Fractional Euler Method Research Articles

FRACTIONAL EULER METHOD; AN EFFECTIVE TOOL FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS

Through this article, a numerical scheme based upon the modified fractional Euler method (MFEM) is introduced to find the numericalsolutions of linear and nonlinear systems of fractional differential equations (SFDEs) as well as nonlinear multi-order fractionaldifferential equations (MOFDEs). The fractional derivatives are defined by Caputo. The proposed algorithm is very simple and providesthe solutions directly without linearization, perturbations or any other assumptions. Illustrating examples with numerical comparisonsbetween the proposed algorithm and the exact and/or fourth order Runge Kutta method (RK4) are given to reveal the efficiency and theaccuracy of our algorithm.

Results on initial value problems for random fuzzy fractional functional differential equations

In this paper random fuzzy fractional functional differential equations (RFFFDEs) with Caputo generalized Hukuhara differentiability are introduced. We present existence and uniqueness results for RFFFDEs using the idea of successive approximations. The behaviour of solutions when the data of the equation are subjected to errors is discussed. Furthermore, the solution to random fuzzy fractional functional initial value problem under Caputo-type fuzzy fractional derivatives by a modified fractional Euler method (MFEM) is presented. The results are illustrated with examples.

Absolute Stability for a Fractional Numerical Algorithm

In this paper, analysis of absolute stability is presented for a Fractional Euler Method found in literature. Both interval and region for absolute stability shown graphically have been found equivalent to absolute stability for classical Euler Method in case of integer-order ordinary differential equations.

Fractional Differential Systems: A Fuzzy Solution Based on Operational Matrix of Shifted Chebyshev Polynomials and Its Applications

In this paper, a new formula of fuzzy Caputo fractional-order derivatives $(0 < v \leq 1)$ in terms of shifted Chebyshev polynomials is derived. The proposed approach introduces a shifted Chebyshev operational matrix in combination with a shifted Chebyshev tau technique for the numerical solution of linear fuzzy fractional-order differential equations. The main advantage of the proposed approach is that it simplifies the problem alike in solving a system of fuzzy algebraic linear equations. An approximated error bound between the exact solution and the proposed fuzzy solution with respect to the number of fuzzy rules and solution errors is derived. Furthermore, we also discuss the convergence of the proposed method from the fuzzy perspective. Experimentally, we show the strength of the proposed method in solving a variety of fractional differential equation models under uncertainty encountered in engineering and physical phenomena (i.e., viscoelasticity, oscillations, and resistor–capacitor (RC) circuits). Comparisons are also made with solutions obtained by the Laguerre polynomials and the fractional Euler method.

Solving interval-valued fractional initial value problems under Caputo gH-fractional differentiability

In this paper interval-valued fractional differential equations (IFDEs) under Caputo generalized Hukuhara differentiability are introduced. We present existence and uniqueness results for IFDEs with a Krasnoselskii–Krein-type condition. The solution to interval-valued fractional initial value problems under Caputo-type interval-valued fractional derivatives by a modified fractional Euler method (MFEM) and a modified Adams–Bashforth–Moulton method (MABMM) is also presented.

Numerical Investigations on Hybrid Fuzzy Fractional Differential Equations by Improved Fractional Euler Method

In this paper, the improved Euler method is used for solving hybrid fuzzy fractional differential equations (HFFDE) of order $q \in (0, 1) $ under Caputo-type fuzzy fractional derivatives. This method is based on the fractional Euler method and generalized Taylor's formula. The accuracy and efficiency of the proposed method is demonstrated by solving numerical examples.

Fuzzy fractional functional integral and differential equations

This paper is devoted to considering the existence and uniqueness results for fuzzy fractional functional integral equations employing the contraction principle. Moreover, the approach is followed to prove the existence and uniqueness of solutions for the fuzzy fractional functional initial value problem under Caputo-type fuzzy fractional derivatives. Finally, the solution to fuzzy fractional functional initial value problem under Caputo-type fuzzy fractional derivatives by a modified fractional Euler method (MFEM) is presented. The method is illustrated by solving some examples.

Modified fractional Euler method for solving Fuzzy sequential Fractional Initial Value Problem under H-differentiability

In this paper, the solution to Fuzzy sequential Fractional Initial Value Problem [FFIVP] under Caputo type fuzzy fractional derivatives by a modified fractional Euler method is presented. The Caputo-type fuzzy fractional derivatives are defined based on Hukuhara difference and strongly generalized fuzzy differentiability. The modified fractional Euler method based on a generalized Taylor's formula and a modified trapezoidal rule is used for solving initial value problem under fuzzy sequential fractional differential equation of order $0< \beta< 1$. Solving two examples of linear and nonlinear FFIVP illustrates the method.

The Finite Difference Methods for Fractional Ordinary Differential Equations

Fractional finite difference methods are useful to solve the fractional differential equations. The aim of this article is to prove the stability and convergence of the fractional Euler method, the fractional Adams method and the high order methods based on the convolution formula by using the generalized discrete Gronwall inequality. Numerical experiments are also presented, which verify the theoretical analysis.

Modified fractional Euler method for solving Fuzzy Fractional Initial Value Problem

In this paper, the solution to Fuzzy Fractional Initial Value Problem [FFIVP] under Caputo-type fuzzy fractional derivatives by a modified fractional Euler method is presented. The Caputo-type fuzzy fractional derivatives are defined based on Hukuhara difference and strongly generalized fuzzy differentiability. The modified fractional Euler method based on a generalized Taylor’s formula and a modified trapezoidal rule is used for solving initial value problem under fuzzy fractional differential equation of order β∈0,1. Solving two examples of linear and nonlinear FFIVP illustrates the method.