reproducing kernel method

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.

In this paper, a reliable new scheme is presented based on combining Reproducing Kernel Method (RKM) with a practical technique for the nonlinear problem to solve the System of Singularly Perturbed Boundary Value Problems (SSPBVP). The Gram-Schmidt orthogonalization process is removed in the present RKM. However, we provide error estimation for the approximate solution and its derivative. Based on the present algorithm in this paper, can also solve linear problem. Several numerical examples demonstrate that the present algorithm does have higher precision.

In this paper, a mixed reproducing kernel function (RKF) is introduced. The kernel function consists of piecewise polynomial kernels and polynomial kernels. On the basis of the mixed RKF, a new numerical technique is put forward for solving non-linear boundary value problems (BVPs) with nonlocal conditions. Compared with the classical RKF-based methods, our method is simpler since it is unnecessary to convert the original equation to an equivalent equation with homogeneous boundary conditions. Also, it is not required to satisfy the homogeneous boundary conditions for the used RKF. Finally, the higher accuracy of the method is shown via several numerical tests.

This paper is concerned with a technique for solving Volterra integro-dierential equationsin the reproducing kernel Hilbert space. In contrast with the conventional reproducing kernelmethod, the Gram-Schmidt process is omitted here and satisfactory results are obtained.The analytical solution is represented in the form of series. An iterative method is given toobtain the approximate solution. The convergence analysis is established theoretically. Theapplicability of the iterative method is demonstrated by testing some various examples.

This paper is concerned with a technique for solving Fredholm integro-di erential equations in the reproducing kernel Hilbert space. In contrast with the conventional reproducing kernel method, the Gram-Schmidt process is omitted here and satisfactory results are obtained. The analytical solution is represented in the form of series. An iterative method is given to obtain the approximate solution. The convergence analysis is established theoretically. The applicability of the iterative method is demonstrated by testing some various examples.

In this letter, the numerical scheme of nonlinear Volterra-Fredholm integro-differential equations is proposed in a reproducing kernel Hilbert space (RKHS). The method is constructed based on the reproducing kernel properties in which the initial condition of the problem is satis ed. The nonlinear terms are replaced by its Taylor series. In this technique, the nonlinear Volterra-Fredholm integro-differential equations are converted to nonlinear differential equations. The exact solution is represented in the form of series in the reproducing Hilbert kernel space. The approximation solution is expressed by n-term summation of reproducing kernel functions and it is converge to the exact solution. Some numerical examples are given to show the accuracy of the method.

This paper is concerned with a technique for solving Volterra integral equations in the reproducing kernel Hilbert space. In contrast with the conventional reproducing kernel method, the Gram-Schmidt process is omitted here and satisfactory results are obtained.The analytical solution is represented in the form of series.An iterative method is given to obtain the approximate solution.The convergence analysis is established theoretically. The applicability of the iterative method is demonstrated by testing some various ýexamples.

In this paper we discuss about nonlinear pseudoparabolic equations with nonlocal boundary conditions and their results. An effective error estimation for this method altough has not yet been discussed. The aim of this paper is to fill this gap.

In this paper we propose a relatively new semi-analytical technique to approximate the solution of nonlinear multi-order fractional differential equations (FDEs). We present some results concerning to the uniqueness of solution of nonlinear multi-order FDEs and discuss the existence of solution for nonlinear multi-order FDEs in reproducing kernel Hilbert space (RKHS). We further give an error analysis for the proposed technique in different reproducing kernel Hilbert spaces and present some useful results. The accuracy of the proposed technique is examined by comparing with the exact solution of some test examples.

The aim of this paper is to present a numerical method for singularly perturbed convection-diffusion problems with a delay. The method is a combination of the asymptotic expansion technique and the reproducing kernel method (RKM). First an asymptotic expansion for the solution of the given singularly perturbed delayed boundary value problem is constructed. Then the reduced regular delayed differential equation is solved analytically using the RKM. An error estimate and two numerical examples are provided to illustrate the effectiveness of the present method. The results of numerical examples show that the present method is accurate and efficient.