Reproducing Kernel Method Research Articles

The reproducing kernel for the reaction-diffusion model with a time variable fractional order

The variable-order fractional calculus has become a useful mathematical frame-work to describe a complex reaction-diffusion process. It is very hard to solve the problem, and there is almost no analytical method available in open literature. In this article, the reproducing kernel method is proposed for this purpose, and some examples show that the method is of high precision.

Solving system of second-order BVPs using a new algorithm based on reproducing kernel Hilbert space

In this study, a new and reliable technique based on Reproducing Kernel Method (RKM) is introduced to solve the linear and nonlinear system of second-order boundary value problem. We propose that RKM without Gram-Schmidt orthogonalization process significantly reduces the CPU time and increases the accuracy of the approximate solutions. The convergence theorem and error analysis are also verified. An algorithm is presented to indicate the steps of implementing the proposed scheme. Since the system of second-order boundary value problem is solvable with the numerical commands in Mathematica software, we examined these commands for numerical examples, compared the results with the present method, and demonstrated the higher precision of the present algorithm.

A new reproducing kernel method with higher convergence order for solving a Volterra–Fredholm integral equation

In the paper, linear Volterra–Fredholm integral equations of the second kind are considered in a reproducing kernel space. The approximate solution is gotten by constructing a uniquely solvable system of linear algebraic equations. Our important contributions are to provide convergence order theorems of the scheme using spline function theories and propose a new scheme with high convergence order for solving the approximate solutions to oscillation and non-oscillation of exact solutions.

On the MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing: two reliable methods

In this paper, a Lie-group integrator based on $$GL_4(\mathbb {R})$$ and the reproducing kernel functions has been constructed to investigate the flow characteristics in an electrically conducting second-grade fluid over a stretching sheet. Accurate initial values can be achieved when the target equation is matched precisely, and then, we can apply the group preserving scheme (GPS) to get a rather accurate results. On the other hand, the reproducing kernel method (RKM) is successfully applied to the underlying equation with convergence analysis. We show exact and approximate solutions by series in the reproducing kernel space. We use a bounded linear operator in the reproducing kernel space to get the solutions by the reproducing kernel method. Comparison of these two methods demonstrates the power and reliability. Finally, effects of magnetic parameter, viscoelastic parameter, stagnation-point flow, and stretching of the sheet parameters are illustrated.

Numerical solution of time-fractional Kawahara equation using reproducing kernel method with error estimate

We present a new approach depending on reproducing kernel method (RKM) for time-fractional Kawahara equation with variable coefficient. This approach consists of obtaining an orthonormal basis function on specific Hilbert spaces. In this regard, some special Hilbert spaces are defined. Kernel functions of these special spaces are given and basis functions are obtained. The approximate solution is attained as serial form. Convergence analysis, error estimation and stability analysis are presented after obtaining the approximate solution. To show the power and effect of the method, two examples are solved and the results are given as table and graphics. The results demonstrate that the presented method is very efficient and convenient for Kawahara equation with fractional order.

A Hybrid Method for Singularly Perturbed Convection–Diffusion Equation

In this research, we present a hybrid method which combination of simplified reproducing kernel method (SRKM) and asymptotic expansion for solving singularly perturbed convection–diffusion problems. According to the hybrid method, firstly asymptotic expansion formed on boundary layer domain and then terminal value problem solved via SRKM on regular domain. To apply the SRKM, some special Hilbert spaces are defined and related reproducing kernel functions are given. Then, the approximate solution is obtained as serial form. A new algorithm of SRKM is presented for nonlinear problem. Also, convergence analysis of approximate solution is given. Some linear and nonlinear problems are solved by hybrid method. The obtained outcomes indicate that the hybrid method is reliable and very effective for solving singularly perturbed convection–diffusion equation.

A reproducing kernel method for solving heat conduction equations with delay

We attempt to propose a numerical algorithm for the one-dimensional conduction equations with delay. Firstly, a novel reproducing kernel space satisfying the delay condition is constructed, then an approximating solution space is derived by using the orthogonal projection. In fact, the proposed scheme is a collocation method. The unique solvability as well as uniform convergence of the new scheme is discussed. Two numerical experiments are provided to illustrate our theory.

Reproducing kernel Hilbert space method based on reproducing kernel functions for investigating boundary layer flow of a Powell–Eyring non-Newtonian fluid

ABSTRACTIn this work, the boundary layer flow of a Powell–Eyring non-Newtonian fluid over a stretching sheet has been investigated by a reproducing kernel method. Reproducing kernel functions are used to obtain the solutions. The approximate solutions are demonstrated, and the proposed technique is compared with some well-known methods. Convergence analysis of the technique is presented. The accuracy of the reproducing kernel method has been proved.

Numerical simulation of time-fractional partial differential equations arising in fluid flows via reproducing Kernel method

Purpose The subject of the fractional calculus theory has gained considerable popularity and importance due to their attractive applications in widespread fields of physics and engineering. The purpose of this paper is to present results on the numerical simulation for time-fractional partial differential equations arising in transonic multiphase flows, which are described by the Tricomi and the Keldysh equations of Robin functions types. Design/methodology/approach Those resulting mathematical models are solved by using the reproducing kernel method, which provide appropriate solutions in term of infinite series formula. Convergence analysis, error estimations and error bounds under some hypotheses, which provide the theoretical basis of the proposed method are also discussed. Findings The dynamical properties of these numerical solutions are discussed and the profiles of several representative numerical solutions are illustrated. Finally, the prospects of the gained results and the method are discussed through academic validations. Originality/value In this paper and for the first time: the authors presented results on the numerical simulation for classes of time-fractional PDEs such as those found in the transonic multiphase flows. The authors applied the reproducing kernel method systematically for the numerical solutions of time-fractional Tricomi and Keldysh equations subject to Robin functions types.

Numerical solution of multi-Pantograph delay boundary value problems via an efficient approach with the convergence analysis

This present investigation is contemplated to provide Legendre spectral collocation method for solving multi-Pantograph delay boundary value problems (BVPs). In this regard, an equivalent integral form of such BVPs has been considered. The proposed method is based on Legendre–Gauss collocation nodes and Legendre–Gauss quadrature rule. Convergence analysis associated to the presented scheme has been provided to show its applicability theoretically. Some numerical examples are given to demonstrate the efficiency, accuracy, and versatility of our method. Numerical results confirm the theoretical predictions and are superior with respect to several recent numerical methods including Hermite collocation approach, Laguerre collocation technique and the reproducing kernel method.

Treatment of near-incompressibility in meshfree and immersed-particle methods

We propose new projection methods for treating near-incompressibility in small and large deformation elasticity and plasticity within the framework of particle and meshfree methods. Using the $$\overline{\mathbf {B}}$$ and $$\overline{\mathbf {F}}$$ techniques as our point of departure, we develop projection methods for the conforming reproducing kernel method and the immersed-particle or material point-like methods. The methods are based on the projection of the dilatational part of the appropriate measure of deformation onto lower-dimensional approximation spaces, according to the traditional $$\overline{\mathbf {B}}$$ and $$\overline{\mathbf {F}}$$ approaches, but tailored to meshfree and particle methods. The presented numerical examples exhibit reduced stress oscillations and are free of volumetric locking and hourglassing phenomena.

A least square point of view to reproducing kernel methods to solve functional equations

In this paper we discuss and present a least square and a QR point of view to reproducing kernel methods to approximate solutions to some linear and nonlinear functional equations. The procedure we discuss here may includes ordinary, partial differential, and integral equations. We also give new proofs to some known results on this subject. The most interesting contribution is that the proposed algorithm may work even when we know a reproducing kernel but nothing more about the associated reproducing kernel Hilbert space, including the inner product structure.

Fuzzy reproducing kernel space method for solving fuzzy boundary value problems

In the beginning, we describe the fuzzy inner product space and the fuzzy Hilbert space. Our goal is to use the fuzzy reproducing kernel method to solve the second-order fuzzy boundary value problem. The fuzzy convergence analysis of introduced method is discussed in detail. We present some examples in the end.

The combined reproducing kernel method and Taylor series for handling nonlinear Volterra integro-differential equations with derivative type kernel

The reproducing kernel method is applied to Volterra nonlinear integro-differential equations. In this technique, the nonlinear term is replaced by its Taylor series. 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. Some numerical examples are solved in two different spaces and parameters of n. Measurements of the experimental data is an indications of stability and convergence on the reproducing kernel.

Construction of fractional power series solutions to fractional stiff system using residual functions algorithm

A powerful analytical approach, namely the fractional residual power series method (FRPS), is applied successfully in this work to solving a class of fractional stiff systems. The methodology of the FRPS method gets a Maclaurin expansion of the solution in rapidly convergent form and apparent sequences based on the Caputo sense without any restriction hypothesis. This approach is tested on a fractional stiff system with nonlinearity ranging. Meanwhile, stability and convergence study are presented in the domain of interest. Illustrative examples justify that the proposed method is analytically effective and convenient, and it can be implemented in a large number of engineering problems. A numerical comparison for the experimental data with another well-known method, the reproducing kernel method, is given. The graphical consequences illuminate the simplicity and reliability of the FRPS method in the determination of the RPS solutions consistently.

Existence of Unique Solutions to the Telegraph Equation in Binary Reproducing Kernel Hilbert Spaces

We demonstrate the existence of a unique solution to a nonhomogeneous telegraph initial/boundary value problem on the unit square in an appropriate binary reproducing kernel Hilbert space which depends on the smoothness of the driver. Examples are given to illustrate the numerical effectiveness of the reproducing kernel method when properly applied and the aberrations which can occur when no solution exists in the space.

Two computational approaches for solving a fractional obstacle system in Hilbert space

The primary motivation of this paper is to extend the application of the reproducing-kernel method (RKM) and the residual power series method (RPSM) to conduct a numerical investigation for a class of boundary value problems of fractional order 2α, 0<alphaleq1, concerned with obstacle, contact and unilateral problems. The RKM involves a variety of uses for emerging mathematical problems in the sciences, both for integer and non-integer (arbitrary) orders. The RPSM is combining the generalized Taylor series formula with the residual error functions. The fractional derivative is described in the Caputo sense. The representation of the analytical solution for the generalized fractional obstacle system is given by RKM with accurately computable structures in reproducing-kernel spaces. While the methodology of RPSM is based on the construction of a fractional power series expansion in rapidly convergent form and apparent sequences of solution without any restriction hypotheses. The recurrence form of the approximate function is selected by a well-posed truncated series that is proved to converge uniformly to the analytical solution. A comparative study was conducted between the obtained results by the RKM, RPSM and exact solution at different values of α. The numerical results confirm both the obtained theoretical predictions and the efficiency of the proposed methods to obtain the approximate solutions.

Piecewise reproducing kernel method for linear impulsive delay differential equations with piecewise constant arguments

In this paper, we introduce a piecewise reproducing kernel method for impulsive delay differential equations with piecewise constant arguments. The method is an improved reproducing kernel method. Compared with the classical reproducing kernel method, the solutions obtained using the present method can give good approximations for a larger time interval. Some numerical examples are used to show the effectiveness and simplicity of the method.

Approximation scheme for handling coupled systems of differential equations within reproducing kernel method

This paper proposes an efficient numerical method to obtain analytical-numerical solutions for a class of system of boundary value problems. This new algorithm is based on a reproducing kernel Hilbert space method. The analytical solution is calculated in the form of series in reproducing kernel space with easily computable components. In addition, convergence analysis for this method is discussed. In this sense, some numerical examples are given to show the effectiveness and performance of the proposed method. The results reveal that the method is quite accurate, simple, straightforward, and convenient to handle a various range of differential equations.

J-Integral Evaluation of Cracked Shell Structures Employing Effective Reproducing Kernel Meshfree Modeling

The J-integral evaluation are analyzed employing effective reproducing kernel method. Several numerical examples of cracked shell structure are carried out to investigate the mixed-mode stress resultant intensity factors (SRIFs). It is formulated by the first order shear deformation plate theory. Reproducing kernel (RK) is used to the meshfree interpolant. A diffraction method, visibility criterion and enriched basis are introduced to model the through crack. J-integral is evaluated based on the stress resultants and is decomposed into the symmetric and asymmetric parts for extracting the mixed-mode SRIFs. The stabilized conforming nodal integration (SCNI) and sub-domain stabilized conforming nodal integration (SSCI) are applied to not only the stiffness matrix but also the J-integral. The path-independency of SRIFs are examined. The results of SRIF are compared with reference solutions and commercial FEM software. The comparison reveals that high accuracy on SRIFs from meshfree modeling are obtained.