Numerical solution of nonlinear Volterra-Fredholm-Hammerstein integral equations via collocation method based on radial basis functions

A numerical technique based on the spectral method is presented for the solution of nonlinear Volterra-Fredholm-Hammerstein integral equations. This method is a combination of collocation method and radial basis functions (RBFs) with the differentiation process (DRBF), using zeros of the shifted Legendre polynomial as the collocation points. Different applications of RBFs are used for this purpose. The integral involved in the formulation of the problems are approximated based on Legendre-Gauss-Lobatto integration rule. The results of numerical experiments are compared with the analytical solution in illustrative examples to confirm the accuracy and efficiency of the presented scheme.     

Numerical solution of some classes of integral equations using Bernstein polynomials

This paper is concerned with obtaining approximate numerical solutions of some classes of integral equations by using Bernstein polynomials as basis. The integral equations considered are Fredholm integral equations of second kind, a simple hypersingular integral equation and a hypersingular integral equation of second kind. The method is explained with illustrative examples. Also, the convergence of the method is established rigorously for each class of integral equations considered here.     

Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations

The aim of this article is to present an analytical approximation solution for linear and nonlinear multi-order fractional differential equations (FDEs) by extending the application of the shifted Chebyshev operational matrix. For this purpose, we convert FDE into a counterpart system and then using proposed method to solve the resultant system. Our results in solving four different linear and nonlinear FDE, confirm the accuracy of proposed method.     

Numerical solution of the nonlinear age-structured population models by using the operational matrices of Bernstein polynomials

In this paper a numerical method for solving the nonlinear age-structured population models is presented which is based on Bernstein polynomials approximation. Operational matrices of integration, differentiation, dual and product are introduced and are utilized to reduce the age-structured population problem to the solution of algebraic equations. The method in general is easy to implement, and yields good results. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

A new numerical approach for the solution of nonlinear Fredholm integral equations system of second kind by using Bernstein collocation method

This paper presents an efficient numerical method for finding solutions of the nonlinear Fredholm integral equations system of second kind based on Bernstein polynomials basis. The numerical results obtained by the present method have been compared with those obtained by B‐spline wavelet method. This proposed method reduces the system of integral equations to a system of algebraic equations that can be solved easily any of the usual numerical methods. Numerical examples are presented to illustrate the accuracy of the method. Copyright © 2013 John Wiley & Sons, Ltd.     

Blow-up time for solutions to some nonlinear Volterra integral equations

The problem of the estimating of a blow-up time for solutions of Volterra nonlinear integral equation with convolution kernel is studied. New estimates, lower and upper, are found and, moreover, the procedure for the improvement of the lower estimate is presented. Main results are illustrated by examples. The new estimates are also compared with some earlier ones related to a shear band model.     

The operational matrices of Bernstein polynomials for solving the parabolic equation subject to specification of the mass

Some physical problems in science and engineering are modelled by the parabolic partial differential equations with nonlocal boundary specifications. In this paper, a numerical method which employs the Bernstein polynomials basis is implemented to give the approximate solution of a parabolic partial differential equation with boundary integral conditions. The properties of Bernstein polynomials, and the operational matrices for integration, differentiation and the product are introduced and are utilized to reduce the solution of the given parabolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

Asymptotic properties of solutions to some nonlinear integral equations of convolution type

For a class of nonlinear integral equations of convolution type we give necessary and sufficient conditions for the boundedness of nonnegative solutions. Moreover, conditions for the solution to converge asymptotically to a determined limit are obtained.     

An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations

This paper aims to construct a general formulation for the Jacobi operational matrix of fractional integral operator. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, a reliable and efficient technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the Jacobi integral operational matrix to the fractional calculus. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.     

Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite-dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.     

Bernstein operational matrix of fractional derivatives and its applications

In this paper, Bernstein operational matrix of fractional derivative of order α in the Caputo sense is derived. We also apply this matrix to the collocation method for solving multi-order fractional differential equations. The numerical results obtained by the present method compares favorably with those obtained by various collocation methods earlier in the literature.     

A residual method for solving nonlinear operator equations and their application to nonlinear integral equations using symbolic computation

A derivative-free residual method for solving nonlinear operator equations in real Hilbert spaces is discussed. This method uses in a systematic way the residual as search direction, but it does not use first order information. Furthermore a convergence analysis and numerical results of the new method applied to nonlinear integral equations using symbolic computation are presented.     

Numerical solution of Volterra integral and integro-differential equations of convolution type by using operational matrices of piecewise constant orthogonal functions

In this paper, we use operational matrices of piecewise constant orthogonal functions on the interval [0,1)$[0,1)$ to solve Volterra integral and integro-differential equations of convolution type without solving any system. We first obtain Laplace transform of the problem and then we find numerical inversion of Laplace transform by operational matrices. Numerical examples show that the approximate solutions have a good degree of accuracy.     

A note on blow-up solutions to some nonlinear Volterra integral equations

In this paper we investigate certain nonlinear Volterra integral equations with the power nonlinearity. The basic results provide a criterion involving the kernel and the nonlinearity for the existence of the nontrivial and blow-up solution.     

单纯形上Bernstein多项式的一些性质

设Ｂ_m（ｆ，·）为函数ｆ在ｄ维单纯形σ上的ｎ阶Ｂｅｒｎｓｔｅｉｎ多项式，本文对ｆ∈Ｃ^ｒ（σ）及ｆ∈Ｃ^r+2（σ）给出了ｆ的各阶编导数用Ｂ_n（ｆ，·）相应偏导数逼近的误差估计．同时也考虑了整系数Ｂｅｒｎｓｔｅｉｎ多项式的Ｌ_p模估计     

Numerical solution of the system of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation

The operational Tau method, a well-known method for solving functional equations, is employed to solve a system of nonlinear Volterra integro-differential equations with nonlinear differential part. In addition, an error estimation of the method is presented. Some cases of the mentioned equations are solved as examples to illustrate the ability and reliability of the method. The results reveal that the method is very effective and convenient.     

New enclosure algorithms for the verified solutions of nonlinear Volterra integral equations

This paper is concerned with new algorithms which provide the sharp bounds that are guaranteed to contain the exact solutions of nonlinear Volterra integral equations. We develop new enclosure algorithms based on the interval methods which was first introduced by Moore in [24] together with the Taylor polynomials to improve the accuracy of the scheme by reducing the width of interval solutions. The modified methods calculate a priori bound automatically in parallel with the computation of solutions of integral equations. We will show that the accuracy of the proposed algorithms is dependent on the number of interval subdivisions. Some numerical experiments are also included to demonstrate the validity and applicability of the scheme and showing a marked improvement in comparison with the recent existing numerical results.     

A wavelet operational method for solving fractional partial differential equations numerically

Fractional calculus is an extension of derivatives and integrals to non-integer orders, and a partial differential equation involving the fractional calculus operators is called the fractional PDE. They have many applications in science and engineering. However not only the analytical solution existed for a limited number of cases, but also the numerical methods are very complicated and difficult. In this paper, we newly establish the simulation method based on the operational matrices of the orthogonal functions. We formulate the operational matrix of integration in a unified framework. By using the operational matrix of integration, we propose a new numerical method for linear fractional partial differential equation solving. In the method, we (1) use the Haar wavelet; (2) establish a Lyapunov-type matrix equation; and (3) obtain the algebraic equations suitable for computer programming. Two examples are given to demonstrate the simplicity, clarity and powerfulness of the new method.     

Direct method to solve Volterra integral equation of the first kind using operational matrix with block-pulse functions

This article proposes a simple efficient direct method for solving Volterra integral equation of the first kind. By using block-pulse functions and their operational matrix of integration, first kind integral equation can be reduced to a linear lower triangular system which can be directly solved by forward substitution. Some examples are presented to illustrate efficiency and accuracy of the proposed method.