular linear two-point boundary value problems. We estimate the error in the maximum norm. For the sake of obtaining full superconvergence uniformly at all nodal points, we introduce local mesh refinements. Then we extend these results to a class of nonlinear problems. Finally, we present some numerical results which confirm our theoretical conclusions.

A new coupled reduced alternating group explicit method for nonlinear singular two-point boundary value problems on a variable mesh

In this paper, we discuss a new coupled reduced alternating group explicit (CRAGE) and Newton-CRAGE iteration methods to solve the nonlinear singular two-point boundary value problems u″ = f(r, u, u′), 0 < r < 1 subject to given natural boundary conditions u(0) = A1, u(1) = A2 where A1 and A2 are finite constants, along with a third-order numerical method on a geometric mesh. The proposed method is applicable to singular and nonsingular problems. We have discussed the convergence of the CRAGE iteration method in detail. The results obtained from the proposed CRAGE iteration method are compared with the results of the corresponding two-parameter alternating group explicit (TAGE) iteration methods to demonstrate computationally the efficiency of the proposed method.

Positive Solutions of Sturm-Liouville Problems for Second Order Singular and Impulsive Differential Equations

In this paper,we study the existence of multiple positive solutions of nonlinear singular two-point boundary value problems for second-order impulsive dierential equations.The proof is based on the theory of fixed point index in cones.

Solving Singular Boundary Value Problems by Optimal Homotopy Asymptotic Method

In this paper, optimal homotopy asymptotic method (OHAM) for the semianalytic solutions of nonlinear singular two-point boundary value problems has been applied to several problems. The solutions obtained by OHAM have been compared with the solutions of another method named as modified adomain decomposition (MADM). For testing the success of OHAM, both of the techniques have been analyzed against the exact solutions in all problems. It is proved by this paper that solutions of OHAM converge rapidly to the exact solution and show most effectiveness as compared to MADM.

Solving a Class of Singular Two-Point Boundary Value Problems Using New Modified Decomposition Method

We introduce an effective methodology for solving a class of linear as well as nonlinear singular two-point boundary value problems. This methodology is based on a modification of Adomian decomposition method (ADM) and a new two fold integral operator. We use all the boundary conditions to derive an integral equation before establishing the recursive scheme for the solution components of solution. Thus, we develop modified recursive scheme without any undetermined coefficients while computing the successive solution components. This modification also avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. However, most of earlier recursive schemes using ADM do require computation of undetermined coefficients. The approximate solution is obtained in the form of series with easily calculable components. Numerical examples are included to demonstrate the accuracy, applicability, and generality of the present technique. The results reveal that the method is very effective, straightforward, and simple.

Approximate Solutions of Singular Two-Point BVPs Using Legendre Operational Matrix of Differentiation

Exact and approximate analytical solutions of linear and nonlinear singular two-point boundary value problems (BVPs) are obtained for the first time by the Legendre operational matrix of differentiation. Different from other numerical techniques, shifted Legendre polynomials and their properties are employed for deriving a general procedure for forming this matrix. The accuracy of the technique is demonstrated through several linear and nonlinear test examples.

Variational Iteration Method for Nonlinear Singular Two-Point Boundary Value Problems Arising in Human Physiology

The variational iteration method is applied to solve a class of nonlinear singular boundary value problems that arise in physiology. The process of the method, which produces solutions in terms of convergent series, is explained. The Lagrange multipliers needed to construct the correctional functional are found in terms of the exponential integral and Whittaker functions. The method easily overcomes the obstacle of singularities. Examples will be presented to test the method and compare it to other existing methods in order to confirm fast convergence and significant accuracy.

A new superconvergent method for systems of nonlinear singular boundary value problems

A new superconvergent method based on a sextic spline is described and analysed for the solution of systems of nonlinear singular two-point boundary value problems (BVPs). It is well known that the optimal orders of convergence could not be achieved using standard formulation of a sextic spline for the solution of BVPs. Based on the method used in our earlier research papers [J. Rashidinia and M. Ghasemi, B-spline collocation for solution of two-point boundary value problems, J. Comput. Appl. Math. 235 (2011), pp. 2325–2342; J. Rashidinia, M. Ghasemi, and R. Jalilian, An o(h 6) numerical solution of general nonlinear fifth-order two point boundary value problems, Numer. Algorithms 55(4) (2010), pp. 403–428], we construct a new O(h 8) locally superconvergent method for the solution of general nonlinear two-point BVPs up to order 6. The error bounds and the convergence properties of the method have been proved theoretically. Then, the method is extended to solve the system of nonlinear two-point BVPs. Some test problems are given to demonstrate the applicability and the superconvergent properties of the proposed method numerically. It is shown that the method is very efficient and applicable for stiff BVPs too.

An approach for solving singular two point boundary value problems: analytical and numerical treatment

The numerical treatment of two point singular boundary value problems has always been a difficult and challenging task due to the singularity behaviour that occurs at a point. Various efficient numerical methods have been proposed to deal with such boundary value problems. We present a new efficient modification of the Adomian decomposition method for solving singular boundary value problems, both linear and nonlinear. Numerical examples illustrate the efficiency and accuracy of the proposed method. References G. Adomian. A review of the decomposition method and some recent results for nonlinear equation. Math. Comput. Modelling, 3, 1992, 17--43. G. Adomian. Solving frontier problems of physics: the decomposition method. Kluwer Academic Publishers, Boston, 1994. G. Adomian. Solution of the Thomas--Fermi equation. Appl. Math. Lett., 11(3), 1998, 131--133. D. Lesnic. A computational algebraic investigation of the decomposition method for time--dependent problems. Appl. Math. Comput., 119, 2001, 197--206. E. Babolian and J. Biazar. Solving the problem of biological species living together by Adomian decomposition method. Appl. Math. Comput., 129, 2002, 339--343. M. Benabidallah and Y. Cherruault. Application of the Adomian method for solving a class of boundary problems. Kybernetes, 33, 2004, 118--132. E. H. Aly, A. Ebaid and R. Rach. Advances in the Adomian decomposition method for solving two--point nonlinear boundary value problems with Neumann boundary conditions. Compu. Math. Applic., 63, 2012, 1056--1065. B. Jang. Two--point boundary value problems by the extended Adomian decomposition method. J. Comput. Appl. Math., 219, 2008, 253--263. A. M. Wazwaz. Partial differential equations and solitary waves theory. Springer, New York, 2009. M. Kumar and N. Singh. Modified Adomian decomposition method and computer implementation for solving singular boundary value problems arising in various physical problems. Comput. Chem. Eng., 34, 2010, 1750--1760. Y. Cherruault, G. Adomian, K. Abbaoui and R. Rach. Further remarks on convergence of decomposition method. Bio--Medical Comput., 38, 1995, 89--93. M. M. Hosseini and H. Nasabzadeh. On the convergence of Adomian decomposition method. Appl. Math. Comput., 182, 2006, 536--543. A. Ebaid. A new analytical and numerical treatment for singular two--point boundary value problems via the Adomian decomposition method. J. Comput. Appl. Math., 235, 2011, 1914--1924. J. Janus and J. Myjak. A generalized Emden--Fowler equation with a negative exponent. Nonlin. Analy., 23, 1994, 953--970. M. K. Kadalbajoo and V. K. Aggarwal. Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline. Appl. Math. Comput., 160, 2005, 851--863. A. S. V. Ravi Kanth, K. Aruna. Solution of singular two--point boundary value problems using differential transformation method. Phys. Lett. A, 372, 2008, 4671--4673. Sami Bataineh, M. S. M. Noorani and I. Hashim. Approximate solutions of singular two--point bvps by modified homotopy analysis method. Phys. Lett. A, 372, 2008, 4062--4066. A. M. Wazwaz. Adomian decomposition method for a reliable treatment of the Emden--Fowler equation. Appl. Math. Comput., 161, 2005, 543--560. M. Inc, M. Ergut, Y. Cherruault. A different approach for solving singular two-point boundary value problems. Kybernetes, 34, 2005, 934--940. C. Chun. A modified Adomian decomposition method for solving higher-order singular boundary value problems. Z. Naturforsch. A, 65, 2010, 1093--1100. A. M. Wazwaz. The modified decomposition method for analytic treatment of differential equations. Appl. Math. Comput., 173, 2006, 165--176. M. Cui and F. Geng. Solving singular two--point boundary value problem in reproducing kernel space. J. Comput. Appl. Math., 205, 2007, 6--15. S. M. El-Sayed. Integral methods for computing solutions of a class of singular two--point boundary value problems. Appl. Math. Comput., 130, 2002, 235--241. A. Ebaid. Exact solutions for a class of nonlinear singular two-point boundary value problems: The decomposition method. Z. Naturforsch. A, 65, 2010, 145--150.

Exact Solutions for a Class of Nonlinear Singular Two-Point Boundary Value Problems: The Decomposition Method

Many problems in applied mathematics and engineering are usually formulated as singular two-point boundary value problems. A well-known fact is that the exact solutions in closed form of such problems were not obtained in many cases. In this paper, the exact solutions for a class of nonlinear singular two-point boundary value problems are obtained to the first time by using Adomian decomposition method

Positive solutions of nonlinear singular two-point boundary value problems for second-order impulsive differential equations

In this paper, we study the positive solutions of nonlinear singular two-point boundary value problems for second-order impulsive differential equations. The existence of one or two positive solutions are established by using the fixed point index theorem in cone.

Cubic spline polynomial for non-linear singular two-point boundary value problems

We present a cubic spline polynomial for the solution of non-linear singular two-point boundary value problems. The quesilinearization technique is used to reduce the non-linear problem to a sequence of linear problems. The resulting sets of differential equations are modified at the singular point and are treated by using cubic spline for finding the numerical solution. The method is tested on three physical model problems from the literature.

An non-uniform mesh cubic spline TAGE method for non-linear singular two-point boundary value problems

We report a non-uniform mesh cubic spline method of accuracy for the solution of non-linear singular two-point boundary value problems. The application of two-parameter alternating group explicit (TAGE) and Newton–TAGE iteration methods which are suitable for use on parallel computers is discussed. The error analysis for TAGE iteration method is discussed in detail. The numerical results confirm the utility of proposed cubic spline TAGE iteration methods.

The Decomposition Method For Solving Of A Class Of Singular Two-Point Boundary Value Problems

In this paper, a class of nonlinear singular two-point boundary value problems are solved by using Adomian's decomposition methods. The approximate solution of this problem is calculated in the form of series with easily computable components. Finally, we give a nonlinear numerical example.