Numerical solution of fredholm fractional integro-differential equation with right-sided caputo’s derivative using bernoulli polynomials operational matrix of fractional derivative

In this article, fractional integro-differential equation (FIDE) of Fredholm type involving right-sided Caputo’s fractional derivative with multi-fractional orders is considered. Analytical expressions of the expansion coefficient ck by Bernoulli polynomials approximation have been derived for both approximation of single- and double-variable function. The Bernoulli polynomials operational matrix of right-sided Caputo’s fractional derivative Pα −;B is derived. By approximating each term in the Fredholm FIDE with right-sided Caputo’s fractional derivative in terms of Bernoulli polynomials basis, the equation is reduced to a system of linear algebraic equation of the unknown coefficients ck. Solving for the coefficients produces the approximate solution for this special type of FIDE

Homotopy Perturbation Transform method for solving the partial and the time-fractional differential equations with variable coefficients

In this paper, we present the exact solutions of the Parabolic-like equations and Hyperbolic-like equations with variable coefficients, by using Homotopy perturbation transform method (HPTM). Finally, we extend the results to the time-fractional differential equations. Keywords: Caputo’s fractional derivative, fractional differential equations, homotopy perturbation transform method, hyperbolic-like equation, Laplace transform, parabolic-like equation

Positive solutions of two-point boundary value problems of nonlinear fractional differential equation at resonance

This paper is concerned with a kind of nonlinear fractional differential boundary value problem at resonance with Caputo's fractional derivative. Our main approach is the recent Leggett-Williams norm-type theorem for coincidences due to O'Regan and Zima. The most interesting point is the acquisition of positive solutions for fractional differential boundary value problem at resonance. Moreover, an example is constructed to show that our result here is valid

A Method to Solve One-dimensional Nonlinear Fractional Differential Equation Using B-Polynomials

In this article, the fractional Bhatti-Polynomial bases are applied to solve one-dimensional nonlinear fractional differential equations (NFDEs). We derive a semi-analytical solution from a matrix equation using an operational matrix which is constructed from the terms of the NFDE using Caputo’s fractional derivative of fractional B-polynomials (B-polys). The results obtained using the prescribed method agree well with the analytical and numerical solutions presented by other authors. The legitimacy of this method is demonstrated by using it to calculate the approximate solutions to four NFDEs. The estimated solutions to the differential equations have also been compared with other known numerical and exact solutions. It is also noted that for solving the NFDEs, the present method provides a higher order of precision compared to the various finite difference methods. The current technique could be effortlessly extended to solving complex linear, nonlinear, partial, and fractional differential equations in multivariable problems

Existence of positive solutions to a coupled system of fractional hybrid differential equations

where Dα is the Caputo’s fractional derivative of order α ,1 0 and the functions f : j × R × R → R , f (0,0) = 0 and g : j × R× R → R satisfy certain conditions. The proof of the existence theorem is based on a coupled fixed-point theorem of Krasnoselskii type, which extends a fixed-point theorem of Burton. Finally, our results are illustrated by providing a counter example

DGJ method for fractional initial-value problems

In this paper, a new iterative method (DGJM) is used to solve the nonlinear fractional initial-value problems(fIVPs). The fractional derivative is described in the Caputo sense. Approximate analytical solutions of the fIVPs are obtained. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the approach

Efficient schemes on solving fractional integro-differential equations

Fractional integro-differential equation (FIDE) emerges in various modelling of physical phenomena. In most cases, finding the exact analytical solution for FIDE is difficult or not possible. Hence, the methods producing highly accurate numerical solution in efficient ways are often sought after. This research has designed some methods to find the approximate solution of FIDE. The analytical expression of Genocchi polynomial operational matrix for left-sided and right-sided Caputo’s derivative and kernel matrix has been derived. Linear independence of Genocchi polynomials has been proved by deriving the expression for Genocchi polynomial Gram determinant. Genocchi polynomial method with collocation has been introduced and applied in solving both linear and system of linear FIDE. The numerical results of solving linear FIDE by Genocchi polynomial are compared with certain existing methods. The analytical expression of Bernoulli polynomial operational matrix of right-sided Caputo’s fractional derivative and the Bernoulli expansion coefficient for a two-variable function is derived. Linear FIDE with mixed left and right-sided Caputo’s derivative is first considered and solved by applying the Bernoulli polynomial with spectral-tau method. Numerical results obtained show that the method proposed achieves very high accuracy. The upper bounds for th