Direct operational matrix approach for weakly singular Volterra integro-differential equations: application in theory of anomalous diffusion

In the current paper, we present an efficient direct scheme for weakly singular Volterra integro-differential equations arising in the theory of anomalous diffusion. The behavior of the system demonstrating the anomalous diffusion is significant for small times. The method is based on operational matrices of Chebyshev and Legendre polynomials with some techniques to reduce the total errors of the already existing schemes. The proposed scheme converts these equations into a linear system of algebraic equations. The main advantages of the method are high accuracy, simplicity of performing, and low storage requirement. The main focus of this study is to obtain an analytical explicit expression to estimate the error. Numerical results confirm the superiority and applicability of our scheme in comparison with other methods in the literature

Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity

An integro-differential equation, modeling dynamic fractional order viscoelasticity, with a Mittag-Leffler type convolution kernel is considered. A discontinuous Galerkin method, based on piecewise constant polynomials is formulated for temporal semidiscretization of the problem. Stability estimates of the discrete problem are proved, that are used to prove optimal order a priori error estimates. The theory is illustrated by a numerical example.Comment: 16 pages, 2 figure

A new approach for solving nonlinear Thomas-Fermi equation based on fractional order of rational Bessel functions

In this paper, the fractional order of rational Bessel functions collocation method (FRBC) to solve Thomas-Fermi equation which is defined in the semi-infinite domain and has singularity at $x = 0$ and its boundary condition occurs at infinity, have been introduced. We solve the problem on semi-infinite domain without any domain truncation or transformation of the domain of the problem to a finite domain. This approach at first, obtains a sequence of linear differential equations by using the quasilinearization method (QLM), then at each iteration solves it by FRBC method. To illustrate the reliability of this work, we compare the numerical results of the present method with some well-known results in other to show that the new method is accurate, efficient and applicable

Numerical solution of fractional Fredholm integro-differential equations by spectral method with fractional basis functions

This paper presents an efficient spectral method for solving the fractional Fredholm integro-differential equations. The non-smoothness of the solutions to such problems leads to the performance of spectral methods based on the classical polynomials such as Chebyshev, Legendre, Laguerre, etc, with a low order of convergence. For this reason, the development of classic numerical methods to solve such problems becomes a challenging issue. Since the non-smooth solutions have the same asymptotic behavior with polynomials of fractional powers, therefore, fractional basis functions are the best candidate to overcome the drawbacks of the accuracy of the spectral methods. On the other hand, the fractional integration of the fractional polynomials functions is in the class of fractional polynomials and this is one of the main advantages of using the fractional basis functions. In this paper, an implicit spectral collocation method based on the fractional Chelyshkov basis functions is introduced. The framework of the method is to reduce the problem into a nonlinear system of equations utilizing the spectral collocation method along with the fractional operational integration matrix. The obtained algebraic system is solved using Newton's iterative method. Convergence analysis of the method is studied. The numerical examples show the efficiency of the method on the problems with smooth and non-smooth solutions in comparison with other existing methods

New Solutions for System of Fractional Integro-Differential Equations and Abel’s Integral Equations by Chebyshev Spectral Method

Chebyshev spectral method based on operational matrix is applied to both systems of fractional integro-differential equations and Abel’s integral equations. Some test problems, for which the exact solution is known, are considered. Numerical results with comparisons are made to confirm the reliability of the method. Chebyshev spectral method may be considered as alternative and efficient technique for finding the approximation of system of fractional integro-differential equations and Abel’s integral equations

Fractional Bernstein operational matrices for solving integro-differential equations involved by Caputo fractional derivative

The present work is devoted to developing two numerical techniques based on fractional Bernstein polynomials, namely fractional Bernstein operational matrix method, to numerically solving a class of fractional integro-differential equations (FIDEs). The first scheme is introduced based on the idea of operational matrices generated using integration, whereas the second one is based on operational matrices of differentiation using the collocation technique. We apply the Riemann–Liouville and fractional derivative in Caputo’s sense on Bernstein polynomials, to obtain the approximate solutions of the proposed FIDEs. We also provide the residual correction procedure for both methods to estimate the absolute errors. Some results of the perturbation and stability analysis of the methods are theoretically and practically presented. We demonstrate the applicability and accuracy of the proposed schemes by a series of numerical examples. The numerical simulation exactly meets the exact solution and reaches almost zero absolute error whenever the exact solution is a polynomial. We compare the algorithms with some known analytic and semi-analytic methods. As a result, our algorithm based on the Bernstein series solution methods yield better results and show outstanding and optimal performance with high accuracy orders compared with those obtained from the optimal homotopy asymptotic method, standard and perturbed least squares method, CAS and Legendre wavelets method, and fractional Euler wavelet method

A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus

We present a new approach based on linear integro-differential operators with logarithmic kernel related to the Hadamard fractional calculus in order to generalize, by a parameter $\nu \in (0,1]$, the logarithmic creep law known in rheology as Lomnitz law (obtained for $\nu=1$). We derive the constitutive stress-strain relation of this generalized model in a form that couples memory effects and time-varying viscosity. Then, based on the hereditary theory of linear viscoelasticity, we also derive the corresponding relaxation function by solving numerically a Volterra integral equation of the second kind. So doing we provide a full characterization of the new model both in creep and in relaxation representation, where the slow varying functions of logarithmic type play a fundamental role as required in processes of ultra slow kinetics.Comment: 15 pages, 2 figures, to appear in Chaos, Solitons and Fractals (2017