Sinc-Galerkin Method Research Articles

Sinc-Galerkin method and a higher-order method for a 1D and 2D time-fractional diffusion equations

In this article, a new numerical algorithm for solving a 1-dimensional (1D) and 2-dimensional (2D) time-fractional diffusion equation is proposed. The Sinc-Galerkin scheme is considered for spatial discretization, and a higher-order finite difference formula is considered for temporal discretization. The convergence behavior of the methods is analyzed, and the error bounds are provided. The main objective of this paper is to propose the error bounds for 2D problems by using the Sinc-Galerkin method. The proposed method in terms of convergence is studied by using the characteristics of the Sinc function in detail with optimal rates of exponential convergence for 2D problems. Some numerical experiments validate the theoretical results and present the efficiency of the proposed schemes.

The numerical solution of Fredholm integral equation of the first kind using the Sinc-Galerkin method

Sinc-Galerkin scheme is one of the new techniques used in solving numerical problems involving integral equations. This method has been shown to be a powerful numerical tool for finding fast and accurate solutions. In this article, a numerical solution of Fredholm integral equation of the first kind which arise in a large number of problems in applied mathematics will be discussed, for this result, we choose Sinc functions as basis functions and Galerkin method to estimate a solution for an unknown function in this equation. So, in this paper, some properties of the Sinc-Galerkin method required for our subsequent development are given and are utilized to reduce integral equation of the first kind to some linear algebraic equations. Then convergence with exponential rate is proved by a theorem to guarantee applicability of numerical technique. Finally, numerical examples are included to confirm applicability and justify rapid convergence of our method.

Solving Boundary Value Problems by Sinc Method and Geometric Sinc Method

This paper introduces an efficient numerical method for approximating solutions to geometric boundary value problems. We propose the multiplicative sinc–Galerkin method, tailored specifically for solving multiplicative differential equations. The method utilizes the geometric Whittaker cardinal function to approximate functions and their geometric derivatives. By reducing the geometric differential equation to a system of algebraic equations, we achieve computational efficiency. The method not only proves to be computationally efficient but also showcases a valuable symmetric property, aligning with inherent patterns in geometric structures. This symmetry enhances the method’s compatibility with the often-present symmetries in geometric boundary value problems, offering both computational advantages and a deeper understanding of geometric calculus. To demonstrate the reliability and efficiency of the proposed method, we present several examples with both homogeneous and non-homogeneous boundary conditions. These examples serve to validate the method’s performance in practice.

Analysis of a sinc-Galerkin Method for the Fractional Laplacian

Analysis of a sinc-Galerkin Method for the Fractional Laplacian

Identification of an Inverse Source Problem in a Fractional Partial Differential Equation Based on Sinc-Galerkin Method and TSVD Regularization

Abstract In this paper, using Sinc-Galerkin method and TSVD regularization, an approximation of the quasi-solution to an inverse source problem is obtained. To do so, the solution of direct problem is obtained by the Sinc-Galerkin method, and this solution is applied in a least squares cost functional. Then, to obtain an approximation of the quasi-solution, we minimize the cost functional by TSVD regularization. Error analysis and convergence of the proposed method are investigated. In addition, at the end, four numerical examples are given in details to show the efficiency and accuracy of the proposed method.

Sinc Numerical Methods for Time Nonlocal Parabolic Equation

In recent years, more and more researchers have paid attention to the study of non-local problems. The numerical method for initial-boundary value problems of time nonlocal parabolic equations is established in this paper. The time nonlocal operator is discretized by finite difference method, and spatial differential operators is discretized by Sinc-Galerkin method. Then fully discrete scheme (D-SD scheme) for solving one-dimensional time nonlocal parabolic equation is obtained. Numerical example shows the effectiveness and superiority of the scheme for solving non-local problems.

Sinc-Galerkin method for solving system of singular perturbed reaction-diffusion problems

In this paper, the system of singularly Perturbed Reaction-Diffusion p roblems w hich a re commonly used in physics and chemistry branches of science, were investigated. Sinc-Galerkin Method was used to obtaining the solution of problems. Because of there is no article about Sinc-Galerkin Method related to singularly perturbed Reaction-Diffusion problems in liter- ature, the efficiency of the method was shown via this problem. There are important results that occurred after our research and application. Sinc Galerkin Method which was used in this paper as the main solution method gave better results according to parameter robust method and asymptotical initial value method. The figures and the tables show this competence and low errors.

Approximate analytical solutions of a class of non-linear fractional boundary value problems with conformable derivative

In this paper, it is presented that an approximate solution of a class of non-linear differential equations with conformable derivative under boundary conditions by using sinc-Galerkin method that is not used to approximately solve the class of the considered equation in the literature. In the method, the solution function is expressed as a finite series in terms of composite translated sinc functions and the unknown coefficients. The problem is reduced into a non-linear matrix-vector system via sinc grid points, and when this system is solved by using Newton?s method, the unknown coefficients of the solution function are easily obtained. Also, error analysis and some test problems are presented to illustrate the applicability and accuracy of the proposed method.

The Crank-Nicolson-type Sinc-Galerkin method for the fourth-order partial integro-differential equation with a weakly singular kernel

In this work, a Sinc-Galerkin method is considered and analyzed for solving the fourth-order partial integro-differential equation with a weakly singular kernel. The time derivative and Riemann-Liouville fractional integral term are approximated via the Crank-Nicolson method and the trapezoidal convolution quadrature rule, respectively. Then a fully discrete scheme is formulated via using the Sinc-Galerkin approximation. The exponential convergence rate in space of proposed method are derived. In addition, some properties of the Toeplitz matrix generated by the composite Sinc function at the Sinc node are extended to the cases of arbitrary order in the preliminary knowledge. Finally, some numerical examples are calculated to verify the accuracy and effectiveness of our method.

Sinc-Galerkin method for solving the time fractional convection\u2013diffusion equation with variable coefficients

In this paper, a new numerical algorithm for solving the time fractional convection–diffusion equation with variable coefficients is proposed. The time fractional derivative is estimated using the L_{1} formula, and the spatial derivative is discretized by the sinc-Galerkin method. The convergence analysis of this method is investigated in detail. The numerical solution is 2-alpha order accuracy in time and exponential rate of convergence in space. Finally, some numerical examples are given to show the effectiveness of the numerical scheme.

Comparison Methods for Solving Non-Linear Sturm–Liouville Eigenvalues Problems

In this paper, we present a comparative study between Sinc–Galerkin method and a modified version of the variational iteration method (VIM) to solve non-linear Sturm–Liouville eigenvalue problem. In the Sinc method, the problem under consideration was converted from a non-linear differential equation to a non-linear system of equations, that we were able to solve it via the use of some iterative techniques, like Newton’s method. The other method under consideration is the VIM, where the VIM has been modified through the use of the Laplace transform, and another effective modification has also been made to the VIM by replacing the non-linear term in the integral equation resulting from the use of the well-known VIM with the Adomian’s polynomials. In order to explain the advantages of each method over the other, several issues have been studied, including one that has an application in the field of spectral theory. The results in solutions to these problems, which were included in tables, showed that the improved VIM is better than the Sinc method, while the Sinc method addresses some advantages over the VIM when dealing with singular problems.

High order WSGL difference operators combined with Sinc-Galerkin method for time fractional Schrödinger equation

ABSTRACT In this paper, high order time discretization schemes for time fractional Schrödinger equations in one and two dimensions are proposed. Our schemes are based on high order weighted and shifted Grünwald-Letnikov (WSGL) difference operators for the time fractional derivatives, Sinc-Galerkin methods are used for the space variables. The stability of time semi-discrete schemes is analysed with the help of -transform. For the fully discretization schemes, the standard Sinc-Galerkin method and symmetric Sinc-Galerkin method are established by selecting proper weight functions. Finally, we apply our numerical schemes to solve one-dimensional and two-dimensional time fractional Schrödinger equations, verify the validity of present numerical schemes.

Numerical methods for the time fractional diffusion equation

In this paper, a differential analog of the Caputo fractional derivative called the L2 − 1σ formula is considered. The time semi-discrete scheme of the time fractional diffusion equation is obtained by using the L2 − 1σ formula to discretize the time derivative. The Sinc-Collocation method and the Sinc-Galerkin method are used to discretize the spatial derivatives, and the numerical schemes of the time fractional diffusion equation are constructed. Finally, the validity of the numerical schemes of this paper is verified by numerical example.

Computational Sinc-scheme for extracting analytical solution for the model Kuramoto-Sivashinsky equation

The present article is designed to supply two different numerical<br />solutions for solving Kuramoto-Sivashinsky equation. We have made<br />an attempt to develop a numerical solution via the use of<br />Sinc-Galerkin method for Kuramoto-Sivashinsky equation, Sinc<br />approximations to both derivatives and indefinite integrals reduce<br />the solution to an explicit system of algebraic equations. The fixed<br />point theory is used to prove the convergence of the proposed<br />methods. For comparison purposes, a combination of a Crank-Nicolson<br />formula in the time direction, with the Sinc-collocation in the<br />space direction is presented, where the derivatives in the space<br />variable are replaced by the necessary matrices to produce a system<br />of algebraic equations. In addition, we present numerical examples<br />and comparisons to support the validity of these proposed<br />methods.

Convergence of numerical schemes for the solution of partial integro-differential equations used in heat transfer

Integro-differential equations play an important role in may physical phenomena. For instance, it appears in fields like fluid dynamics, biological models and chemical kinetics. One of the most important physical applications is the heat transfer in heterogeneous materials, where physician are looking for efficient methods to solve their modeled equations. The difficulty of solving integro-differential equations analytically made mathematician to search about efficient methods to find an approximate solution. The present article is designed to supply numerical solution of a parabolic Volterra integro-differential equation under initial and boundary conditions. We have made an attempt to develop a numerical solution via the use of Sinc-Galerkin method, the convergence analysis via the use of fixed point theory has been discussed, and showed to be of exponential order. For comparison purposes, we approximate the solution of integro-differential equation using Adomian decomposition method. Sometimes, the Adomian decomposition method is a highly efficient technique used to approximate analytical solution of differential equations, applicability of Adomian decomposition method to partial integro-differential equations has not been studied in details previously in the literatures. In addition, we present numerical examples and comparisons to support the validity of these proposed methods.

Sinc-Galerkin solution to eighth-order boundary value problems

Many problems that arise in astrophysics, hydrodynamic and hydromagnetic stability, fluid dynamics, astronomy, beam and long wave theory are modeled as eighth-order boundary-value problems. In this paper we show that the sinc-Galerkin method is an efficient and accurate numerical scheme for solving these problems. The inner product approximations to eighth and seventh order derivatives and the corresponding error estimates are derived. Compared to the non-polynomial spline technique and the reproducing kernel space method, we show that the sinc-Galerkin method provides more accurate results. In addition, for problems with singular solutions or singular source functions, the sinc-Galerkin method is shown to maintain the exponential convergence rate.

Sinc-Galerkin method for solving hyperbolic partial differential equations

In this work, we consider the hyperbolic equations to determine the approximate solutions via Sinc-Galerkin Method (SGM). Without any numerical integration, the partial differential equation transformed to an algebraic equation system. For the numerical calculations, Maple is used. Several numerical examples are investigated and the results determined from the method are compared with the exact solutions. The results are illustrated both in the table and graphically.

On the sinc-Galerkin method for triharmonic boundary-value problems

A numerical scheme is developed to provide an approximate solution to the triharmonic boundary value problem. Based on the sinc inner product approximations, a direct discretization of the triharmonic operator Δ3 reduces the problem to a generalized Sylvester equation. Numerical examples illustrate the pertinent features of the sinc-Galerkin method where very accurate results are obtained even when singularities occur at the boundaries.

Approximation by Sinc-Galerkin method for a class of fractional boundary value problems

Approximation by Sinc-Galerkin method for a class of fractional boundary value problems

Application of Sinc-Galerkin method for solving a nonlinear inverse parabolic problem

In this paper, using Sinc-Galerkin and Levenberg–Marquardt methods a stable numerical solution is obtained to a nonlinear inverse parabolic problem. Due to this, this problem is reduced to a parameter approximation problem. To approximate unknown parameters, we consider an optimization problem where objective function is minimized by Levenberg–Marquardt method. This objective function is obtained by using Sinc-Galerkin method and the overposed measured data. Finally, some numerical examples are given to demonstrate the accuracy and reliability of the proposed method.