Finding the one-loop soliton solution of the short-pulse equation by means of the homotopy analysis method

The homotopy analysis method is applied to the short-pulse equation in order to find an analytic approximation to the known exact solitary upright-loop solution. It is demonstrated that the approximate solution agrees well with the exact solution. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear solitary waves

Solitary smooth hump solutions of the Camassa-Holm equation by means of the homotopy analysis method

The homotopy analysis method is used to find a family of solitary smooth hump solutions of the Camassa-Holm equation. This approximate solution, which is obtained as a series of exponentials, agrees well with the known exact solution. This paper complements the work of Wu & Liao [Wu W, Liao S. Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos, Solitons & Fractals 2005;26:177-85] who used the homotopy analysis method to find a different family of solitary wave solutions

Solitary-wave solutions of the Degasperis-Procesi equation by means of the homotopy analysis method

The homotopy analysis method is applied to the Degasperis-Procesi equation in order to find analytic approximations to the known exact solitary-wave solutions for the solitary peakon wave and the family of solitary smooth-hump waves. It is demonstrated that the approximate solutions agree well with the exact solutions. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear solitary waves

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

(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

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

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