Fixed Point Analysis for Cauchy‐Type Variable‐Order Fractional Differential Equations With Finite Delay

Efficiently solving fractional delay differential equations of variable order via an adjusted spectral element approach

This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. All of the results in this study are established with the help of generalized intervals and piecewise constant functions. We convert the Riemann–Liouville fractional variable-order problem to equivalent standard Riemann–Liouville problems of fractional-constant orders. Finally, two examples are constructed to illustrate the validity of the observed results.

In this paper, we approximate the solution of fractional differential equations with delay using a new approach based on artificial neural networks. We consider fractional differential equations of variable order with the Mittag-Leffler kernel in the Liouville-Caputo sense. With this new neural network approach, an approximate solution of the fractional delay differential equation is obtained. Synaptic weights are optimized using the Levenberg-Marquardt algorithm. The neural network effectiveness and applicability were validated by solving different types of fractional delay differential equations, linear systems with delay, nonlinear systems with delay and a system of differential equations, for instance, the Newton-Leipnik oscillator. The solution of the neural network was compared with the analytical solutions and the numerical simulations obtained through the Adams-Bashforth-Moulton method. To show the effectiveness of the proposed neural network, different performance indices were calculated.

In this paper, generalized fractional-order Bernoulli wavelet functions based on the Bernoulli wavelets are constructed to obtain the numerical solution of problems of anomalous infiltration and diffusion modeling by a class of nonlinear fractional differential equations with variable order. The idea is to use Bernoulli wavelet functions and operational matrices of integration. Firstly, the generalized fractional-order Bernoulli wavelets are constructed. Secondly, operational matrices of integration are derived and utilize to convert the fractional differential equations (FDE) into a system of algebraic equations. Finally, some numerical examples are presented to demonstrate the validity, applicability and accuracy of the proposed Bernoulli wavelet method.

This paper investigated the existence and stability of solutions for boundary value problems involving Caputo fractional differential equations of variable order. Unlike constant-order models, variable-order equations allow the fractional order to change over time, enabling more flexible and accurate modeling of complex systems with evolving dynamics and memory. Using Darbo's fixed point theorem and the Kuratowski measure of noncompactness, we established new existence results for solutions within a Banach space of continuous functions. Our approach treated the variable order as piecewise constant, transforming the problem into a sequence of more manageable constant-order subproblems. Furthermore, we demonstrated Ulam-Hyers stability of the solutions, ensuring that small perturbations in the system did not lead to significant deviations in the results. To validate the theoretical findings, we provided a detailed example supported by numerical simulations. These results offered a solid foundation for future applications in science and engineering where system dynamics evolve over time.

This article studies the convergence of a numerical method for solving an initial-boundary value problem of a fractional differential equation with a variable order of the fractional derivative. In the generalized fractional differential filtration equation with a transitional filtration law in heterogeneous porous media, it is assumed that the order of the fractional derivative depends on the spatial variable. The main attention is paid to the development and theoretical justification of a method that provides high accuracy and efficiency of calculations with a variable order of the fractional derivative. For the numerical solution, an approximation was developed that combines the finite difference method for the time derivative and the finite element method for the spatial variable. The fractional derivative of variable order in the sense of Caputo is approximated by a formula of second order in time. The convergence of the constructed method is proven with order O(τ2+hk+1) for the case α(x)ϵ(0,1). The results of computational experiments for various functions of the order of the fractional derivative are presented, confirming the reliability of the theoretical analysis. The conclusions drawn emphasize the importance and relevance of the further development of numerical methods for fractional differential equations of variable order in modern mathematics and applied sciences, including the modeling of complex processes.

The semigroup properties of the Riemann–Liouville fractional integral have played a key role in dealing with the existence of solutions to differential equations of fractional order. Based on some results of some experts’, we know that the Riemann–Liouville variable order fractional integral does not have semigroup property, thus the transform between the variable order fractional integral and derivative is not clear. These judgments bring us extreme difficulties in considering the existence of solutions of variable order fractional differential equations. In this work, we will introduce the concept of approximate solution to an initial value problem for differential equations of variable order involving the derivative argument on half-axis. Then, by our discussion and analysis, we investigate the unique existence of approximate solution to this initial value problem for differential equation of variable order involving the derivative argument on half-axis. Finally, we give examples to illustrate our results.

In this manuscript, the existence, uniqueness, and stability of solutions to the multiterm boundary value problem of Caputo fractional differential equations of variable order are established. All results in this study are established with the help of the generalized intervals and piece-wise constant functions, we convert the Caputo fractional variable order to an equivalent standard Caputo of the fractional constant order. Further, two fixed point theorems due to Schauder and Banach are used, the Ulam–Hyers stability of the given Caputo variable order is examined, and finally, we construct an example to illustrate the validity of the observed results. In literature, the existence of solutions to the variable-order problems is rarely discussed. Therefore, investigating this interesting special research topic makes all our results novel and worthy.

In this manuscript, we examine both the existence, uniqueness, and the stability of solutions to the boundary value problem (BVP) of Hadamard fractional differential equations of variable order by converting it into an equivalent standard Hadamard (BVP) of the fractional constant order with the help of the generalized intervals and the piecewise constant functions. All results in this study are established using Krasnoselskii fixed‐point theorem and the Banach contraction principle. Further, the Ulam–Hyers stability of the given problem is examined, and finally, we construct an example to illustrate the validity of the observed results.

In this paper, we discuss the existence and uniqueness of solutions for nonlinear fractional differential equations of variable order with fractional antiperiodic boundary conditions. The main results are obtained by using fixed point theorem.

In this paper, we present the solution of nonlinear fractional partial differential equations by using the Homotopy Perturbation Aboodh Transform Method (HPATM) and Homotopy Decomposition Method (HDM). The Two methods introduced an efficient tool for solving a wide class of linear and nonlinear fractional differential equations. The results shown that the (HDM) has an advantage over the (HPATM) that it takes less time and using only the inverse operator to solve the nonlinear problems and there is no need to use any other inverse transform as in the case of (HPATM).

In this work, we introduce a analytical framework for proving the existence and uniqueness of solutions to nonlinear fractional differential equations (NFDEs) of order $ \nu \in (1, 2] $, subject to irregular boundary conditions, in a Banach space. Several fundamental definitions, theorems, and auxiliary lemmas are presented for the analysis of the proposed problem. We establish sufficient conditions for existence and uniqueness for the proposed system by applying Krasnosel'skii's fixed-point theorem and the Banach contraction principle. Finally, an illustrative example is constructed to validate the main theoretical results and demonstrate the effectiveness and applicability of the proposed approach, showing how the generalized framework can be systematically applied to analyze fractional boundary value problems.

This paper deals with a nonlinear fractional differential equation with integral boundary condition of the following form: Dαt x(t) = f(t, x(t),Dβ t x(t)), t ∈ (0, 1), x(0) = 0, x(1) = 1 0 g(s)x(s)ds, where 1 < α ≤ 2, 0 < β < 1. Our results are based on the Schauder fixed point theorem and the Banach contraction principle.

In this paper, we establish the existence of solutions for a class of nonlinear neutral fractional differential equations with terminal condition and Hilfer-Katugampola fractional derivative. The arguments are based upon the Banach contraction principle, and Krasnoselskii?s fixed point theorem. An example is included to show the applicability of our results.

In this manuscript, we examine both the existence and the stability of solutions of the boundary value problems of Hadamard-type fractional differential equations of variable order. New outcomes are obtained in this paper based on the Darbo’s fixed point theorem (DFPT) combined with Kuratowski measure of noncompactness (KMNC). We construct an example to illustrate the validity of the observed results.