A sinc-collocation method for initial value problems (original) (raw)
On using sinc collocation approach for solving a parabolic PDE with nonlocal boundary conditions
Journal of Nonlinear Sciences and Applications
This work suggests a simple method based on a sinc approximation at sinc nodes for solving parabolic partial differential equations with nonlocal boundary conditions. Sinc approximation are typified by errors of the form O e −k/h , where k > 0 is a constant and h is a step size. Some numerical examples are utilized to reveal the efficaciousness and precision of this method. The suggested method is flexible, easy to programme and efficient.
A Collocation Method for Second Order Boundary Value Problems
This paper is an extension of Mamadu and Ojobor (2017) were the efficiency of the collocation method was considered based on the type of basis function in developing the scheme. Here, we investigate the convergence of the method as applied to second order boundary value problems (BVPs) at the various collocation points: Gauss-Lobatto (G-L), Gauss-Chebychev (G-C) and Gauss-Radau (G – R) collocation points. Also, the class of Chebychev polynomials of the first kind have been adopted as basis function. We have employed Maple 18 software in our analysis and computations. Introduction Let the generalized form of a differential equation be given as í µí°¿ í µí±¦ í µí±¥ = í µí± í µí±¥ , í µí¼ í µí±¦ í µí± 1 = í µí±, í µí¼ í µí±¦ í µí± 2 = í µí±, (1.0) where í µí»¼, í µí¼ and í µí¼ are considered as differential operators. Differential equation are often applied in the construction and development of most mathematical models such as predictive control in AP monitor (Hedengen et. al., 2014), temperature distribution in cylindrical conductor (Fortini et. al., 2008), dynamic optimization (cizinar et. al., 2015), etc. Modeling is the bridge between the subject and real-life situations for students realization. Differential equations model real-life situations, and provide the real-life answers with the help of computer calculations and graphics. Investigation into methods for solving these problems has been on the increase in recent years. Obviously, many methods (analytical or numerical methods) have been developed and implemented by many researchers. Of these methods, the numerical methods seem to be more popular than their analytic counterpart due to their adequate approximation of the analytic solution in a rapid converging series. Popular numerical methods include; the Tau method (Adeniyi, 2004), orthogonal collocation method (Mamadu and Ojobor, 2017), Tau-Collocation method (Mamadu and Njoseh, 2016), Variation iteration decomposition method (Ojobor and Mamadu, 2017), Elzaki transform method (Mamadu and Njoseh, 2017), Power series approximation method (Njoseh and Mamadu, 2016a), Modified power series approximation method (Njoseh and Mamadu, 2017), etc. However, the collocation method remains one of the best numerical method due to its level of simplicity and accuracy. Moreover, the efficiency of the method is dependent on the class of basis function and the collocation point adopted in constructing the scheme. There exist different types of basis functions that can be adopted to construct the scheme such as; canonical polynomials, Chebychev polynomials, Bernoulli polynomials, Lagrange polynomials (Fox and Pascal, 1968; Lanczos, 1938). And, the different collocations that can be adopted include; the equally spaced points; Gauss-lobotto points, Gauss-Chebychev points, Gauss-Radau point, etc. These points improves better than one another in terms of convergence.
CONVERGENCE ANALYSIS OF THE SINC COLLOCATION METHOD FOR INTEGRO-DIFFERENTIAL EQUATIONS SYSTEM
In this paper, a numerical solution for a system of linear Fredholm integro-differential equations by means of the sinc method is considered. This approximation reduces the system of integro-differential equations to an explicit system of algebraic equations. The exponential convergence rate O(e −k √ N) of the method is proved. The analytical results are illustrated with numerical examples that exhibit the exponential convergence rate.
Discretization of nonlinear models by Sinc collocation-interpolation methods
Computers & Mathematics with Applications, 1996
This paper deals with the solution of initial-boundary value problems for nonlinear evolution equations in one and two space dimensions. The solution technique is based on collocationinterpolation methods which use Sinc functions. The paper aints to provide a guideline towards a large number of nonlinear problems of interest in applied sciences. has to be computed by the solution of the initial-boundary value problem, where independent variables in (1.1) are normalized in a fashion that t E [0, 1], x E [0, 1], y E [0, el, where g is of the order of the unity. In the applications, it is also useful normalizing the variable u in order to obtain u E [-1, 1].
Modified Sinc-Galerkin Method for Nonlinear Boundary Value Problems
Journal of Mathematics, 2013
This paper presents a modified Galerkin method based on sinc basis functions to numerically solve nonlinear boundary value problems. The modifications allow for the accurate approximation of the solution with accurate derivatives at the endpoints. The algorithm is applied to well-known problems: Bratu and Thomas-Fermi problems. Numerical results demonstrate the clear advantage of the suggested modifications in obtaining accurate numerical solutions as well as accurate derivatives at the endpoints.
A “sinc-Galerkin” method of solution of boundary value problems
Mathematics of Computation, 1979
This paper illustrates the application of a "Sinc-Galerkin" method to the approximate solution of linear and nonlinear second order ordinary differential equations, and to the approximate solution of some linear elliptic and parabolic partial differential equations in the plane. The method is based on approximating functions and their derivatives by use of the Whittaker cardinal function. The DE is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products, the evaluation of which does not require any numerical integration. Using n function evaluations, the error in the final approximation to the solution of the DE is O ( e − c n 1 / 2 d ) O({e^{ - c{n^{1/2d}}}}) , where c is independent of n, and d denotes the dimension of the region on which the DE is defined. This rate of convergence is optimal in the class of n-point methods which assume that the solution is analytic in the interior of the interval, and which ignore possib...
The double exponential sinc collocation method for singular Sturm-Liouville problems
Journal of Mathematical Physics, 2016
Sturm-Liouville problems are abundant in the numerical treatment of scientific and engineering problems. In the present contribution, we present an efficient and highly accurate method for computing eigenvalues of singular Sturm-Liouville boundary value problems. The proposed method uses the double exponential formula coupled with sinc collocation method. This method produces a symmetric positive-definite generalized eigenvalue system and has exponential convergence rate. Numerical examples are presented and comparisons with single exponential sinc collocation method clearly illustrate the advantage of using the double exponential formula.
Modelling and Simulation in Engineering
In this paper, we investigate the numerical approximation solution of parabolic and hyperbolic equations with variable coefficients and different boundary conditions using the space-time localized collocation method based on the radial basis function. The method is based on transforming the original d -dimensional problem in space into d + 1 -dimensional one in the space-time domain by combining the d -dimensional vector space variable and 1 -dimensional time variable in one d + 1 -dimensional variable vector. The advantages of such formulation are (i) time discretization as implicit, explicit, θ -method, method-of-line approach, and others are not applied; (ii) the time stability analysis is not discussed; and (iii) recomputation of the resulting matrix at each time level as done for other methods for solving partial differential equations (PDEs) with variable coefficients is avoided and the matrix is computed once. Two different formulations of the d -dimensional problem as a d + ...