Numerical investigation of Differential Biological-Models via GA-Kansa Method Inclusive Genetic Strategy

In this paper, we use Kansa method for solving the system of differential equations in the area of biology. One of the challenges in Kansa method is picking out an optimum value for Shape parameter in Radial Basis Function to achieve the best result of the method because there are not any available analytical approaches for obtaining optimum Shape parameter. For this reason, we design a genetic algorithm to detect a close optimum Shape parameter. The experimental results show that this strategy is efficient in the systems of differential models in biology such as HIV and Influenza. Furthermore, we prove that using Pseudo-Combination formula for crossover in genetic strategy leads to convergence in the nearly best selection of Shape parameter.Comment: 42 figures, 23 page

New Ultraspherical Wavelets Spectral Solutions for Fractional Riccati Differential Equations

We introduce two new spectral wavelets algorithms for solving linear and nonlinear fractional-order Riccati differential equation. The suggested algorithms are basically based on employing the ultraspherical wavelets together with the tau and collocation spectral methods. The main idea for obtaining spectral numerical solutions depends on converting the differential equation with its initial condition into a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. For the sake of illustrating the efficiency and the applicability of our algorithms, some numerical examples including comparisons with some algorithms in the literature are presented

Two Hybrid Methods for Solving Two-Dimensional Linear Time-Fractional Partial Differential Equations

A computationally efficient hybridization of the Laplace transform with two spatial discretization techniques is investigated for numerical solutions of time-fractional linear partial differential equations in two space variables. The Chebyshev collocation method is compared with the standard finite difference spatial discretization and the absolute error is obtained for several test problems. Accurate numerical solutions are achieved in the Chebyshev collocation method subject to both Dirichlet and Neumann boundary conditions. The solution obtained by these hybrid methods allows for the evaluation at any point in time without the need for time-marching to a particular point in time

A Modified Generalized Laguerre-Gauss Collocation Method for Fractional Neutral Functional-Differential Equations on the Half-Line

The modified generalized Laguerre-Gauss collocation (MGLC) method is applied to obtain an approximate solution of fractional neutral functional-differential equations with proportional delays on the half-line. The proposed technique is based on modified generalized Laguerre polynomials and Gauss quadrature integration of such polynomials. The main advantage of the present method is to reduce the solution of fractional neutral functional-differential equations into a system of algebraic equations. Reasonable numerical results are achieved by choosing few modified generalized Laguerre-Gauss collocation points. Numerical results demonstrate the accuracy, efficiency, and versatility of the proposed method on the half-line

Generalized Spectral Decomposition for Stochastic Non Linear Problems

International audienceWe present an extension of the Generalized Spectral Decomposition method for the resolution of non-linear stochastic problems. The method consists in the construction of a reduced basis approximation of the Galerkin solution and is independent of the stochastic discretization selected (polynomial chaos, stochastic multi-element or multiwavelets). Two algorithms are proposed for the sequential construction of the successive generalized spectral modes. They involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems. Compared to the classical Galerkin method, the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes. The methodology is detailed and tested on two model problems, the one-dimensional steady viscous Burgers equation and a two-dimensional non-linear diffusion problem. These examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space