(R1491) Numerical Solution of the Time-space Fractional Diffusion Equation with Caputo Derivative in Time by a-polynomial Method

In this paper, a novel type of polynomial is defined which is equipped with an auxiliary parameter a. These polynomials are a combination of the Chebyshev polynomials of the second kind. The approximate solution of each equation is assumed as the sum of these polynomials and then, with the help of the collocation points, the unknown coefficients of each polynomial, as well as auxiliary parameter, is obtained optimally. Now, by placing the optimal value of a in polynomials, the polynomials are obtained without auxiliary parameter, which is the restarted step of the present method. The time discretization is performed on fractional partial differential equations by L1 method. In the following, the convergence theorem of the method is proved

Existence of extremal solutions for fuzzy polynomials and their numerical solutions

In this paper, we consider the existence of a solution for fuzzy polynomials anx^n + an−1x^n−1 + · · · + a1x + a0 = x, where ai, i = 0, 1, 2, · · · , n and x are positive fuzzy numbers satisfying certain conditions. To this purpose, we use fixed point theory, applying results such as the well-known fixed point theorem of Tarski, presenting some results regarding the existence of extremal solutions to the above equation.Peer Reviewe

The He\u27s Variational Iteration Method for Solving the Integro-differential Parabolic Problem with Integral Conditions

In this paper, the variational iteration method is applied for finding the solution of an Integro-differential parabolic problem with integral conditions. Convergence of the proposed method is also discussed. Finally, some numerical examples are given to show the effectiveness of the proposed method

(R1951) Numerical Solution for a Class of Nonlinear Emden-Fowler Equations by Exponential Collocation Method

In this research, exponential approximation is used to solve a class of nonlinear Emden-Fowler equations. This method is based on the matrix forms of exponential functions and their derivatives using collocation points. To demonstrate the usefulness of the method, we apply it to some different problems. The numerical approximate solutions are compared with available (existing) exact (analytical) solutions to show the accuracy of the proposed method. The method has been checked with several examples to show its validity and reliability. The reported examples illustrate that the method is reasonably efficient and accurate

The Lie-group method based on radial basis functions for solving nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation in arbitrary domains

The aim of this paper is to introduce a new numerical method for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation. This method is combination of group preserving scheme (GPS) with radial basis functions (RBFs), which takes advantage of two powerful methods, one as geometric numerical integration method and the other meshless method. Thus, we introduce this method as the Lie-group method based on radial basis functions (LG–RBFs). In this method, we use Kansas approach to approximate the spatial derivatives and then we apply GPS method to approximate first-order time derivative. One of the important advantages of the developed method is that it can be applied to problems on arbitrary geometry with high dimensions. To demonstrate this point, we solve nonlinear GBBMB equation on various geometric domains in one, two and three dimension spaces. The results of numerical experiments are compared with analytical solutions and the method presented in Dehghan et al. (2014) to confirm the accuracy and efficiency of the presented method

Numerical Solution of Fractional Partial Differential Equations with Normalized Bernstein Wavelet Method

In this paper, normalized Bernstein wavelets are presented. Next, the fractional order integration and Bernstein wavelets operational matrices of integration are derived and finally are used for solving fractional partial differential equations. The operational matrices merged with the collocation method are used in order to convert fractional problems to a number of algebraic equations. In the suggested method the boundary conditions are automatically taken into consideration. An assessment of the error of function approximation based on the normalized Bernstein wavelet is also presented. Some numerical instances are given to manifest the versatility and applicability of the suggested method. Founded numerical results are correlated with the best reported results in the literature and the analytical solutions in order to prove the accuracy and applicability of the suggested method

Numerical Solution of Fuzzy Fractional Differential Equation By Haar Wavelet

In this paper, we deal with a wavelet operational method based on Haar wavelet to solve the fuzzy fractional differential equation in the Caputo derivative sense. To this end, we derive the Haar wavelet operational matrix of the fractional order integration. The given approach provides an efficient method to find the solution and its upper bond error. To complete the discussion, the convergence theorem is subsequently expressed in detail. So far, no paper has used the Harr wavelet method using generalized difference and fuzzy derivatives, and this is the first time we have done so. Finally, the presented examples reflect the accuracy and efficiency of the proposed method

The Traveling Wave Solution of the Fuzzy Linear Partial Differential Equation

In this paper we are going to obtain fuzzy traveling wave solutions for fuzzy linear partial differential equations by considering the type of generalized Hukuhara differentiability. In particular, the fuzzy traveling wave solutions for fuzzy Advection equation, fuzzy linear Diffusion equation, fuzzy Convection-Diffusion-Reaction equation, and fuzzy Klein-Gordon equation are obtained