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It is known that Genocchi polynomials have some advantages over classical orthogonal polynomials in approximating function, such as lesser terms and smaller coefficients of individual terms. In this paper, we apply a new operational matrix via Genocchi polynomials to solve fractional integro-differential equations (FIDEs). We also derive the expressions for computing Genocchi coefficients of the integral kernel and for the integral of product of two Genocchi polynomials. Using the matrix approach, we further derive the operational matrix of fractional differentiation for Genocchi polynomial as well as the kernel matrix. We are able to solve the aforementioned class of FIDE for the unknown function f(x). This is achieved by approximating the FIDE using Genocchi polynomials in matrix representation and using the collocation method at equally spaced points within interval [0,1]. This reduces the FIDE into a system of algebraic equations to be solved for the Genocchi coefficients of the solution f(x). A few numerical examples of FIDE are solved using those expressions derived for Genocchi polynomial approximation. Numerical results show that the Genocchi polynomial approximation adopting the operational matrix of fractional derivative achieves good accuracy comparable to some existing methods. In certain cases, Genocchi polynomial provides better accuracy than the aforementioned methods.

An effective collocation method based on Genocchi operational matrix for solving generalized fractional pantograph equations with initial and boundary conditions is presented. Using the properties of Genocchi polynomials, we derive a new Genocchi delay operational matrix which we used together with the Genocchi operational matrix of fractional derivative to approach the problems. The error upper bound for the Genocchi operational matrix of fractional derivative is also shown. Collocation method based on these operational matrices is applied to reduce the generalized fractional pantograph equations to a system of algebraic equations. The comparison of the numerical results with some existing methods shows that the present method is an excellent mathematical tool for finding the numerical solutions of generalized fractional pantograph equations.

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Abstract:In this paper, two-dimensional Genocchi polynomials and the Ritz-Galerkin method were developed to investigate the Fractional Diffusion Wave Equation (FDWE) and the Fractional Klein-Gordon Equation (FKGE). A satisfier function that satisfies all the initial and boundary conditions was used. A linear system of algebraic equations was obtained for the considered equation with the help of two-dimensional Genocchi polynomials along with the Ritz-Galerkin method. The FDWE and FKGE, including the nonlinear case, were reduced to solve the linear system of the algebraic equation. Hence, the proposed method was able to greatly reduce the complexity of the problems and provide an accurate solution. The effectiveness of the proposed technique is demonstrated through several examples.

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In this paper, we develop a new, simple, and accurate scheme to obtain approximate solution for nonlinear differential equation in the sense of Caputo-Fabrizio operator. To derive this new predictor-corrector scheme, which suits on Caputo-Fabrizio operator, firstly, we obtain the corresponding initial value problem for the differential equation in the Caputo-Fabrizio sense. Hence, by fractional Euler method and fractional trapeziodal rule, we obtain the predictor formula as well as corrector formula. Error analysis for this new method is derived. To test the validity and simplicity of this method, some illustrative examples for nonlinear differential equations are solved. KEYWORDSAdams-Bashforth-Moulton method, Caputo fractional derivative, Caputo-Fabrizio operator, nonlinear differential equation, predictor-corrector scheme INTRODUCTIONIn 2015, Michele Caputo and Mauro Fabrizio introduced a new definition of Caputo derivative, 1 which was called Caputo-Fabrizio fractional derivative. The main characteristics of this new definition are including regular kernel, having two different representations for temporal and spatial variables and as memory operator. Although it was verified that the operator does not fit the usual concepts neither for fractional nor for integer derivative/integral, 2 the new operator was successfully applied in describing the control movement of waves on the area of shallow water, 3 unsteady flows of an incompressible Maxwell fluid, 4 anomalous diffusion phenomena, 5 diffusion and the diffusion-advection equation, 6 epidemiological model for computer viruses, 7 model of circadian rhythms, 8 and many more phenomena. In this research direction, some existing numerical methods have been extended to the problem defined in the sense of this new Caputo-Fabrizio operator. Among them is the combination of the Laplace transform and homotopy analysis method to solve the so-called fractional partial differential equation problem with Caputo-Fabrizio operator. 9 Other methods include multiqaudric (MQ)-RBF collocation method, 10 discretization scheme, 11 fractional Adams-Bashforth method, 12 and three-step fractional Adams-Bashforth method. 13 However, since Caputo-Fabrizio operator is relatively new, there are still relatively limited works that have been done to obtain the simple, reliable, and accurate solution for the problem defined in Caputo-Fabrizio operator.On the other hand, predictor-corrector scheme had been extended to solve fractional differential equations. 14 Since that, some modification was done to obtain a better accurate scheme or more suitable to different problems, such as Abbreviation used: Fractional ordinary differential equation (FODE).