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In the current work, a new reproducing kernel method (RKM) for solving nonlinear forced Duffing equations with integral boundary conditions is developed. The proposed collocation technique is based on the idea of RKM and the orthonormal Bernstein polynomials (OBPs) approximation together with the quasi-linearization method. In our method, contrary to the classical RKM, there is no need to use the Gram-Schmidt orthogonalization procedure and only a few nodes are used to obtain efficient numerical results. Three numerical examples are included to show the applicability and efficiency of the suggested method. Also, the obtained numerical results are compared with some results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, the properties of Chebyshev polynomials and the Gauss–Legendre quadrature rule are employed to construct a new operational matrix of distributed-order fractional derivative. This operational matrix is applied for solving some problems such as distributed-order fractional differential equations, distributed-order time-fractional diffusion equations and distributed-order time-fractional wave equations. Our approach easily reduces the solution of all these problems to the solution of some set of algebraic equations. We also discuss the error analysis of approximation distributed-order fractional derivative by using this operational matrix. Finally, to illustrate the efficiency and validity of the presented technique five examples are given. Abbreviations: DFDEs: distributed-order fractional differential equations; DTFDEs: distributed-order time-fractional diffusion equations; DTFWEs: distributed-order time-fractional wave equations; OMFD: operational matrix of fractional derivative; SCP: shifted Chebyshev polynomial [ABSTRACT FROM AUTHOR]

An important equation usually used in modeling neuronal dynamics is cable equation. In this work, a numerical method for the fractional cable equation which involves two Riemann-Liouville fractional derivatives is proposed. Our computational technique is based on collocation idea where a combination of Bernoulli polynomials and Sinc functions are used to approximate the solution to this problem. The constructed approximation by our method convert the fractional cable equation into a set of algebraic equations. Also, we provide two numerical examples to confirm the accuracy and effectiveness of the present method. [ABSTRACT FROM AUTHOR]

This work presents a new approximation approach to solve the linear/nonlinear distributed order fractional differential equations using the Chebyshev polynomials. Here, we use the Chebyshev polynomials combined with the idea of the collocation method for converting the distributed order fractional differential equation into a system of linear/nonlinear algebraic equations. Also, fractional differential equations with initial conditions can be solved by the present method. We also give the error bound of the modified equation for the present method. Moreover, four numerical tests are included to show the effectiveness and applicability of the suggested method. [ABSTRACT FROM AUTHOR]

In this paper, for the first time, the shifted Legendre operational matrix of distributed order fractional derivative has been derived. Also, this new operational matrix is used together with tau method for approximation of solutions of linear distributed order fractional differential equations and diffusion equations with distributed order in time. Moreover, eight numerical examples are implemented in order to show the validity and reliability of the suggested methods. [ABSTRACT FROM AUTHOR]

In this paper, a direct collocation method based on rational Legendre functions is proposed for solving the magneto-hydrodynamic (MHD) boundary layer flow over a nonlinear stretching sheet. Here, we use rational Legendre–Gauss–Radau nodes and transformed Hermite–Gauss nodes as interpolation points. We present the comparison of this work with some other numerical results. Moreover, residual norm shows that the present solutions are accurate and applicable. [ABSTRACT FROM AUTHOR]

A new sinc-Gauss-Jacobi collocation method for solving the fractional Volterra's population growth model in a closed system is proposed. This model is a nonlinear fractional Volterra integro-differential equation where the integral term represents the effects of toxin. The fractional derivative is considered in the Liouville-Caputo sense. In the proposed method, we first convert fractional Volterra's population model to an equivalent nonlinear fractional differential equation, and then the resulting problem is solved using collocation method. The proposed collocation technique is based on sinc functions and Gauss-Jacobi quadrature rule. In this approach, the problem is reduced to a set of algebraic equations. The obtained numerical results of the present method are compared with some well-known results in the literature to show the applicability and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

An important class of fluids commonly used in industries is non-Newtonian fluids. In this paper, two numerical techniques based on rational Legendre functions and Chebyshev polynomials are presented for solving the flow of a third-grade fluid in a porous half space. This problem can be reduced to a nonlinear two-point boundary value problem on semi-infinite interval. Our methods are utilized to reduce the computation of this problem to some algebraic equations. The comparison of the results with the other methods and residual norm show very good accuracy and rate of convergence of our approach. [ABSTRACT FROM AUTHOR]

An axisymmetric viscous flow, generated by two large parallel plates slowly approaching each other is investigated. The steady nonlinear governing equations are converted into a fourth-order nonlinear differential equation using integrability condition. The resulting nonlinear boundary value problem is solved using quintic B-spline collocation and sinc-collocation methods. The approach consists of reducing the problem to a set of algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the techniques and a comparison is made with existing results in the literature. [ABSTRACT FROM PUBLISHER]

The Legendre pseudospectral method is developed for the numerical solution of nonlinear Duffing equation involving both integral and non-integral forcing terms. By using differentiation matrix, the problem is reduced to the solution of a system of algebraic equations. The method is general, easy to implement, and yields very accurate results. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed computational procedure. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this work, the problem of parameter identification in a parabolic partial differential equation is discussed. The shifted Legendre-tau method is presented for finding the unknown function and also the unknown coefficient in the one-dimensional diffusion model. The method consists of expanding the required approximate solution as the elements of a shifted Legendre polynomial and by using operational matrices we reduce the problem to a set of algebraic equations. Several examples are given to demonstrate the validity and applicability of the technique and a comparison is made with existing results. The method is easy to implement and yields very accurate results. [ABSTRACT FROM PUBLISHER]

A numerical method for solving the linear and non-linear fractional integro-differential equations of Volterra type is presented. The fractional derivative is described in the Caputo sense. The method is based upon Legendre approximations. The properties of Legendre polynomials together with the Gaussian integration method are utilized to reduce the fractional integro-differential equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results. [ABSTRACT FROM PUBLISHER]

The Exp-function method with the aid of symbolic computational system can be used to obtain the generalized solitary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics. In this paper, we study the analytic treatment of the Zakharov-Kuznetsov (ZK) equation, the modified ZK equation, and the generalized forms of these equations. Exact solutions with solitons and periodic structures are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of the simultaneous determination of time-dependent coefficients in a one-dimensional partial differential equation. The main aim is to apply the tau technique to determine unknown coefficients in a time-dependent partial differential equation. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique. [ABSTRACT FROM AUTHOR]

Abstract: A homotopy perturbation method (HPM) is proposed to solve non-linear systems of second-order boundary value problems. HPM yields solutions in convergent series forms with easily computable terms, and in some cases, yields exact solutions in one iteration. Moreover, this technique does not require any discretization, linearization or small perturbations and therefore reduces the numerical computations a lot. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. The results reveal that the method is very effective, straightforward and simple. [Copyright &y& Elsevier]

A numerical procedure is developed to obtain the solution of a mathematical model for capillary formation in tumor angiogenesis. The proposed method is based on the shifted Legendre tau technique. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution as a shifted Legendre polynomial with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative example is included to demonstrate the validity and applicability of the presented technique. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: The Chebyshev finite difference method is used for finding the solution of the ordinary differential equations which arise from problems of calculus of variations. Our approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a non-uniform finite difference scheme. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. The method is easy to implement and yields very accurate results. [Copyright &y& Elsevier]

A Chebyshev finite difference method has been proposed in order to solve linear and nonlinear second-order Fredholm integro-differential equations. The approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a nonuniform finite difference scheme. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. [ABSTRACT FROM AUTHOR]

Abstract: The Chebyshev finite difference method is presented for solving a nonlinear system of second-order boundary value problems. Our approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a non-uniform finite difference scheme. Some numerical results are also given to demonstrate the validity and applicability of the presented technique and a comparison is made with the existing results. The method is easy to implement and yields very accurate results. [Copyright &y& Elsevier]