Accurate spectral solutions of first and second-order initial value problems by the ultraspherical wavelets-Gauss collocation method

In this paper, we present an ultraspherical wavelets-Gauss collocation method for obtaining direct solutions of first- and second-order nonlinear differential equations subject to homogenous and nonhomogeneous initial conditions. The properties of ultraspherical wavelets are used to reduce the differential equations with their initial conditions to systems of algebraic equations, which then must be solved by using suitable numerical solvers. The function approximations are spectral and have been chosen in such a way that make them easy to calculate the expansion coefficients of the thought-for solutions. Uniqueness and convergence of the proposed function approximation is discussed. Four illustrative numerical examples are considered and these results are comparing favorably with the analytic solutions and proving more accurate than those discussed by some other existing techniques in the literature

An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method

In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.Comment: 34 pages, 13 figures, Published in "Computer Physics Communications

General theory for integer-type algorithm for higher order differential equations

Based on functional analysis, we propose an algorithm for finite-norm solutions of higher-order linear Fuchsian-type ordinary differential equations (ODEs) P(x,d/dx)f(x)=0 with P(x,d/dx):=[\sum_m p_m (x) (d/dx)^m] by using only the four arithmetical operations on integers. This algorithm is based on a band-diagonal matrix representation of the differential operator P(x,d/dx), though it is quite different from the usual Galerkin methods. This representation is made for the respective CONSs of the input Hilbert space H and the output Hilbert space H' of P(x,d/dx). This band-diagonal matrix enables the construction of a recursive algorithm for solving the ODE. However, a solution of the simultaneous linear equations represented by this matrix does not necessarily correspond to the true solution of ODE. We show that when this solution is an l^2 sequence, it corresponds to the true solution of ODE. We invent a method based on an integer-type algorithm for extracting only l^2 components. Further, the concrete choice of Hilbert spaces H and H' is also given for our algorithm when p_m is a polynomial or a rational function with rational coefficients. We check how our algorithm works based on several numerical demonstrations related to special functions, where the results show that the accuracy of our method is extremely high.Comment: Errors concerning numbering of figures are fixe

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