Iterative Reproducing Kernel Method Research Articles

Numerical investigation for Caputo-Fabrizio fractional Riccati and Bernoulli equations using iterative reproducing kernel method

The pivotal aim of present work is to design and implement an effective numerical approach, the iterative reproducing-kernel algorithm, to provide numerical solutions to the fractional Riccati and Bernoulli differential equations in the Caputo-Fabrizio sense. To achieve this, we develop an advanced operational algorithm based on creating a complete orthonormal function system utilizing the reproducing-kernel function in order to formulate the solution in the form of uniformly convergent Fourier series. Additionally, error and stability analysis of the proposed method are studied in the appropriate space W2,2(Δ). On the other hand, some meaningful numerical applications are included to demonstrate the feasibility and reliability of this approach under the qualitative effect of the Caputo-Fabrizio fractional derivative. In numerical viewpoints, the achieved results using the proposed approach indicate a high level of accuracy and rare technical skills, which qualifies it to deal with many fractional models emerging in various mathematical and physical disciplines within the frame of Caputo-Fabrizio derivatives.

A novel iterative solution for time-fractional Boussinesq equation by reproducing kernel method

In this study, iterative reproducing kernel method (RKM) will be applied in order to observe the effect of the method on numerical solutions of fractional order Boussinesq equation. Hilbert spaces and their kernel functions, linear operators and base functions which are necessary to obtain the reproducing kernel function are clearly explained. Iterative solution is constituted in a serial form by using reproducing kernel function. Then convergence of RKM solution is shown with lemma and theorem. Two problems, “good” Boussinesq and generalized Boussinesq equations, are examined by using RKM for different fractional values. Results are presented with tables and graphics.

Numerical Solutions of Fractional Systems of Two-Point BVPs by Using the Iterative Reproducing Kernel Algorithm

We propose an efficient computational method, namely, the iterative reproducing kernel method for the approximate solution of fractional-order systems of two-point time boundary-value problems in the Caputo sense. Two extended inner-product spaces are constructed in which the boundary conditions are satisfied for the analyzed systems. The reproducing kernel functions are constructed to get an accurate algorithm for the investigation of fractional systems. The developed procedure is based on generating the orthonormal basis with an aim to formulate the solution via the evolution of the algorithm. The analytic solution is represented in the form of a series in the reproducing kernel Hilbert space with readily computed components. In this connection, some numerical examples are presented to show the good performance and applicability of the developed algorithm. The numerical results indicate that the proposed algorithm is a powerful tool for the solution of fractional models encountered in various fields of sciences and engineering.

An iterative kernel based method for fourth order nonlinear equation with nonlinear boundary condition

This article discusses an iterative reproducing kernel method with respect to its effectiveness and capability of solving a fourth-order boundary value problem with nonlinear boundary conditions modeling beams on elastic foundations. Since there is no method of obtaining reproducing kernel which satisfies nonlinear boundary conditions, the standard reproducing kernel methods cannot be used directly to solve boundary value problems with nonlinear boundary conditions as there is no knowledge about the existence and uniqueness of the solution. The aim of this paper is, therefore, to construct an iterative method by the use of a combination of reproducing kernel Hilbert space method and a shooting-like technique to solve the mentioned problems. Error estimation for reproducing kernel Hilbert space methods for nonlinear boundary value problems have yet to be discussed in the literature. In this paper, we present error estimation for the reproducing kernel method to solve nonlinear boundary value problems probably for the first time. Some numerical results are given out to demonstrate the applicability of the method.

Simplified iterative reproducing kernel method for handling time-fractional BVPs with error estimation

Abstract In this article, we introduce a novel numerical scheme, the iterative reproducing kernel method (IRKM), for providing numerical approximate solutions of a certain class of time-fractional boundary value problem within favorable aspects of the reproducing kernel Hilbert space in Caputo sense. The algorithm methodology is based on generating an orthonormal basis from the reproducing kernel function to formulate the solution in form of uniformly convergent series, accordance with the constraint conditions in the space ω 5 [ 0 , 1 ] . Error estimates in the ω 5 -norm are obtained as well as numerical experiments are described to represent the hypothesis, and to confirm the design procedure of the proposed algorithm. The numerical results indicate that the IRKA is a significant development tool for handling such issues arising in computer, physics and engineering fields.

An iterative reproducing kernel method in Hilbert space for the multi-point boundary value problems

Abstract In this paper, an iterative method is proposed to solve the nonlinear Bitsadze–Samarskii boundary value problems with multi-point boundary conditions. The algorithm is based on the reproducing kernel Hilbert space method. We use an iterative scheme to overcome the nonlinearity of the problem. The convergence and error estimate of the iterative scheme are established. The reproducing kernel Hilbert space method is used to generate an approximation of the linearized problem. In fact, the reproducing kernel Hilbert space method is combined with an iterative scheme to approximate the solution and an error estimate of the approximate solution is derived. In order to show the efficiency and versatility of the proposed method, some numerical results are reported. The comparison of numerical results with the analytical solution and the best results reported in the literature confirms the good accuracy and applicability of the proposed method.

Using the iterative reproducing kernel method for solving a class of nonlinear fractional differential equations

ABSTRACTIn previous works, we have devoted to using the reproducing kernel methods solving integer order differential equations, based on the review of previous works, in this paper, we mainly present a method for solving a class of higher order fractional differential equations with general boundary value problems by using Taylor formula into reproducing kernel space. Its analytical solution is represented in the form of series. The analytical solution and approximate solution obtained by this method is given and it is uniformly converge to the exact solution and its corresponding derivatives. The numerical examples are studied to demonstrate the accuracy of the present method.

Iterative Reproducing Kernel Method for Solving Second-Order Integrodifferential Equations of Fredholm Type

We present an efficient iterative method for solving a class of nonlinear second-order Fredholm integrodifferential equations associated with different boundary conditions. A simple algorithm is given to obtain the approximate solutions for this type of equations based on the reproducing kernel space method. The solution obtained by the method takes form of a convergent series with easily computable components. Furthermore, the error of the approximate solution is monotone decreasing with the increasing of nodal points. The reliability and efficiency of the proposed algorithm are demonstrated by some numerical experiments.

Solution of a Linear Third order Multi-Point Boundary Value Problem using RKM

Aims: In this paper an approximate method for the solution of third-order differential equation with two and three point boundary condition is developed using iterative reproducing kernel method. Methods: The third order boundary value problem is converted into integro-differential equation of second order two point boundary value problem. The reproducing kernel method which takes the form of a convergent series with easily computable components is used for the solution of second order two point boundary value problem. Results: Six numerical examples are given to demonstrate the efficiency of the present method. The results obtained are better than the existing methods developed in [19,20,21,22,23]. Conclusions: In this paper, the solution of linear and nonlinear third order (two and three point) boundary value problem is determined. For the solution of third order three point boundary value problem reproducing kernel method is proposed and obtained a good accuracy in absolute errors. As the reproducing kernel method cannot solve the third order three-point boundary value problems directly, so the third order boundary value problem is converted to second order two point boundary value problem after absorbing the nonlocal condition at . The method developed is compared with those developed by Li et al. [19], El-Salam et al. [20], Khan and Aziz [21], Li and Wu [22] and Wu and Li [23]. As observed in Example 3, 4, 5 and 6 that the method obtained in this paper is better than [19]-[23]. Results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for third-order linear as well as nonlinear boundary value problem.

Nonlinear Functional Analysis of Boundary Value Problems: Novel Theory, Methods, and Applications

and Applied Analysis 3 1-homogeneous operator in a Banach space and then demonstrate its application in establishing the existence of positive solutions for p-Laplacian boundary value problems under certain conditions. (xi) In the paper titled “Existence of solutions for nonhomogeneous A-harmonic equations with variable growth,” the authors establish a theorem for the existence of weak solutions for nonhomogeneous A-harmonic equations in subspace and then give three examples to demonstrate its application. (xii) In the paper titled “Multiple solutions for degenerate elliptic systems near resonance at higher eigenvalues,” the authors study the degenerate semilinear elliptic system in an open bounded domain with smooth boundary, and some multiplicity results of solutions are obtained for the system near resonance at certain eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory. (xiii) In the paper titled “A regularity criterion for the Navier-Stokes equations in the multiplier spaces,” the authors establish a regularity criterion in terms of the pressure gradient for weak solutions to the NavierStokes equations in a special class. The third set of papers, including four papers, deal with several boundary value problems for highly nonlinear ordinary differential equations. (i) In the paper titled “Positive solutions for second-order singular semipositone differential equations involving Stieltjes integral conditions,” the authors investigate the existence of positive solutions for second-order singular differential equations with a negatively perturbed term, by means of the fixed-point theory in cones. (ii) In the paper titled “Positive solutions for Sturm-Liouville boundary value problems in a Banach Space,” the sufficient conditions for the existence of single and multiple positive solutions for a second-order SturmLiouville boundary value problem are established in a Banach space, by using the fixed-point theorem of strict set contraction operators in the frame of the ODE technique. (iii) In the paper titled “Positive solutions of a nonlinear fourth-order dynamic eigenvalue problem on time scales,” the authors study a nonlinear fourth-order dynamic eigenvalue problem on time scales and obtain the existence and nonexistence of positive solutions when 0 λ, respectively, for some λ, by using the Schauder fixed-point theorem and the upper and lower solution method. (iv) In the paper titled “Bifurcation analysis for a predatorprey model with time delay and delay-dependent parameters,” a class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. By using the normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifur-cating from Hopf bifurcations are obtained. The fourth set of papers focus on finding the approximate and numerical solutions of various complex nonlinear boundary value problems. (i) In the paper titled “On spectral homotopy analysis method for solving linear Volterra and Fredholm integrodifferential equations,” a spectral homotopy analysis method (SHAM) is proposed to solve linear Volterra integrodifferential equations, and some examples are given to test the efficiency and the accuracy of the proposed method. (ii) In the paper titled “The solution of a class of singularly perturbed two-point boundary value problems by the iterative reproducing kernel method,” the authors establish an iterative reproducing kernel method (IRKM) for solving singular perturbation problems with boundary layers and give two numerical examples to demonstrate the effectiveness of the method. (iii) In the paper titled “A Galerkin solution for Burgers’ equation using cubic B-spline finite elements,” a Galerkin method using cubic B-splines is set up to find the numerical solutions of Burgers’ equation, and the method is shown to be capable of solving Burgers’ equation accurately for values of viscosity ranging from very small to very large. (iv) In the paper titled “Forward-backward splitting methods for accretive operators in Banach spaces,” the authors introduce two iterative forward-backward splitting methods with relaxations to find zeros of the sum of two accretive operators in Banach spaces and prove the weak and strong convergence of these methods under mild conditions, and also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem. Yong Hong Wu Lishan Liu Benchawan Wiwatanapataphee Shaoyong Lai Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in

The Solution of a Class of Singularly Perturbed Two-Point Boundary Value Problems by the Iterative Reproducing Kernel Method

In (Wang et al., 2011), we give an iterative reproducing kernel method (IRKM). The main contribution of this paper is to use an IRKM (Wang et al., 2011), in singular perturbation problems with boundary layers. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate that the method is simple and effective.

Iterative reproducing kernel method for a beam equation with third-order nonlinear boundary conditions

Purpose: This paper investigates an analytical approximate solution of a fourth-order differential equation with nonlinear boundary conditions modeling beams on elastic foundations using iterative reproducing kernel method. Methods: The solution obtained using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve the problems since there is no method of obtaining a reproducing kernel satisfying nonlinear boundary conditions. The aim of this paper is to fill this gap. Results: Several illustrative examples are given to demonstrate the effectiveness of the present method. Conclusions: Results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for the beam equation with third-order nonlinear boundary conditions.

Iterative reproducing kernel method for nonlinear oscillator with discontinuity

In this paper, iterative reproducing kernel method is applied to obtain the analytical approximate solution of a nonlinear oscillator with discontinuities. The solution obtained by using the method takes the form of a convergent series with easily computable components. An illustrative example is given to demonstrate the effectiveness of the present method. The results obtained using the scheme presented here show that the numerical scheme is very effective and convenient for the nonlinear oscillator with discontinuities.