Solutions Of Fractional Differential Equations Research Articles

Caputo delta weakly fractional difference equations

In this paper, solutions of fractional difference equations with Caputo-type delta-based fractional difference operator of order $$\mu \sim 1$$ are compared with solutions of corresponding limit difference equations with usual first-order forward difference. It is shown that the limit initial value problems differ substantially when $$\mu \rightarrow 1^-$$ and $$\mu \rightarrow 1^+$$ . To derive convergence results, Gronwall type inequalities are proved for suitable fractional sum inequalities of general noninteger order. An illustrative example is also given.

ANALYSIS OF DYNAMIC BEHAVIOUR OF FRACTIONAL ORDER SIR EPIDEMIC MODEL OF CHILDHOOD DISEASES USING RVIM

In the given article, the solution of a Mathematical Model of epidemic in childhood diseases is presented. Reconstruction of Variational Iteration Method (RVIM) is used to analyze the model which inculcates the Laplace Transform to reconstruct VIM. This technique is generally used to get solutions of fractional differential equations of linear and non linear form. The result obtained has been simulated and discussed through graphs.

Novel approximate solution for fractional differential equations by the optimal variational iteration method

In this work, the optimal variational iteration method (OVIM) is used to solve partial and ordinary fractional differential equations. The standard variational iteration method (VIM) is reassessed by introducing a parameter that accelerates convergence. The value of the convergence acceleration coefficient is determined by calculating the residual of the parameter within the L2 norm. The numerical results proved that the proposed method gives much better and more accurate results than the standard VIM method, as the effect of adding the parameter to the VIM method and its calculation method was clear in improving the method and increasing the convergence of the solution, where we solved several examples of ordinary and partial fractional differential equations. In addition, the fractional Kawahara and foam drainage equations have also been solved. All results were obtained using the Mathematica®12 program. The results were also compared with the optimal Adomian decomposition method (ADM), and the proposed method showed higher efficiency and better accuracy.

Analytical solution of fractional differential equations by Akbari–Ganji’s method

According to the various and extensive applications of fractional calculus in a range of fields, such as engineering, biology, image processing, material science and economics, researchers have discovered new, simpler-to-use and more accurate approaches to solve these complicated problems today. In this article an analytical method called Akbari–Ganji’s method is proposed to solve nonlinear fractional differential equations. This method was used to solve nonlinear differential equations before but not fractional differential equations. In this method, first, the solution of the differential equation is considered as a polynomial with constant coefficients. Then, with the help of the initial conditions of the problem and also the Akbari–Ganji’s method’s boundary conditions, the final solution of the differential equation obtained by having the constant coefficients of the polynomial. The method was validated by comparing the answers of two nonlinear fractional differential equations to other methods. Some other examples also provided to confirm the strength and validity of the method.

The Existence and Uniqueness of Riccati Fractional Differential Equation Solution and Its Approximation Applied to an Economic Growth Model

This work proposes and investigates the existence and uniqueness of solutions of Riccati Fractional Differential Equation (RFDE) with constant coefficients using Banach’s fixed point theorem. This theorem is the uniqueness theorem of a fixed point on a contraction mapping of a norm space. Furthermore, the combined theorem of the Adomian Decomposition Method (ADM) and Kamal’s Integral Transform (KIT) is used to convert the solution of the Fractional Differential Equation (FDE) into an infinite polynomial series. In addition, the terms of an infinite polynomial series can be decomposed using ADM, which assumes that a function can be decomposed into an infinite polynomial series and nonlinear operators can be decomposed into an Adomian polynomial series. The final result of this study is to find a solution of the RFDE approach to the economic growth model with a quadratic cost function using the combined ADM and KIT. The results showed that the RFDE solution on the economic growth model using the combined ADM and KIT showed a very good performance. Furthermore, the numerical solution of RFDE on the economic growth model is presented at the end of this work.

Pricing of spread and exchange options in a rough jump–diffusion market

Asset dynamics with rough volatility recently received a great deal of attention in finance because they are consistent with empirical observations. This article provides a detailed analysis of the impact of roughness on prices of spread and exchange options. We consider a bivariate extension of the rough Heston model with jumps and derive the joint characteristic functions of asset log-returns under the risk neutral measure and under a measure using a risky asset as numeraire. These characteristic functions are expressed in terms of solutions of fractional differential equations (FDE’s). To infer these FDE’s, we rewrite the rough model as an infinite dimensional Markov process and propose a finite dimensional approximation. Next, we show that characteristic functions of log-returns admit a representation in terms of forward differential equations. FDE’s are obtained by passing to the limit. Spread and exchange options are valued with a two or a one dimensional discrete Fourier Transform. The numerical illustration reveals that considering a rough instead of Brownian volatility does not systematically increase exchange option prices.

A unified fixed point approach to study the existence of solutions for a class of fractional boundary value problems arising in a chemical graph theory.

A theory of chemical graphs is a part of mathematical chemistry concerned with the effects of connectedness in chemical graphs. Several researchers have studied the solutions of fractional differential equations using the concept of star graphs. They employed star graphs because their technique requires a central node with links to adjacent vertices but no edges between nodes. The purpose of this paper is to extend the method’s range by introducing the concept of an octane graph, which is an essential organic compound having the formula C8H18. In this manner, we analyze a graph with vertices annotated by 0 or 1, which is influenced by the structure of the chemical substance octane, and formulate a fractional boundary value problem on each of the graph’s edges. We use the Schaefer and Krasnoselskii fixed point theorems to investigate the existence of solutions to the presented boundary value problems in the framework of the Caputo fractional derivative. Finally, two examples are provided to highlight the importance of our results in this area of study.

Existence of Triple Positive Solutions to a Four-Point Boundary Value Problem of a Fractional Differential Equations

In this paper, the existence result of at least triple positive solutions to a boundary value problem of a fractional differential equations is achieved by means of the Avery-Peterson fixed point theorem and the careful analysis of the associate Green's function in which the derivative of unknown function is involved in the nonlinear term explicitly. An example illustrating our main result is given. Our results complement the previous work in the area of the positive solutions of fractional differential equations.

On differentiability of solutions of fractional differential equations with respect to initial data

In this paper, we deal with a Cauchy problem for a nonlinear fractional differential equation with the Caputo derivative of order $$\alpha \in (0, 1)$$ . As initial data, we consider a pair consisting of an initial point, which does not necessarily coincide with the inferior limit of the fractional derivative, and a function that determines the values of a solution on the interval from this inferior limit to the initial point. We study differentiability properties of the functional associating initial data with the endpoint of the corresponding solution of the Cauchy problem. Stimulated by recent results on the dynamic programming principle and Hamilton–Jacobi–Bellman equations for fractional optimal control problems, we examine so-called fractional coinvariant derivatives of order $$\alpha $$ of this functional. We prove that these derivatives exist and give formulas for their calculation.

Exact Solutions of Three Types of Conformable Fractional-Order Partial Differential Equations.

Fractional calculus is widely used in biology, control systems, and engineering, so it has been highly valued by scientists. Fractional differential equations are considered an important mathematical model that is widely used in science and technology to describe physical phenomena more accurately in terms of time memory and spatial interactions. The study of exact solutions of fractional differential equations helps to understand these complex physical phenomena and dynamic processes. The results show that the method is simple and clear. In addition, with the help of MAPL, we provide some 3D maps with accurate solutions.

Positive solutions for fractional differential equation at resonance under integral boundary conditions

Abstract By using the theory of fixed point index and spectral theory of linear operators, we study the existence of positive solutions for Riemann-Liouville fractional differential equations at resonance. Our approach will provide some new ideas for the study of this kind of problem.

Approximate Solution of Fractional Differential Equation by Quadratic Splines

In this article, we consider approximate solutions by quadratic splines for a fractional differential equation with two Caputo fractional derivatives, the orders of which satisfy 1<α<2 and 0<β<1. Numerical computing schemes of the two fractional derivatives based on quadratic spline interpolation function are derived. Then, the recursion scheme for numerical solutions and the quadratic spline approximate solution are generated. Two numerical examples are used to check the proposed method. Additionally, comparisons with the L1–L2 numerical solutions are conducted. For the considered fractional differential equation with the leading order α, the involved undetermined parameters in the quadratic spline interpolation function can be exactly resolved.

On the Stability and Numerical Scheme of Fractional Differential Equations with Application to Biology

The fractional differential equations involving different types of fractional derivatives are currently used in many fields of science and engineering. Therefore, the first purpose of this study is to investigate the qualitative properties including the stability, asymptotic stability, as well as Mittag–Leffler stability of solutions of fractional differential equations with the new generalized Hattaf fractional derivative, which encompasses the popular forms of fractional derivatives with non-singular kernels. These qualitative properties are obtained by constructing a suitable Lyapunov function. Furthermore, the second aim is to develop a new numerical method in order to approximate the solutions of such types of equations. The developed method recovers the classical Euler numerical scheme for ordinary differential equations. Finally, the obtained analytical and numerical results are applied to a biological nonlinear system arising from epidemiology.

Existence and Ulam stability of solutions for Caputo-Hadamard fractional differential equations

In this paper, we study the existence of solutions for fractional differential equations with the Caputo-Hadamard fractional derivative of order 2 (1, 2]. The uniqueness result is proved via Banach’s contraction mapping principle and the existence results are established by using the Schauder’s fixed point theorem. Furthermore, the Ulam-Hyers and Ulam-Hyers-Rassias stability of the proposed equation is employed. Some examples are given to illustrate the results.

Convergence and Stability of a Novel M ‐Iterative Algorithm with an Application

In this manuscript, we propose a novel three - step iteration scheme called M - iteration to approximate the invariant points for the class of weak contractions in the sense of Berinde and obtain that M - iteration strongly converges to one and only one fixed point for Berinde mappings. Proving ‘almost - stability’ of M - iteration, we compare the M - iterative scheme with other chief iterative algorithms, namely. Picard, Mann, Ishikawa, Noor, S, normal-S, Abbas, Thakur, Varat, Ullah and F ∗ and claim that the framed iterative procedure converges to the invariant points of weak contractions at faster rate than other vital algorithms. Some numerical illustrations are adduced to strengthen our claim. We further ascertain data dependency through M - iteration. Finally, we establish the solution of Caputo type fractional differential equation as an application and exhibit that M - fixed point procedure converges to the solution of a fractional differential equation. The obtained results are not only new but, also, extend the scope of previous findings.

Fractional Dynamics with Depreciation and Obsolescence: Equations with Prabhakar Fractional Derivatives

In economics, depreciation functions (operator kernels) are certain decreasing functions, which are assumed to be equal to unity at zero. Usually, an exponential function is used as a depreciation function. However, exponential functions in operator kernels do not allow simultaneous consideration of memory effects and depreciation effects. In this paper, it is proposed to consider depreciation of a non-exponential type, and simultaneously take into account memory effects by using the Prabhakar fractional derivatives and integrals. Integro-differential operators with the Prabhakar (generalized Mittag-Leffler) function in the kernels are considered. The important distinguishing features of the Prabhakar function in operator kernels, which allow us to take into account non-exponential depreciation and fading memory in economics, are described. In this paper, equations with the following operators are considered: (a) the Prabhakar fractional integral, which contains the Prabhakar function as the kernels; (b) the Prabhakar fractional derivative of Riemann–Liouville type proposed by Kilbas, Saigo, and Saxena in 2004, which is left inverse for the Prabhakar fractional integral; and (c) the Prabhakar operator of Caputo type proposed by D’Ovidio and Polito, which is also called the regularized Prabhakar fractional derivative. The solutions of fractional differential equations with the Prabhakar operator and its special cases are suggested. The asymptotic behavior of these solutions is discussed.

The Fractional Differential Equations with Uncertainty by Conformable Derivative

We provide a fractional order fuzzy fractional differential equation q ∈ (0, 1]. A fuzzy fractional integral and a fuzzy conformable derivative are shown and proved. To prove fuzzy solutions for fractional differential equations with fuzzy beginning values and deterministic or fuzzy functions, two alternative techniques are used. The application has been submitted.

Numerical solution of fractional differential equations with Caputo derivative by using numerical fractional predict\u2013correct technique

Fractional differential equations have recently demonstrated their importance in a variety of fields, including medicine, applied sciences, and engineering. The main objective of this study is to propose an Adams-type multistep method for solving differential equations of fractional order. The method is developed by implementing the Lagrange interpolation and taking into account the idea of the Adams–Moulton method for fractional case. The fractional derivative applied in this study is in the Caputo derivative operator. The analysis of the proposed method is presented in terms of order of the method, order of accuracy, and convergence analysis, with the proposed method being proved to converge. The stability of the method is also examined, where the stability regions appear to be symmetric to the real axis for various values of α. In order to validate the competency of the proposed method, several numerical examples for solving linear and nonlinear fractional differential equations are included. The method will be presented in the numerical predict–correct technique for the condition where alpha in (0,1), in which α represents the order of fractional derivatives of D^{alpha }y(t).

The numerical solutions of fractional differential equations with beta-derivative

The numerical solutions of fractional differential equations with beta-derivative

A Survey on the Oscillation of Solutions for Fractional Difference Equations

In this paper, we present a systematic study concerning the developments of the oscillation results for the fractional difference equations. Essential preliminaries on discrete fractional calculus are stated prior to giving the main results. Oscillation results are presented in a subsequent order and for different types of equations. The investigation was carried out within the delta and nabla operators.