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The fractional reaction-diffusion equation has an important physical and theoretical meaning, but its analytical solution poses considerable problems. This paper develops an efficient numerical process, the local radial basis function generated by the finite difference (named LRBF-FD) method, for finding the approximation solution of the time-fractional fourth-order reaction-diffusion equation in the sense of the Riemann-Liouville derivative. The time fractional derivative is approximated using the second-order accurate formulation, while the spatial terms are discretized by means of the LRB-FFD method. The advantage of the local collocation method is to approximate the differential operators by a weighted sum of the values of the function on a local set of nodes (local support) by deriving the RBF expansion. Furthermore, the ill-conditioned matrix resulting from using the global collocation method is avoided. The unconditional stability property and convergence analysis of the time-discrete approach are thoroughly proven and verified numerically. Three numerical examples are presented confirming the theoretical formulation and effectiveness of the new approach.

This paper develops the solution of the two-dimensional time fractional evolution model using finite difference scheme derived from radial basis function (RBF-FD) method. In this discretization process, a finite difference formula is implemented to discrete the temporal variable, while the local RBF-FD formulation is utilized to approximate the spatial variable. The pattern of data distribution in the local support domain is assumed as having a fixed number of nodes. The local RBF-FD is based on the local support domain that leads to a sparsity system and also avoids the ill-conditioning problem caused by global collocation method. The stability and convergence of time-discrete approach in H 1 -norm are discussed by means of the energy method. Numerical results illustrate the proposed method and demonstrate that it provides accurate solutions on regular and irregular computational domains.

This paper introduces a neurodynamics optimization model to compute the solution of mathematical programming with equilibrium constraints (MPEC). A smoothing method based on NPC-function is used to obtain a relaxed optimization problem. The optimal solution of the global optimization problem is estimated using a new neurodynamic system, which, in finite time, is convergent with its equilibrium point. Compared to existing models, the proposed model has a simple structure, with low complexity. The new dynamical system is investigated theoretically, and it is proved that the steady state of the proposed neural network is asymptotic stable and global convergence to the optimal solution of MPEC. Numerical simulations of several examples of MPEC are presented, all of which confirm the agreement between the theoretical and numerical aspects of the problem and show the effectiveness of the proposed model. Moreover, an application to resource allocation problem shows that the new method is a simple, but efficient, and practical algorithm for the solution of real-world MPEC problems.

The cable equation is one useful description for modeling phenomena such as neuronal dynamics and electrophysiology. The time-fractional cable model (TFCM) generalizes the classical cable equation by considering the anomalous diffusion that occurs in the ionic motion present for example in the neuronal system. This paper proposes a novel meshless numerical procedure, the radial basis function-generated finite difference (RBF-FD), to approximate the TFCM involving two fractional temporal derivatives. The time discretization of the TFCM is performed based on the Grunwald–Letnikov expansion. The spatial derivatives are discretized using the RBF-FD. The pattern of data distribution in the support domain is assumed as having a fixed number of points. The RBF-FD is based on the local support domain that leads to a sparsity system and also tackles the ill-conditioning problem caused by global collocation method. The theoretical stability and the convergence analysis of the scheme are also discussed in detail. It is shown that the proposed method is efficient and that the numerical results confirm the theoretical formulation.

The nonlinear modified anomalous sub-diffusion model characterizes processes that become less anomalous as time progresses by including a second fractional time derivative acting on the term of diffusion. This paper introduces a radial basis function-generated finite difference (RBF-FD) method for solving the governing problem. The Grunwald–Letnikov formula with first-order accuracy is implemented to discretize the problem in the time direction, and the spatial variable is discretized using the local RBF-FD method. The convergence and stability of the time discretization scheme are deduced in an appropriate Sobolev space. The data distribution pattern within the support domain is considered to have a constant number of points. The numerical results on regular and irregular domains show the efficiency and high accuracy of the method and confirm the theoretical prediction.

In this paper, a projection-based recurrent neural network is proposed to solve convex quadratic bilevel programming problems (CQBPP). The Karush–Kuhn–Tucker optimal conditions (KKT) of the lower level problem are used to obtain identical one-level optimization problem. A projected dynamical system which its equilibrium point coincides with the global optimal solution of the corresponding optimization problem is presented. Compared to existing models, the proposed model has the least number of variables and a simple structure with low complexity. Analytically, it is demonstrated that the state vector of the suggested neural network model is stable in the sense of Lyapunov and globally convergent to an optimal solution of CQBPP in finite time. Some numerical examples, a supply chain model and an application deals with an environmental problem are discussed in order to confirm the efficiency of the theoretical results and the performance of the model.

Ill-conditioning is drawback of global radial basis functions (RBFs) method. Local RBFs method has overcome this difficulty and has become popular as an alternative methodology. In this paper, we analyze local RBFs method to solve singularly perturbed linear convection–diffusion problems. In some articles, it is investigated numerically but our aim is mainly to prove stability, consistency and convergence of local RBF method for boundary layer problems. It is known that the approximate error bound of Local RBFs method depends on two factors, Local density of knots, h, and the shape parameter, c. It is proven that this method is first order consistent. for verification, an unsteady Magnetohydrodynamic (MHD) free convective heat and mass transfer flow with Thermophoresis past a radiate inclined permeable plate in the presence of variable chemical reaction and temperature dependent viscosity is considered. Additionally some mathematical test problems are approximated. Numerical results reveal that the proposed method is accurate than upwind finite difference method. The optimal shape parameter strategy will also be utilized to approximate significant solutions.

The mathematical modeling in trade and finance issues is the key purpose in the computation of the value and considering option during preferences in contract. This paper investigates the pricing of double barrier options when the price change of the underlying is considered as a fractal transmission system. Due to the outstanding memory effect present in the fractional derivatives, approximating financial options with regards to their hereditary characteristics can be well interpreted and stated. Motivated by the reason mentioned, relatively reliable and also efficient numerical approaches have to be found while facing with fractional differential equations. The main objective of the current paper is to obtain the approximation solution of the time fractional Black–Scholes model of order $$0

In this study, we propose an efficient and accurate numerical technique that is called the rational Chebyshev collocation (RCC) method to solve the two dimensional flow of a viscous fluid in the vicinity of a stagnation point named Hiemenz flow. The Navier-Stokes equations governing the flow, are reduced to a third-order ordinary differential equation of a boundary value problem with a semi-infinite domain by using similarity transformation. The rational Chebyshev method reduces this nonlinear ordinary differential equation to a system of algebraic equations. This technique is a powerful type of the collocation methods for solving the boundary value problems over a semi-infinite interval without truncating it to a finite domain. We also present the comparison of this work with others and show that the present method is more accurate and efficient.

Ill-conditioning problem of Global Radial Basis Functions (GRBFs) is a fundamental limitation for approximation of differential equation using this method. Most recently, some researchers have applied Local Radial Basis Functions (LRBFs) to approximate singularly perturbed convention diffusion problems. In these kinds of problems, mostly appear boundary and (or) interior layers. Existence of these regions causes instability and oscillation solutions in numerical methods. To avoid these problems, upwind LRBFs are used. In other words, convective part and diffusion part of equations discretize through upwind LRBFs and central LRBFs method, respectively. Although this technique stabilizes the scheme but reduces the accuracy, by the way, the selection of upwind direction is more complicated. To overcome these problems, in this paper, we multiply an artificial diffusion to diffusion part and apply central scheme to discretize convective part. Since central method for these kinds of problems are unstable, adding artificial diffusion causes the central method is stabilized. This strategy significantly improve the accuracy. For verification, several numerical examples are considered and compared with other schemes. The comparisons show the superiority of the method to tradition approaches. Results of this study reveal that the proposed method can be successfully applied for singularly perturbed differential equations.

A dynamic model to solve equilibrium constrained optimization problem is proposed.The limit stationary point of the model approximates the solutions of the MPECs.Proposed model has asymptotic stability over its equilibrium point.Numerical simulations and its application in game theory is presented. In this paper, a feedback neural network model is proposed to compute the solution of the mathematical programs with equilibrium constraints (MPEC). The MPEC problem is altered into an identical one-level non-smooth optimization problem, then a sequential dynamic scheme that progressively approximates the non-smooth problem is presented. Besides asymptotic stability, it is proven that the limit equilibrium point of the suggested dynamic model is a solution for the original MPEC problem. Numerical simulation of various types of MPEC problems shows the significance of the results. Moreover, the scheme is applied to compute the StackelbergCournotNash equilibria.

In this paper, a new local meshless approach based on radial basis functions (RBFs) is presented to price the options under the Black–Scholes model. The global RBF approximations derived from the conventional global collocation method usually lead to ill-conditioned matrices. Employing the idea of local approximants of the finite difference (FD) method and combining it with the radial basis function (RBF) method can result in a local meshless approach such as RBF-FD. It removes the difficulty of ill-conditionness of the original method. The new proposed approach is unconditionally stable as it is shown by Von-Neumann stability analysis. It is fast and produces high accurate results as shown in numerical experiments. Moreover, we took into account the variation of shape parameter and analyzed numerically the behavior of the RBF-FD method.

The price variation of the correlated fractal transmission system is used to deduce the fractional Black–Scholes model that has an $$\alpha $$ -order time fractional derivative. The fractional Black–Scholes model is employed to price American or European call and put options on a stock paying on a non-dividend basis. Upon encountering fractional differential equations, the efficient and relatively reliable numerical schemes must be obtained for their solution due to fractional derivatives being non-local. The present paper is aimed at determining the numerical solution of the time fractional Black–Scholes model (TFBSM) with boundary conditions for a problem of European option pricing involved with the method of radial basis functions (RBFs), which is a truly meshfree scheme. The TFBSM is discretized in the temporal sense based on finite difference scheme of order $${\mathcal {O}}(\delta t^{2-\alpha })$$ for $$ 0< \alpha

A nonlinear wave phenomenon is one of the significant fields of scientific research, which a lot of researchers in the past have deliberated about mathematical models clarifying the treatment. The nonlinear Korteweg-de Vries-Benjamin-Bona-Mahony-Burgers (KdV-BBM-B) model plays an essential role in numerous subjects of engineering and science. This model is numerically approximated by means of the combination of the radial basis function (RBF) and the finite difference (RBF-FD), RBF-pseudospectral (RBF-PS) that are stated in this paper. These methods are meshless, and no obligation is required to linearize the nonlinear parts. The radial kernels are employed to convert the PDE into a system of ODEs. This ODE system is solved with the help of the higher-order ODE solver. In addition, it is indicated that the present numerical techniques are stable. To evaluate the validity and applicability of the aforesaid techniques, the error norms and three invariants I1, I2 and I3 are calculated and displayed both numerically and graphically. The obtained results are compared with previous schemes found in the literature.

Evolution equations containing fractional derivatives can offer efficient mathematical models for determination of anomalous diffusion and transport dynamics in multi-faceted systems that cannot be precisely modeled by using normal integer order equations. In recent times, researches have found out that lots of physical processes illustrate fractional order characteristics that alters with time or space. The continuum of order in the fractional calculus permits the order of the fractional operator be accounted for as a variable. In the current research work, radial basis functions (RBFs) approximation is utilized for solving fractional mobile-immobile advection-dispersion (TF-MIM-AD) model in a bounded domain which is applied for explaining solute transport in both porous and fractured media. In this approach, firstly, the discretization process of the aforesaid equation with of convergence order $$\mathcal {O}(\delta t^{})$$ in the t-direction is described via the finite difference scheme for $$ 0< \alpha

The time fractional Klein–Kramers model (TFKKM) is obtained by incorporating the subdiffusive mechanisms into the Klein–Kramers formalism. The TFKKM can efficiently express subdiffusion while an external force field is present in the phase space. The model describes the escape of a particle over a barrier and has a significant role in examining a variety of systems including slow (subdiffusion) dynamics. This paper describes a hybrid algorithm adopting the local radial basis functions based finite difference (LRBF–FD) for the numerical solution of the TFKKM. The time discretization is accomplished via the Grunwald-Letnikov formulation with second-order accuracy and the spatial derivatives are discretized by the LRBF–FD. The LRBF–FD is based on the local support domain that causes to a more sparse matrix and overcomes the ill-conditioning associated with the global collocation. The convergence and stability analysis of the time-discrete algorithm are deduced using the energy method. The feasibility and applicability of the LRBF–FD are demonstrated by using irregular domains. Numerical results are compared with analytical solution and with those obtained by other techniques to verify the accuracy and validity of the LRBF–FD.

In this paper, two-dimensional Schrodinger equations are solved by differential quadrature method. Key point in this method is the determination of the weight coefficients for approximation of spatial derivatives. Multiquadric (MQ) radial basis function is applied as test functions to compute these weight coefficients. Unlike traditional DQ methods, which were originally defined on meshes of node points, the RBFDQ method requires no mesh-connectivity information and allows straightforward implementation in an unstructured nodes. Moreover, the calculation of coefficients using MQ function includes a shape parameter c. A new variable shape parameter is introduced and its effect on the accuracy and stability of the method is studied. We perform an analysis for the dispersion error and different internal parameters of the algorithm are studied in order to examine the behavior of this error. Numerical examples show that MQDQ method can efficiently approximate problems in complexly shaped domains.

In this note, the solution of nonlinear integral equations was discussed using radial basis functions (RBFs) method. This method will represent the solution of nonlinear integral equation by interpolating the RBFs based on Legendre-Gauss-Lobatto (LGL) nodes and weights. Zeros of the shifted Legendre polynomials are used as the collocation points. The method is used to some examples to illustrate the accuracy and the implementation of the method.

In this paper, we develop a new local meshless approach based on radial basis functions (RBFs) to solve the Black---Scholes equation. The global RBF approximations derived from conventional global collocation method usually lead to ill-conditioned matrices. The new scheme employs the idea of the finite difference method to localize them. It removes the difficulty of ill-conditioning of the original method. The new proposed approach is unconditionally stable as it is shown by Von-Neumann stability analysis. As well as it is fast and it produces accurate results as shown in numerical experiments.

The fractal mobile/immobile model of the solute transport is based on the assumption that the waiting times in the immobile region follow a power-law, and this leads to the application of fractional time derivatives. The model covers a wide family of systems that include heat diffusion and ocean acoustic propagation. This paper develops an efficient computational technique, stemming from the radial basis function-generated finite difference (RBF-FD), to solve the fractal mobile-immobile transport model (FMTM). The time fractional derivative of the FMTM is discretized via the shifted Grunwald-Letnikov formula with second-order accuracy. On the other hand, the spatial derivative is approximated using the local RBF-FD method. The main benefit of the local collocation technique is that we only need to consider discretization points present in each of the sub-domains around the collocation point. The stability and convergence analysis of the proposed method are proven via the energy method in the L2 space. The numerical results for the FMTM on regular and irregular domains confirm the theoretical formulation and efficiency of the proposed scheme.

Stability, convergence and application of radial basis function finite difference (RBF-FD) scheme is studied for solving the reaction-diffusion equations (RDEs). We show that the explicit RBF-FD method is stable, and stability condition depends on the shape parameter of related radial basis function.The generalized multiquadric (GMQ) is applied as radial basis function and weight coefficients are explicitly presented for equispaced node distribution. Also, two methods are presented to compute the optimal shape parameter. The combination of these methods with the GMQ-FD method will produce two efficient algorithms for numerical solution of RDEs: the variable GMQ-FD (VGMQ-FD) and the constant GMQ-FD (CGMQ-FD). We test the scheme on traveling wave and compare its accuracy with the conventional finite difference method (FDM).

One of the most popular meshless methods is constructed by radial kernels as basis called radial basis function method. It has a unique feature which affects significantly on accuracy and stability of approximation: existence of a free parameter known as shape parameter that can be chosen constantly or variably. Several techniques for selecting a variable shape parameter have been presented in the older works. Our study focuses on investigating the deficiency of these techniques and we introduce two new alternative strategies called hybrid shape parameter and binary shape parameter strategies based on the advantages of older studies. The proposed approaches produce the more accurate results as shown in numerical results where they are compared with random shape parameter strategy for interpolating one-dimensional and two- dimensional functions as well as in approximating the solution of Poisson equation.

The current paper deals with elaborating a novel framework for solving a class of linear matrix differential equations. To this end, the operational matrices of integration and the product based on the shifted Bernoulli polynomials are presented and a general procedure for forming this matrices is given. The properties of this matrices are exploited to reduce the main problem to a linear matrix equation. Numerical experiments are reported to demonstrate the applicably and efficiency of the propounded technique.

Shape parameters play an important role in radial basis function RBF approximations. Therefore, the choice of them is an active field in numerical analysis research. In this paper, first we review the available strategies in the literature for selecting shape parameters. Then, we introduce an alternative approach called hybrid strategy for scaling the RBFs. This strategy is constructed based on the advantages of the older ones and discards their disadvantages. The efficiency of the new strategy is demonstrated by comparing the effects of different strategies on approximating the eigenvalues of ordinary and partial differential equations with different boundary conditions.

In this paper, a meshfree method based on radial basis functions (RBFs) is developed to approximate the eigenvalues of Stokes equations in primitive variables in a square domain. To avoid the inaccuracy near the boundaries, the collocation on boundary technique is applied. This approach leads to more accurate solutions in comparisons with finite element methods. To investigate the role of shape parameter in approximation, some discussion on shape parameter is presented.

This paper provides a robust convergence checking method for nonlinear differential equations of fractional order with consideration of homotopy perturbation technique. The differential operators are taken in the Caputo sense. Some theorems to prove the existence and uniqueness of the series solutions are presented. Results show that the proposed theoretical analysis is accurate. © 2011 Elsevier Inc.

The fractional Zakharov–Kuznetsov equations are increasingly used in modeling various kinds of weakly nonlinear ion acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field. This has led to a significant interest in the study of these equations. In this work, solitary pattern solutions of fractional Zakharov–Kuznetsov equations are investigated by means of the homotopy perturbation method with consideration of Jumarie’s derivatives. The effects of fractional derivatives for the systems under consideration are discussed. Numerical results and a comparison with exact solutions are presented.

In this paper, we consider a partial integro-differential equation (PIDE) problem with a free boundary, arising in an American option model when the stock price follows a diffusion process with jump components. We use a front-fixing transformation of the underlying asset variable to fix the free boundary conditions and approximate the integral term by the Laguerre polynomials. We use the Radial basis functions (RBF) method to achieve an implicit nonlinear system of first order equations and apply the Crank–Nicholson scheme. We apply the Predictor–Corrector method, to deal with the system of nonlinear equations. The proposed method is stable and the results are in agreement with those obtained by other numerical methods in literature.

This paper outlines a reliable strategy to use the homotopy perturbation method based on Jumarie’s derivative for solving fractional differential equations. In this framework, compact structures of fourth-order fractional diffusion-wave equations are considered as prototype examples. Moreover, convergence of the proposed approach for these types of equations is investigated. Results show that the response expressions are Mittag-Leffler stable.

The fractional advection-dispersion equation (FADE) known as its non-local dispersion, is used in groundwater hydrology and has been proven to be a reliable tool to model the transport of passive tracers carried by fluid flow in a porous media. In this paper, compact structures of FADE are investigated by means of the homotopy perturbation method with consideration of a promising scheme to calculate nonlinear terms. The problems are formulated in the Jumarie sense. Analytical and numerical results are presented.

In this study, the homotopy perturbation method is used to solve fractional flux advectiondispersion equation. The problem is formulated in the Caputo sense. The results reveal that the present method is very effective and convenient.

The homotopy perturbation method is applied to the generalized fourth-order fractional diffusion–wave equations. The problem is formulated in the Caputo sense. Moreover, a reliable scheme for calculating nonlinear operators is proposed. The results reveal that the present method is very effective and convenient.

In this paper, a family of high-order compact finite difference methods in combination preconditioned methods are used for solution of the Diffusion-Convection equation. We developed numerical methods by replacing the time and space derivatives by compact finite-difference approximations. The system of resulting nonlinear finite difference equations are solved by preconditioned Krylov subspace methods. Numerical results are given to verify the behavior of high-order compact approximations in combination preconditioned methods for stability, convergence. Also, the accuracy and efficiency of the proposed scheme are considered.

This paper presents meshless method using RBF collocation scheme for the coupled Schrodinger-KdV equations. Instead of traditional mesh oriented methods such as finite element method (FEM) or finite difference method (FDM), this method requires only a scattered set of nodes in the domain. For this scheme, error estimates and stability analysis are studied. L 2 and L ∞ error norms between the results and exact solution is used as a performance measure. Moreover the results of numerical experiments are presented, and are compared with the findings of Finite Element method, finite difference Crank–Nicolson (CN) scheme and analytical solution to confirm the good accuracy of the presented scheme.

In this paper, we propose a simple general form of high-order approximation of O (c 2 + ch 2 + h 4 ) to solve the two-dimensional parabolic equation αuxx + βu yy = F (x, y, t, u, ux, u y, ut ) ,w hereα and β are positive constants. We apply the compact form for solving diffusion–convection equation. The results of numerical experiments are presented and compared with analytical solutions to confirm the higher accuracy of the presented scheme. © 2010 Elsevier B.V. All rights reserved.

In this paper, the He's variational iteration method (VIM) based on a reliable modiflcation of Adomian algorithm has been used to obtain so- lutions of the nonlinear boundary value problems (BVP). Comparison of the result obtained by the present method with that obtained by Adomian method (A. M. Wazwaz, Found Phys. Lett. 13 (2000) 493 and G. L. Liu, Modern Math- ematical and Mechanics, (1995) 643 ) reveals that the present method is very efiective and convenient.

In this paper we have used the homotopy analysis method (HAM) to obtain solutions of multi-term linear and nonlinear diffusion–wave equations of fractional order. The fractional derivative is described in the Caputo sense. Some illustrative examples have been presented.

A numerical solution for solving non-linear Fredholm integral equations is presented. The method is based upon homotopy perturbation theory. The result reveal that the modified homotopy perturbation method (MHPM) is very effective and convenient.

In this paper, a novel learning strategy for radial basis function networks (RBFN) is proposed. By adjusting the parameters of the hidden layer, including the RBF centers and widths, the weights of the output layer are adapted by local optimization methods. A new local optimization algorithm based on a combination of the gradient and Newton methods is introduced. The efficiency of some local optimization methods to update the weights of RBFN is studied in solving systems of nonlinear integral equations.

In this paper, we present a modification to homotopy perturbation method for solving some non-linear Fredholm integral equations of the first kind. Solved problems reveal that the proposed method is very effective and simple and in some cases it gives the exact solution rather than the approximated one.

In this study, we use the spectral collocation method using Chebyshev polynomials for spatial derivatives and fourth order Runge–Kutta method for time integration to solve the generalized Burger’s–Huxley equation (GBHE). To reduce round-off error in spectral collocation (pseudospectral) method we use preconditioning. Firstly, theory of application of Chebyshev spectral collocation method with preconditioning (CSCMP) and domain decomposition on the generalized Burger’s–Huxley equation presented. This method yields a system of ordinary differential algebric equations (DAEs). Secondly, we use fourth order Runge–Kutta formula for the numerical integration of the system of DAEs. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.

In this study, we use the spectral collocation method using Chebyshev polynomials for spatial derivatives and fourth order Runge–Kutta method for time integration to solve the generalized Burger’s–Fisher equation (B–F). Firstly, theory of application of Chebyshev spectral collocation method (CSCM) and domain decomposition on the generalized Burger’s–Fisher equation is presented. This method yields a system of ordinary differential algebraic equations (DAEs). Secondly, we use fourth order Runge–Kutta formula for the numerical integration of the system of DAEs. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.

This paper is about the construction of a simple method to approximate the solution of system of linear Fredholm integral equations of the second kind based on Adomian’s decomposition method. Easy computations rather than successive integrations are used with simple algorithm. Some solved problems are given to show the efficiency of the method. The convergence of the method is also considered.

In this paper, exact and numerical solutions are obtained for the generalized Zakharov equation (GZE) by the well known variational iteration method (VIM). This method is based on Lagrange multipliers for identification of optimal values of parameters in a functional. Using this method creates a sequence which tends to the exact solution of the problem.

In this paper, He’s variational iteration method is employed successfully for solving parabolic partial differential equations with Dirichlet boundary conditions. In this method, the solution is calculated in the form of a convergent series with an easily computable component. This approach does not need linearization, weak nonlinearity assumptions or perturbation theory. The results reveal that the method is very effective and convenient.

In this study, we use the spectral collocation method with preconditioning to solve various nonlinear Schrödinger equations. To reduce round-off error in spectral collocation method we use preconditioning. We study the numerical accuracy of the method. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.

Homotopy perturbation method (HPM) is applied to construct a new family of Newton-like iterative methods for solving system of non-linear equations. Comparison of the result obtained by the present method with Newton–Raphson method reveals that the accuracy and fast convergence of the new method.

Homotopy perturbation method is applied to the numerical solution for solving eighth-order boundary value problems. Comparison of the result obtained by the present method with that obtained by modified decomposition method [M. Mesrovic, The modified decomposition method for eighth-order boundary value problems, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.11.015] reveals that the present method is very effective and convenient.

A numerical method based on modified homotopy perturbation method (HPM) is proposed for solving nonlinear algebric equations. It is shown that the proposed method has third-order convergence. To assess its validity and accuracy, the method is applied to solve several test problems.