共查询到20条相似文献,搜索用时 432 毫秒
1.
K. Parand M. Shahini Mehdi Dehghan 《Communications in Nonlinear Science & Numerical Simulation》2010,15(2):360-367
In this paper, the numerical solution of the Blasius problem is obtained using the collocation method based on rational Chebyshev functions. The Blasius equation is a nonlinear ordinary differential equation which arises in the boundary layer flow. The method reduces solving the equation to solving a system of nonlinear algebraic equations. The results presented here demonstrate reliability and efficiency of the method. 相似文献
2.
In this paper we propose, a collocation method for solving the Blasius equation. The Blasius equation is a third-order nonlinear ordinary differential equation. This approach is based on a rational scaled generalized Laguerre function collocation method. We also present the comparison of this work with some well-known results and show that the present solution is accurate. 相似文献
3.
Harendra Singh Fahimeh Akhavan Ghassabzadeh Emran Tohidi Carlo Cattani 《Mathematical Methods in the Applied Sciences》2020,43(9):5941-5952
In this paper, the Legendre spectral collocation method (LSCM) is applied for the solution of the fractional Bratu's equation. It shows the high accuracy and low computational cost of the LSCM compared with some other numerical methods. The fractional Bratu differential equation is transformed into a nonlinear system of algebraic equations for the unknown Legendre coefficients and solved with some spectral collocation methods. Some illustrative examples are also given to show the validity and applicability of this method, and the obtained results are compared with the existing studies to highlight its high efficiency and neglectable error. 相似文献
4.
Kamal Rashedi 《Mathematical Methods in the Applied Sciences》2023,46(2):1752-1771
In this work, we present a spectral method for recovering an unknown time-dependent lower-order coefficient and unknown wave displacement in a nonlinear Klein–Gordon equation with overdetermination at a boundary condition. We apply the initial and boundary conditions to construct the satisfier function and use this function in a transformation to convert the main problem to a nonclassical hyperbolic equation with homogeneous initial and boundary conditions. Then, we utilize the orthonormal Bernstein basis functions to approximate the solution of the reformulated problem and use a direct technique based on the operational matrices of integration and differentiation of these basis functions together with the collocation technique to reduce the problem to a system of nonlinear algebraic equations. Regarding the perturbed measurements, the method takes advantage of the mollification method in order to derive stable numerical derivatives. Numerical simulations for solving several test examples are presented to show the applicability of the proposed method for obtaining accurate and stable results. 相似文献
5.
In this paper, we present a collocation method to obtain the approximate solutions of continuous population models for single and interacting species. By using the Bessel polynomials and collocation points, this method transforms population model into a matrix equation. The matrix equation corresponds to a system of nonlinear equations with the unknown Bessel coefficients. The reliability and efficiency of the proposed scheme are demonstrated by two numerical examples and performed on the computer algebraic system Maple. 相似文献
6.
Kerli Orav-Puurand 《Numerical Functional Analysis & Optimization》2013,34(3-4):352-370
In a weakly singular integral equation of the second kind, we perform a smoothing change of variables and solve the resulting equation either by the collocation or by the product integration method based on a “central part” interpolation by polynomials on the uniform grid. Optimal convergence order of both methods is established. The latter method is hopeful due to especially simple and cheap assembling of the algebraic system of equations. 相似文献
7.
Ay?egül Akyüz-Da?c?o?lu Handan Çerdi˙k-Yaslan 《Applied mathematics and computation》2011,217(12):5658-5666
By the use of the Chebyshev series, a direct computational method for solving the higher order nonlinear differential equations has been developed in this paper. This method transforms the nonlinear differential equation into the matrix equation, which corresponds to a system of nonlinear algebraic equations with unknown Chebyshev coefficients, via Chebyshev collocation points. The solution of this system yields the Chebyshev coefficients of the solution function. An algorithm for this nonlinear system is also proposed in this paper. The method is valid for both initial-value and boundary-value problems. Several examples are presented to illustrate the accuracy and effectiveness of the method. 相似文献
8.
Mohammad Ramezani 《Mathematical Methods in the Applied Sciences》2019,42(14):4640-4663
In this work, we present numerical analysis for nonlinear multi‐term time fractional differential equation which involve Caputo‐type fractional derivatives for . The proposed method is based on utilization of fractional B‐spline basics in collocation method. The scheme can be readily obtained efficient and quite accurate with less computational work numerical result. The proposal approach transform nonlinear multi‐term time fractional differential equation into a suitable linear system of algebraic equations which can be solved by a suitable numerical method. The numerical experiments will be verify to demonstrate the effectiveness of our method for solving one‐ and two‐dimensional multi‐term time fractional differential equation. 相似文献
9.
K. Parand A. R. Rezaei A. Taghavi 《Mathematical Methods in the Applied Sciences》2010,33(17):2076-2086
This paper aims to compare rational Chebyshev (RC) and Hermite functions (HF) collocation approach to solve Volterra's model for population growth of a species within a closed system. This model is a nonlinear integro‐differential equation where the integral term represents the effect of toxin. This approach is based on orthogonal functions, which will be defined. The collocation method reduces the solution of this problem to the solution of a system of algebraic equations. We also compare these methods with some other numerical results and show that the present approach is applicable for solving nonlinear integro‐differential equations. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献
10.
İbrahim Çelik 《Mathematical Methods in the Applied Sciences》2016,39(3):366-377
In this paper, new and efficient numerical method, called as Chebyshev wavelet collocation method, is proposed for the solutions of generalized Burgers–Huxley equation. This method is based on the approximation by the truncated Chebyshev wavelet series. By using the Chebyshev collocation points, algebraic equation system has been obtained and solved. Approximate solutions of the generalized Burgers–Huxley equation are compared with exact solutions. These calculations demonstrate that the accuracy of the Chebyshev wavelet collocation solutions is quite high even in the case of a small number of grid points. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
11.
Functionally-fitted methods are generalizations of collocation techniques to integrate an equation exactly if its solution is a linear combination of a chosen set of basis functions. When these basis functions are chosen as the power functions, we recover classical algebraic collocation methods. This paper shows that functionally-fitted methods can be derived with less restrictive conditions than previously stated in the literature, and that other related results can be derived in a much more elegant way. The novelty in our approach is to fully retain the collocation framework without reverting back into derivations based on cumbersome Taylor series expansions. AMS subject classification (2000) 65L05, 65L06, 65L20, 65L60 相似文献
12.
In this paper, we propose a Tau method for solving the singular Lane–Emden equation—a nonlinear ordinary differential equation on a semi‐infinite interval. We applied collocation, Galerkin, and Tau methods for solving this problem, and according to the results, the solution of Tau method is the most accurate. The operational derivative and product matrices of the modified generalized Laguerre functions are presented. These matrices, in conjunction with the Tau method, are then utilized to reduce the solution of the Lane–Emden equation to that of a system of algebraic equations. We also present a comparison of this work with some well‐known results and show that the present solution is highly accurate. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
13.
Şuayip Yüzbaşı 《Mathematical Methods in the Applied Sciences》2012,35(8):885-893
In this paper, we will develop the Bessel collocation method to find approximate solutions of the Chen system, which is a three‐dimensional system of ODEs with quadratic nonlinearities. This scheme consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions by means of the Bessel polynomials with unknown coefficients. By help of the collocation points and the matrix operations of derivatives, the unknown coefficients of the Bessel polynomials are calculated. The accuracy and efficiency of the proposed approach are demonstrated by two numerical examples and performed with the aid of a computer code written in MAPLE. In addition, comparisons between our method and the homotopy perturbation method numerical solutions are made with the accuracy of solutions. Copyright © 2012 John Wiley & Sons, Ltd. 相似文献
14.
In this paper, we develop symplectic and multi-symplectic wavelet collocation methods to solve the two-dimensional nonlinear Schrödinger equation in wave propagation problems and the two-dimensional time-dependent linear Schrödinger equation in quantum physics. The Hamiltonian and the multi-symplectic formulations of each equation are considered. For both formulations, wavelet collocation method based on the autocorrelation function of Daubechies scaling functions is applied for spatial discretization and symplectic method is used for time integration. The conservation of energy and total norm is investigated. Combined with splitting scheme, splitting symplectic and multi-symplectic wavelet collocation methods are also constructed. Numerical experiments show the effectiveness of the proposed methods. 相似文献
15.
Şuayip Yüzbaşı 《Applied Mathematical Modelling》2012,36(12):5876-5890
In this study, we will obtain the approximate solutions of the HIV infection model of CD4+T by developing the Bessel collocation method. This model corresponds to a class of nonlinear ordinary differential equation systems. Proposed scheme consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions by means of the Bessel polynomials with unknown coefficients. The unknown coefficients of the Bessel polynomials are computed using the matrix operations of derivatives together with the collocation method. The reliability and efficiency of the proposed approach are demonstrated in the different time intervals by a numerical example. All computations have been made with the aid of a computer code written in Maple 9. 相似文献
16.
Dulat Dzhumabaev Elmira Bakirova Sandugash Mynbayeva 《Mathematical Methods in the Applied Sciences》2020,43(4):1788-1802
A nonlinear loaded differential equation with a parameter on a finite interval is studied. The interval is partitioned by the load points, at which the values of the solution to the equation are set as additional parameters. A nonlinear boundary value problem for the considered equation is reduced to a nonlinear multipoint boundary value problem for the system of nonlinear ordinary differential equations with parameters. For fixed parameters, we obtain the Cauchy problems for ordinary differential equations on the subintervals. Substituting the values of the solutions to these problems into the boundary condition and continuity conditions at the partition points, we compose a system of nonlinear algebraic equations in parameters. A method of solving the boundary value problem with a parameter is proposed. The method is based on finding the solution to the system of nonlinear algebraic equations composed. 相似文献
17.
18.
Mehdi Tatari Behnam Sepehrian Maryam Alibakhshi 《Numerical Methods for Partial Differential Equations》2012,28(1):248-262
In this article, we consider the problem of solving Burgers‐Fisher equation. The approximate solution is found using the radial basis functions collocation method. Also for solving of the resulted nonlinear system of equations, we proposed a predictor corrector method based on the fixed point iterations. The numerical tests show that this method is accurate and efficient for finding a closed form approximation of the solution of nonlinear partial differential equations. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 248–262, 2012 相似文献
19.
In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms. 相似文献
20.
Dina Mostafa Mahmoud A. Zaky Ramy M. Hafez Ahmed S. Hendy Mohamed A. Abdelkawy Ahmed A. Aldraiweesh 《Mathematical Methods in the Applied Sciences》2023,46(1):656-674
We present a class of orthogonal functions on infinite domain based on Jacobi polynomials. These functions are generated by applying a tanh transformation to Jacobi polynomials. We construct interpolation and projection error estimates using weighted pseudo-derivatives tailored to the involved mapping. Then, using the nodes of the newly introduced tanh Jacobi functions, we develop an efficient spectral tanh Jacobi collocation method for the numerical simulation of nonlinear Schrödinger equations on the infinite domain without using artificial boundary conditions. The applicability and accuracy of the solution method are demonstrated by two numerical examples for solving the nonlinear Schrödinger equation and the nonlinear Ginzburg–Landau equation. 相似文献