Modeling and Solution of Reaction-Diffusion Equations by Using the Quadrature and Singular Convolution Methods.

In the present work, polynomial, discrete singular convolution and sinc quadrature techniques are employed as the new techniques to derive accurate and efficient numerical solutions for the reaction-diffusion equations. Three models, Fitzhugh-Nagumo, Newell-Whitehead-Segel, and tumor growth models, were presented. The equations of three models are reduced to nonlinear ordinary differential equations by using different quadrature schemes. Then, Runge-Kutta fourth-order method is employed to solve nonlinear ordinary differential equations. In addition, the MATLAB program is used to solve these problems. Comparisons between the new methods and the existing ones are included, demonstrating the ease of implementation and efficiency. Also, the calculated results are supported by four various statistical errors. It is found that the rate of error reaches ≤ 10-6 in discrete singular convolution depending on regularized Shannon kernel which is better than others. Further, a parametric analysis is presented to discuss the influence of diffusion and reaction parameters on the solution.

Free vibration of a piezoelectric nanobeam resting on nonlinear Winkler-Pasternak foundation by quadrature methods.

This work introduces a numerical scheme for free vibration analysis of elastically supported piezoelectric nanobeam. Based on Hamilton principle, governing equations of the problem are derived. The problem is formulated for linear and nonlinear Winkler-Pasternak foundation type. Three differential quadrature techniques are employed to reduce the problem to an Eigen-value problem. The reduced system is solved iteratively. The natural frequencies of the beam are obtained. Numerical analysis is implemented to investigate computational characteristics affecting convergence, accuracy and efficiency of the proposed scheme. The obtained results agreed with the previous analytical and numerical ones. Furthermore, a parametric study is introduced to show influence of supporting conditions, two different electrical boundary conditions, material characteristics, foundation parameters, temperature change, external electric voltage, nonlocal parameter and beam length-to-thickness ratio on the values of natural frequencies and mode shapes of the problem.