Analyze implicit fractional differential equations using the AB-Caputo fractional derivative

The aim of the present analysis is to apply the Adomian decomposition method and He's variational method for the approximate analytical solution of a nonlinear ordinary fractional differential equation. The solutions obtained by the above two methods have been numerically evaluated and presented in the form of tables and also compared with the exact solution. It was found that the results obtained by the above two methods are in excellent agreement with the exact solution. Finally, a surface plot of the approximate solutions of the fractional differential equation by the above two methods is drawn for 0⩽t⩽2 and 1<α⩽2.

Realizing the behavior of the solution in the asymptotic situations is essential for repetitive applications in the control theory and modeling of the real-world systems. This study discusses a robust and definitive attitude to find the interval approximate asymptotic solutions of fractional differential equations (FDEs) with the Atangana-Baleanu (A-B) derivative. In fact, such critical tasks require to observe precisely the behavior of the noninterval case at first. In this regard, we initially shed light on the noninterval cases and analyze the behavior of the approximate asymptotic solutions, and then, we introduce the A-B derivative for FDEs under interval arithmetic and develop a new and reliable approximation approach for fractional interval differential equations with the interval A-B derivative to get the interval approximate asymptotic solutions. We exploit Laplace transforms to get the asymptotic approximate solution based on the interval asymptotic A-B fractional derivatives under interval arithmetic. The techniques developed here provide essential tools for finding interval approximation asymptotic solutions under interval fractional derivatives with nonsingular Mittag-Leffler kernels. Two cases arising in the real-world systems are modeled under interval notion and given to interpret the behavior of the interval approximate asymptotic solutions under different conditions as well as to validate this new approach. This study highlights the importance of the asymptotic solutions for FDEs regardless of interval or noninterval parameters.

&lt;abstract&gt;&lt;p&gt;In the past, the existence and uniqueness of the solutions of fractional differential equations have been investigated by many researchers theoretically in various approaches in the literature. In this paper, there is no discussion of the existence of solutions for the nonlinear differential equations with fractal fractional operators. The objective of this work is to present novel contraction approaches, notably the $ \varpropto $-$ \psi $-contraction $ \varpropto $-type of the $ \tilde{\texttt{F}} $-contraction, within the context of $ \hat{F} $-metric and orbital metric spaces. The aim of this study is to illustrate certain fixed point theorems that offer a new and direct approach to establish the existence and uniqueness of the solution to the general partial differential equations by employing the fractal fractional operators.&lt;/p&gt;&lt;/abstract&gt;

Differential equations of fractional order are widely used in physics, chemistry as well as engineering fields, this is the main reason that the approximate solution of fractional differential equations becomes a hot topic. In this paper, a numerical scheme for a class of fractional boundary value problems (FBVPs) is presented. In this approach, the FBVPs are expressed in terms of Caputo's fractional derivative. This scheme is based on exponential spline functions consisting of a polynomial part of degree one and an exponential part. For convergence analysis of this method, it is assumed that the exact solution of fractional boundary value problem belongs to a class of $$C^{6}$$C6-functions. Numerical examples are considered to illustrate the practical usefulness of this method and comparison show that this scheme is more accurate than the existing method Zahra and Elkholy (Numer Algorithms 59:373---391, 2012).

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

We propose an Adomian decomposition method to solve a class of nonlinear differential equations of fractional‐order with modified Caputo derivatives and integral boundary conditions. Our approach uses the integral boundary conditions to derive an equivalent nonlinear Volterra integral equation before establishing existence and uniqueness of solution and a recursion scheme for the solution. The convergence of the method is proved and an error analysis given. Two numerical examples are solved by obtaining a rapidly converging sequence of analytical functions to the solution.

In this paper, we investigate the existence of solutions for implicit impulsive fractional order differential equations with non-local conditions. An example is included to prove the applicability of the results.

This paper is devoted to studying on the locations of zeros of related integral operators and the solutions of some ordinary and fractional differential equations. We generalize Sturm and Picone's theorems and Leighton and Levin's criteria. Moreover, we share some oscillation and disconjugacy criteria for the solutions of ordinary second‐order Sturm–Liouville and fractional differential equations. Finally, we introduce some properties of the solutions of fractional differential equations.

In this paper, we establish sufficient conditions for the existence and uniqueness of solutions for a boundary value problem of fractional differential equations with nonlocal and average type integral boundary conditions. The Leray–Schauder nonlinear alternative, Krasnoselskii’s fixed point theorem and Banach’s fixed point theorem together with Holder inequality are applied to construct proofs for the main results. Examples illustrating the obtained results are also presented.

The solutions of fractional differential equations (FDEs) have a natural singularity at the initial point. The accuracy of their numerical solutions is lower than the accuracy of the numerical solutions of FDEs whose solutions are differentiable functions. In the present paper we propose a method for improving the accuracy of the numerical solutions of ordinary linear FDEs with constant coefficients which uses the fractional Taylor polynomials of the solutions. The numerical solutions of the two-term and three-term FDEs are studied in the paper.

In this paper, we study the existence of solutions for fractional differential equations of arbitrary order with multi-point multi-term Riemann-Liouville type integral boundary conditions involving two indices. The Riemann-Liouville type integral boundary conditions considered in the problem address a more general situation in contrast to the case of a single index. Our results are based on standard fixed point theorems. Some illustrative examples are also presented. MSC:26A33, 34A08.

This work is devoted to the noncontinuability to infinity of solutions for superlinear fractional differential equations of arbitrary order, considering the Riemann–Liouville derivative. As an application, we establish the asymptotic behavior of nonoscillatory solutions of such equations.

This paper introduces a new concept of dual periodic boundary conditions. One of these conditions relates to the endpoints of an interval, while the other one involves two nonlocal positions within the interval. Under these boundary conditions, we establish the criteria for the existence of solutions for a fractional differential equation with p-Laplacian operator by using Mawhin's continuation theorem. An example is constructed for illustrating the obtained results.

"By using the fixed-point theorems, we give sufficient conditions for the existence and uniqueness of solutions for the nonlocal fractional boundary value problem of nonlinear Riesz-Caputo differential equation. The boundedness assumption on the nonlinear term is replaced by growth conditions or by a continuous function. Finally, some examples are presented to illustrate the applications of the obtained results. Keywords: Fractional boundary value problem, Riesz-Caputo fractional derivative, existence and uniqueness, fixed point, nonlocal conditions. "

&lt;abstract&gt;&lt;p&gt;We investigate the existence of solutions for a Caputo fractional differential equation with periodic boundary condition. Using the positivity of Green's function of the corresponding linear equation, we show the existence of positive solutions by using Krasnosel'skii fixed point theorem. Meanwhile, by using monotone iterative method and lower and upper solutions method, we also discuss the existence of extremal solutions for a special case.&lt;/p&gt;&lt;/abstract&gt;