B‐spline collocation algorithms for numerical solution of the RLW equation

Both sextic and septic B‐spline collocation algorithms are presented for the numerical solutions of the RLW equation. Numerical results resolve the fine structure of the single solitary wave propagation, two and three solitary waves interaction, and evolution of solitary waves. Comparison of the numerical results is done by the results of some earlier schemes mentioned in the article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 581–607, 2011     

Cosine expansion‐based differential quadrature algorithm for numerical solution of the RLW equation

The differential quadrature method based on cosine expansion is applied to obtain numerical solutions of the RLW equation. The propagation of single solitary wave is studied to validate the efficiency of the algorithm. Then, test problems including interaction of two and three solitary waves, undulation, and evolution of solitary waves are implemented. Solutions are compared with earlier results. Discrete conservation quantities are computed for test experiments. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

B‐spline collocation algorithm for numerical solution of the generalized Burger's‐Huxley equation

The cubic B‐spline collocation scheme is implemented to find numerical solution of the generalized Burger's–Huxley equation. The scheme is based on the finite‐difference formulation for time integration and cubic B‐spline functions for space integration. Convergence of the scheme is discussed through standard convergence analysis. The proposed scheme is of second‐order convergent. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are compared with other results given in literature. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Numerical solution of the Rosenau–KdV–RLW equation by operator splitting techniques based on B‐spline collocation method

In the present study, the operator splitting techniques based on the quintic B‐spline collocation finite element method are presented for calculating the numerical solutions of the Rosenau–KdV–RLW equation. Two test problems having exact solutions have been considered. To demonstrate the efficiency and accuracy of the present methods, the error norms L₂ and L_∞ with the discrete mass Q and energy E conservative properties have been calculated. The results obtained by the method have been compared with the exact solution of each problem and other numerical results in the literature, and also found to be in good agreement with each other. A Fourier stability analysis of each presented method is also investigated.     

Operator splitting for numerical solutions of the RLW Equation

In this study, the numerical behavior of the one-dimensional Regularized Long Wave (RLW) equation has been sought by the Strang splitting technique with respect to time. For this purpose, cubic B-spline functions are used with the finite element collocation method. Then, single solitary wave motion, the interaction of two solitary waves and undular bore problems have been studied and the effectiveness of the method has been investigated. The new results have been compared with those of some of the previous studies available in the literature. The stability analysis has also been taken into account by the von Neumann method.     

An optimal sixth‐order quartic B‐spline collocation method for solving Bratu‐type and Lane‐Emden–type problems

This paper is concerned with the numerical solutions of Bratu‐type and Lane‐Emden–type boundary value problems, which describe various physical phenomena in applied science and technology. We present an optimal collocation method based on quartic B‐spine basis functions to solve such problems. This method is constructed by perturbing the original problem and on a uniform mesh. The method has been tested by four nonlinear examples. In order to show the advantage of the new method, numerical results are compared with those obtained by some of the existing methods, such as normal quartic B‐spline collocation method and the finite difference method (FDM). It has been observed that the order of convergence of the proposed method is six, which is two orders of magnitude larger than the normal quartic B‐spline collocation method. Moreover, our method gives highly accurate results than the FDM.     

A meshfree method for the numerical solution of the RLW equation

In this paper, we present a meshfree technique for the numerical solution of the regularized long wave (RLW) equation. This approach is based on a global collocation method using the radial basis functions (RBFs). Different kinds of RBFs are used for this purpose. Accuracy of the new method is tested in terms of _L2$L_2$ and _L∞$L_{\infty }$ error norms. In case of non-availability of the exact solution, performance of the new method is compared with existing methods. Stability analysis of the method is established. Propagation of single and double solitary waves, wave undulation, and conservation properties of mass, energy and momentum of the RLW equation are discussed.     

Four techniques based on the B‐spline expansion and the collocation approach for the numerical solution of the Lane–Emden equation

Four numerical techniques based on the linear B‐spline functions are presented for the numerical solution of the Lane–Emden equation. Some properties of the B‐spline functions are presented and are utilized to reduce the solution of the Lane–Emden equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the new techniques. Copyright © 2013 John Wiley & Sons, Ltd.     

A high‐order B‐spline collocation scheme for solving a nonhomogeneous time‐fractional diffusion equation

A high‐accuracy numerical approach for a nonhomogeneous time‐fractional diffusion equation with Neumann and Dirichlet boundary conditions is described in this paper. The time‐fractional derivative is described in the sense of Riemann‐Liouville and discretized by the backward Euler scheme. A fourth‐order optimal cubic B‐spline collocation (OCBSC) method is used to discretize the space variable. The stability analysis with respect to time discretization is carried out, and it is shown that the method is unconditionally stable. Convergence analysis of the method is performed. Two numerical examples are considered to demonstrate the performance of the method and validate the theoretical results. It is shown that the proposed method is of order O(Δx⁴ + Δt^{2 ? α}) convergence, where α ∈ (0,1) . Moreover, the impact of fractional‐order derivative on the solution profile is investigated. Numerical results obtained by the present method are compared with those obtained by the method based on standard cubic B‐spline collocation method. The CPU time for present numerical method and the method based on cubic B‐spline collocation method are provided.     

B‐spline solution of fractional integro partial differential equation with a weakly singular kernel

The main objective of the paper is to find the approximate solution of fractional integro partial differential equation with a weakly singular kernel. Integro partial differential equation (IPDE) appears in the study of viscoelastic phenomena. Cubic B‐spline collocation method is employed for fractional IPDE. The developed scheme for finding the solution of the considered problem is based on finite difference method and collocation method. Caputo fractional derivative is used for time fractional derivative of order α, . The given problem is discretized in both time and space directions. Backward Euler formula is used for temporal discretization. Collocation method is used for spatial discretization. The developed scheme is proved to be stable and convergent with respect to time. Approximate solutions are examined to check the precision and effectiveness of the presented method.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1565–1581, 2017     

Taylor‐Galerkin B‐spline finite element method for the one‐dimensional advection‐diffusion equation

The advection‐diffusion equation has a long history as a benchmark for numerical methods. Taylor‐Galerkin methods are used together with the type of splines known as B‐splines to construct the approximation functions over the finite elements for the solution of time‐dependent advection‐diffusion problems. If advection dominates over diffusion, the numerical solution is difficult especially if boundary layers are to be resolved. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method with emphasis on treatment of boundary conditions. Taylor‐Galerkin methods have been constructed by using both linear and quadratic B‐spline shape functions. Results shown by the method are found to be in good agreement with the exact solution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Operator time‐splitting techniques combined with quintic B‐spline collocation method for the generalized Rosenau–KdV equation

In this article, the generalized Rosenau–KdV equation is split into two subequations such that one is linear and the other is nonlinear. The resulting subequations with the prescribed initial and boundary conditions are numerically solved by the first order Lie–Trotter and the second‐order Strang time‐splitting techniques combined with the quintic B‐spline collocation by the help of the fourth order Runge–Kutta (RK‐4) method. To show the accuracy and reliability of the proposed techniques, two test problems having exact solutions are considered. The computed error norms L₂ and L_∞ with the conservative properties of the discrete mass Q(t) and energy E(t) are compared with those available in the literature. The convergence orders of both techniques have also been calculated. Moreover, the stability analyses of the numerical schemes are investigated.     

Numerical analysis nonlinear multi‐term time fractional differential equation with collocation method via fractional B‐spline

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.     

Numerical solution of Fokker‐Planck equation using the cubic B‐spline scaling functions

In this article a numerical technique is presented for the solution of Fokker‐Planck equation. This method uses the cubic B‐spline scaling functions. The method consists of expanding the required approximate solution as the elements of cubic B‐spline scaling function. Using the operational matrix of derivative, the problem will be reduced to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Numerical investigation of the solution of Fisher's equation via the B‐spline Galerkin method

The Galerkin method is used with quadratic B‐spline base functions to obtain the numerical solutions of Fisher's equation which is a one dimensional reaction‐diffusion equation. To observe the effects of reaction and diffusion, four test problems related to pulse disturbance, step disturbance, super‐speed wave and strong reaction are studied. A comparison is performed between the obtained numerical results and some earlier studies. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

A numerical method for one‐dimensional nonlinear Sine‐Gordon equation using collocation and radial basis functions

In this article, we propose a numerical scheme to solve the one‐dimensional undamped Sine‐Gordon equation using collocation points and approximating the solution using Thin Plate Splines (TPS) radial basis function (RBF). The scheme works in a similar fashion as finite difference methods. The results of numerical experiments are presented and are compared with analytical solutions to confirm the good accuracy of the presented scheme.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Finite element method by using quartic B‐splines

In this article, we discuss a kind of finite element method by using quartic B‐splines to solve Dirichlet problem for elliptic equations. Bivariate spline proper subspace of S(Δ) satisfying homogeneous boundary conditions on Type‐2 triangulations and quadratic B‐spline interpolating boundary functions are primarily constructed. Linear and nonlinear elliptic equations are solved by Galerkin quartic B‐spline finite element method. Numerical examples are provided to illustrate the proposed method is flexible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 818–828, 2011     

Numerical simulation of the nonlinear generalized time‐fractional Klein–Gordon equation using cubic trigonometric B‐spline functions

In this paper, an efficient numerical procedure for the generalized nonlinear time‐fractional Klein–Gordon equation is presented. We make use of the typical finite difference schemes to approximate the Caputo time‐fractional derivative, while the spatial derivatives are discretized by means of the cubic trigonometric B‐splines. Stability and convergence analysis for the numerical scheme are discussed. We apply our scheme to some typical examples and compare the obtained results with the ones found by other numerical methods. The comparison shows that our scheme is quite accurate and can be applied successfully to a variety of problems of applied nature.     

Finite differences and collocation methods for the solution of the two‐dimensional heat equation

In this article, we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two‐dimensional heat equation. We employ, respectively, second‐order and fourth‐order schemes for the spatial derivatives, and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is nonsingular. Numerical experiments carried out on serial computers show the unconditional stability of the proposed method and the high accuracy achieved by the fourth‐order scheme. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 54–63, 2001     

Orthogonal spline collocation scheme for multiterm fractional convection‐diffusion equation with variable coefficients

The orthogonal spline collocation (OSC) technique is an efficient way to solve a wide variety of problems that are modeled by ordinary and partial differential equations. In this article, by using OSC method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established for a class of two‐dimensional multiterm fractional convection‐diffusion reaction equation with variable coefficients. The optimal estimates in H^j (j = 0, 1, 2) norms at each time step are derived. Also, estimate in space is provided. At last, we provide some numerical results to verify the accuracy and efficiency of the proposed algorithm.