Sinc-collocation Method Research Articles

Numerical solution of singular Poisson equations using sinc approximation

Abstract In this article, we use the Sinc approximation to solve the single Poisson problem (where the first derivative or higher order does not have an exact answer on the border of the boundary). We have examined the Sinc Galerkin approximation to solve the single Poisson problem, and finally, to solve the Poisson problem, we will use the sinc collocation method and in this method, we will reach a linear system. By carefully choosing the length of the steps and the number of nodal points, we will solve this system with two methods, with the orthogonalization technique, a numerical approximation will be obtained. Its accuracy can be exponential and of order where is a transaction parameter and c are a constant independent of N. In the final part, we will give some numerical examples of single Poisson problems.

Numerical solution of singular Poisson equations using sinc approximation

Abstract In this article, we use the Sinc approximation to solve the single Poisson problem (where the first derivative or higher order does not have an exact answer on the border of the boundary). We have examined the Sinc Galerkin approximation to solve the single Poisson problem, and finally, to solve the Poisson problem, we will use the sinc collocation method and in this method, we will reach a linear system. By carefully choosing the length of the steps and the number of nodal points, we will solve this system with two methods, with the orthogonalization technique, a numerical approximation will be obtained. Its accuracy can be exponential and of order where is a transaction parameter and c are a constant independent of N. In the final part, we will give some numerical examples of single Poisson problems.

Numerical solution of singularly perturbed singular third order boundary value problems with nonclassical sinc method

In this paper, we employ a nonclassical sinc-collocation method to compute numerical solutions for singularly perturbed singular third-order boundary value problems prevalent in various scientific and engineering domains. Utilizing the sinc approximation offers a strategic advantage in navigating singularities, thus enabling an efficient computational strategy. Our method streamlines the solution process by converting singular boundary value problems into sets of linear equations, thereby improving computational efficiency. Moreover, its straightforward implementation adds to its robustness. We explore the convergence properties and error estimation of our proposed methods in detail. Finally, we provide two illustrative examples that demonstrate the effectiveness of our approach.

Solving the two-dimensional time-dependent Schrödinger equation using the Sinc collocation method and double exponential transformations

Over the last four decades, Sinc methods have occupied an important place in numerical analysis due to their simplicity and great performance. An incorporation of the Sinc collocation method with double exponential transformation is used to solve the two-dimensional time dependent Schrödinger equation. Numerical comparison between the double exponential and single exponential approaches is made to illustrate the superiority of the double exponential Sinc method.

Nonlinear Free Vibration of Anisogrid Lattice Cylindrical Panel Using the 2D Double Exponential Sinc Collocation Method

This study explores the nonlinear free vibrations of anisogrid lattice cylindrical panels under clamped circumstances for the first time. The lattice panel is made of repeating circumferential hexagonal unit cells with helical and circumferential isotropic ribs. The panel’s stiffness and mass are calculated using a continuous model based on orthotropic deep panel theory. The strain–displacement relationships are based on von-Kármán’s geometrically nonlinear theory. Using the Hamilton principle, nonlinear two-dimensional (2D) motion equations are extracted in their strong form. A 2D collocation approach based on the Sinc functions with double exponential (DE) transformation with a high convergence rate is used to solve the governing motion equations. The time variable is then eliminated from the equations using the weighted residual Galerkin procedure. Finally, the system’s nonlinear natural frequencies are computed using an iterative algorithm based on the bilateral displacement control. The influence of unit cell features, such as the angle of helical ribs, the number and thickness of ribs, and the panel dimensions, is examined in detail on the panel’s linear and nonlinear vibration behavior.

Sinc Collocation Method to Simulate the Fractional Partial Integro-Differential Equation with a Weakly Singular Kernel

In this article, we develop an efficient numerical scheme for dealing with fractional partial integro-differential equations (FPIEs) with a weakly singular kernel. The weight and shift Grünwald difference (WSGD) operator is adopted to approximate a time fractional derivative and the Sinc collocation method is applied for discretizing the spatial derivative.The exponential convergence of our proposed method is demonstrated in detail. Finally, numerical evidence is employed to verify the theoretical results and confirm the expected convergence rate.

A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity

The purpose of this paper is to investigate a space-time Sinc-collocation method for solving the fourth-order nonlocal heat model arising in viscoelasticity, which is a class of partial integro-differential equations (PIDEs) whose solution typically exhibits the weak singularities at the initial time. Most of existing approximate methods for solving such kind PIDEs are unbalanced, since most work have used a low order scheme for integrating the temporal variable and a high order numerical framework for discretization of space variables. The current paper contributes to a space-time spectral-order Sinc-collocation method which is balanced in both time and space variables. In order to deal with the initial singularity of solution in the time direction, the Sinc-collocation method with single exponential (SE) transformation is used for the first time, which is an effective method to deal with the singularity of the equation. Also, we fully analyse the error bounds of the method which show the exponential convergence rate in space and time. Meanwhile, we consider several test problems for examining the suggested scheme. The experiment results highlight a good precision of the new Sinc-colloction method and the correctness of the theoretical prediction for this kind nonlocal heat problems.

On the applications of collocation method for numerically analyzing the nonlinear Degasperis–Procesi and Benjamin–Bona–Mahony equations

This research uses the Sinc Collocation method to numerically examine the Degasperis–Procesi and Benjamin–Bona–Mahony equations, achieving a high level of precision and accuracy on computational grounds with a variety of mesh points. The proposed technique involves global collocation using Sinc bases function (SBF) as an activation function. Initially, the time derivative discretization has been performed through finite difference scheme and the remaining spatial derivatives are approximated by [Formula: see text]-weighted scheme to minimize the complexity in numerical computations. The main contribution of research is in modifying the proposed scheme to overcome the complexity involved in the whole process of finding solutions with a cardinal expansion of Sinc functions in nonlinear and singular systems. The solutions of Degasperis–Procesi and Benjamin–Bona–Mahony stiff nonlinear systems are illustrated numerically and graphically and the validation and verification of the obtained results are examined through stability analysis by calculating the spectral radius by bounding the value of [Formula: see text]. A comparative study is presented with some of the well-known numerical techniques, which assured the accuracy and reliability of our technique.

Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

Sinc collocation method for solving linear systems of Fredholm Volterra integro–differential equations of high order with variable coefficients

Numerical solution of system of second-order integro-differential equations using nonclassical sinc collocation method

In this paper, a nonclassical sinc collocation method is constructed for the numerical solution of systems of second-order integro-differential equations of the Volterra and Fredholm types. The novelty of the approach is based on using the new nonclassical weight function for sinc method instead of the classic ones. The sinc collocation method based on nonclassical weight functions is used to reduce the system of integro-differential equations to a system of algebraic equations. Furthermore, the convergence of the method is proposed theoretically, showing that the method converges exponentially. By solving some examples, including problems with a non-smooth solution, the results are compared with other existing results to demonstrate the efficiency of the new method.

Two Reliable Computational Techniques for Solving the MRLW Equation

In this paper, a numerical solution of the modified regularized long wave (MRLW) equation is obtained using the Sinc-collocation method. This approach approximates the space dimension of the solution with a cardinal expansion of Sinc functions. First, discretizing the time derivative of the MRLW equation by a classic finite difference formula, while the space derivatives are approximated by a θ—weighted scheme. For comparison purposes, we also find a soliton solution using the Adomian decomposition method (ADM). The Sinc-collocation method was were found to be more accurate and efficient than the ADM schemes. Furthermore, we show that the number of solitons generated can be approximated using the Maxwellian initial condition. The proposed methods’ results, analytical solutions, and numerical methods are compared. Finally, a variety of graphical representations for the obtained solutions makes the dynamics of the MRLW equation visible and provides the mathematical foundation for physical and engineering applications.

On the solution of lake pollution model by Sinc collocation method

In this paper, Sinc collocation method is employed for finding the approximate solution of the three lakes pollution model. In order to show the effectiveness and accuracy of the Sinc collocation method, several examples are presented in the tables and graphical forms. After observing the tables and graphical forms, it is concluded that Sinc collocation method is applicable with high accuracy to other real world problems.

Sinc collocation method: solution of a class of strongly nonlinear two-point boundary value problems

Sinc collocation method: solution of a class of strongly nonlinear two-point boundary value problems

Sinc collocation method: solution of a class of strongly nonlinear two-point boundary value problems

Sinc collocation method: solution of a class of strongly nonlinear two-point boundary value problems

Numerical solution of Rosenau–KdV equation using Sinc collocation method

The numerical method based on Sinc function is applied for the solution of Rosenau–KdV equation in this paper. The equation is fully-discretized by using the Sinc collocation method for spatial discretization and the forward finite difference for time discretization. The difference scheme is indicated to be conditionally stable using error analysis. The validity and accuracy of this method are verified by serval numerical experiments using single solitary waves, double solitary waves and multiple solitary waves.

A numerical scheme for solving a class of time fractional integro-partial differential equations with Caputo–Fabrizio derivative

In this paper, we find the numerical solution of the time-fractional partial integro-differential equation with Caputo–Fabrizio fractional derivative. The problem is discretized by some finite difference schemes in the time direction, and then the Sinc collocation method is applied to the resulting problems in the spatial direction. The convergence and stability of the method are proved and illustrated by three test examples.

An efficient Sinc-collocation method via the DE transformation for eighth-order boundary value problems

This paper shows the exponential convergence of the Sinc-collocation method based on the double exponential (DE) transformation applied to eighth-order boundary value problems (BVPs). Then using Kantorovich’s theorem, we obtain the exponential convergence of the non-linear eighth-order ordinary differential equation (ODE). Furthermore, we extend the analytical results to the arbitrary even-order case. In the numerical experiment, several linear and nonlinear examples are provided to verify our theoretical analysis. Meanwhile, the solution yielded via DE transformation is compared with those obtained by single exponential (SE) transformation and existing method to demonstrate the high efficiency and accuracy of our method.

A non-classical sinc-collocation method for the solution of singular boundary value problems arising in physiology

In this paper, a non-classical sinc-collocation method is used to find numerical solution of a class of singular second-order boundary value problems arising in physiology. The ability of the sinc approximation to overcome the singularity makes it an efficient method. This method is utilized to reduce the computation of solution of singular boundary value problems to some nonlinear systems of equations. Implementing this method is simple and attractive. Furthermore, convergence of proposed method is discussed by preparing the theorems which show exponential convergence and guarantee the applicability of those. Several examples are solved and numerical results are compared with other existing methods to show the efficiency and applicability of the method.

On the solution of Zabolotskaya–Khokhlov and Diffusion of Oxygen equations using a Sinc collocation method

Recent progress seems to suggest that the use of Sinc collocation method for the numerical treatment of partial differential equations, as a great level of precision and accuracy has been obtained on computational grounds. Sinc functions based on collocation points, provide us, the approximation of functions over all types of problems containing singularities on semi-infinite domain as well as on boundaries. Two partial differential equations are tested by using cardinal expansion of Sinc functions. However, employing Sinc functions to reduce the governing PDEs into system of algebraic equations and using collocations points with matrix operations leads to desired results. Furthermore, this technique ensures higher accuracy and convergence of obtained results of numerical solutions.

Numerical solution of the diffusion problem of distributed order based on the Sinc-collocation method

Numerical solution of the diffusion problem of distributed order based on the Sinc-collocation method