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     

The use of cubic B‐spline scaling functions for solving the one‐dimensional hyperbolic equation with a nonlocal conservation condition

Problems for parabolic partial differential equations with nonlocal boundary conditions have been studied in many articles, but boundary value problems for hyperbolic partial differential equations have so far remained nearly uninvestigated. In this article a numerical technique is presented for the solution of a nonclassical problem for the one‐dimensional wave equation. This method uses the cubic B‐spline scaling functions. Some numerical results are reported to support our study. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

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.     

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.     

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 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     

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     

Solution and stability of a cubic functional equation

In this paper, we investigate the general solution and the stability of a cubic functional equation f（x ＋ ny） ＋ f（x - ny） ＋ f（nx） = n^2 f（x ＋ y） ＋ n^2 f（x - y）＋ （n^3 - 2n^2 ＋ 2）f（x）,where n ≥ 2 is an integer. Furthermore, we prove the stability by the fixed point method.     

Monotonic positive solution of a nonlinear quadratic functional integral equation

The existence of positive monotonic solutions, in the class of continuous functions, for some nonlinear quadratic integral equation have been studied in [4], [5], [6], [7] and [8]. Here we are concerning with a nonlinear quadratic integral equation of Volterra type and we shall prove the existence of at least one L₁-positive monotonic solution for that equation under Carathèodory condition.     

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     

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.     

On quadratic integral equation of fractional orders

We present an existence theorem for a nonlinear quadratic integral equations of fractional orders, arising in the queuing theory and biology, in the Banach space of real functions defined and continuous on a bounded and closed interval. The concept of measure of noncompactness and a fixed point theorem due to Darbo are the main tool in carrying out our proof.     

On existence and asymptotic behaviour of solutions of a functional integral equation

The paper contains a result on the existence and asymptotic behaviour of solutions of a functional integral equation. That result is proved under rather general hypotheses. The main tools used in our considerations are the concept of a measure of noncompactness and the classical Schauder fixed point principle. The investigations of the paper are placed in the space of continuous and tempered functions on the real half-line. We prove an existence result which generalizes several ones concerning functional integral equations and obtained earlier by other authors. The applicability of our result is illustrated by some examples.     

Numerical solution of the nonlinear Fredholm integral equations by positive definite functions

A numerical method for solving the nonlinear Fredholom integral equations is presented. The method is based on interpolation by radial basis functions (RBF) to approximate the solution of the Fredholm nonlinear integral equations. Several examples are given and numerical examples are presented to demonstrate the validity and applicability of the method.     

Global asymptotic stability of solutions of a functional integral equation

Using the technique of measures of noncompactness we prove a theorem on the existence and global asymptotic stability of solutions of a functional integral equation. The investigations are placed in the Banach space of real functions defined, continuous and bounded on an unbounded interval. A few realizations of the result obtained are indicated.     

Numerical solutions of the coupled Burgers’ equation by the Galerkin quadratic B‐spline finite element method

In this paper, a coupled Burgers’ equation has been numerically solved by a Galerkin quadratic B‐spline FEM. The performance of the method has been examined on three test problems. Results obtained by the method have been compared with known exact solution and other numerical results in the literature. A Fourier stability analysis of the method is also investigated. Copyright © 2013 John Wiley & Sons, Ltd.     

B‐spline spectral method for constrained fractional optimal control problems

In this paper, we present a direct B‐spline spectral collocation method to approximate the solutions of fractional optimal control problems with inequality constraints. We use the location of the maximum of B‐spline functions as collocation points, which leads to sparse and nonsingular matrix B whose entries are the values of B‐spline functions at the collocation points. In this method, both the control and Caputo fractional derivative of the state are approximated by B‐spline functions. The fractional integral of these functions is computed by the Cox‐de Boor recursion formula. The convergence of the method is investigated. Several numerical examples are considered to indicate the efficiency of the method.     

An efficient cubic B-spline and bicubic B-spline collocation method for numerical solutions of multidimensional nonlinear stochastic quadratic integral equations

This paper aims to develop a novel numerical approach on the basis of B-spline collocation method to approximate the solution of one-dimensional and two-dimensional nonlinear stochastic quadratic integral equations. The proposed approach is based on the hybrid of collocation method, cubic B-spline, and bi-cubic B-spline interpolation and Itô approximation. Using this method, the problem solving turns into a nonlinear system solution of equations that is solved by a suitable numerical method. Also, the convergence analysis of this numerical approach has been discussed. In the end, examples are given to test the accuracy and the implementation of the method. The results are compared with the results obtained by other methods to verify that this method is accurate and efficient.     

Noncompact‐type Krasnoselskii fixed‐point theorems and their applications

In this paper, we first establish some user‐friendly versions of fixed‐point theorems for the sum of two operators in the setting that the involved operators are not necessarily compact and continuous. These fixed‐point results generalize, encompass, and complement a number of previously known generalizations of the Krasnoselskii fixed‐point theorem. Next, with these obtained fixed‐point results, we study the existence of solutions for a class of transport equations, the existence of global solutions for a class of Darboux problems on the first quadrant, the existence and/or uniqueness of periodic solutions for a class of difference equations, and the existence and/or uniqueness of solutions for some kind of perturbed Volterra‐type integral equations. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical solutions of nonlinear 3‐dimensional Volterra integral‐differential equations with 3D‐block‐pulse functions

This paper studies nonlinear 3‐dimensional Volterra integral‐differential equations, by implementing 3‐dimensional block‐pulse functions. First, we prove a theorem and corollary about sufficient condition for the minimum of mean square error under the block pulse coefficients and uniqueness of solution of the nonlinear Volterra integral‐differential equations. Then, we convert the main problem to a nonlinear system to the 3‐dimensional block‐pulse functions. In addition, illustrative examples are included to demonstrate the validity and applicability of the presented method.