Solutions Of Fractional Differential Equations Research Articles

A Finite Difference-Based Adams-Type Approach for Numerical Solution of Nonlinear Fractional Differential Equations: A Fractional Lotka–Volterra Model as a Case Study

Abstract In this paper, we developed an efficient Adams-type predictor–corrector (PC) approach for the numerical solution of fractional differential equations (FDEs) with a power law kernel. The main idea of the proposed approach is to use a linear approximation to the nonlinear problem and then implement finite difference approximations of derivatives. Numerical comparisons with the fractional Adams method are made and simulation results are demonstrated to evaluate the approximation error of the proposed approach. The efficiency of this approach has been depicted by presenting numerical solutions of some test fractional calculus models. Numerical simulation of a fractional Lotka–Volterra model is provided, as a case study, using the proposed approach. The advantage of the proposed approach lies in its flexibility in providing approximate numerical solutions with high accuracy.

Exploring the Influence of Generalized Kernels on Green’s Function in Fractional Differential Equations

The basic purpose of this article is to define the Green’s function in order to provide the solution of fractional differential equations in the presence of general analytic kernel. Using the technique of Laplace and Fourier transforms, we construct the Green’s function for ordinary and partial fractional differential equations. The presented results will provide the generalization of some models existing in the literature. Some examples are also provided to prove the results for some particular cases.

Existence and Uniqueness of Solution of Fractional Differential Equation Using Non-local Operator

Objectives : To investigate the existence and uniqueness of a solution to the non-local Cauchy problem of order using the Atangana Baleanu (AB) fractional derivative operator. Methods : Using Schauder's fixed point theorem and the Arzela-Ascoli theorem, the study proved the existence of a solution to the given problem. Further, it obtains results for the uniqueness of the solution. Findings : This study proves the existence of a solution to the non-local Cauchy problem under the given conditions. Results are also provided for the uniqueness of the solution. Novelty : A novel approach to fractional differential equations is represented by applying Atangana-Baleanu fractional derivative operator to the non-local Cauchy problem. Schauder's fixed point theorem and Arzela-Ascoli's theorem, are used to show existence and uniqueness. Further, one detailed example has been solved. Keywords: Existence, Uniqueness, Non-local operator, Schauder fixed point theorem, Arzela-Ascoli theorem, Cauchy problem

Multiple solutions for fractional differential equations with $p$-Laplacian through variational method

Multiple solutions for fractional differential equations with $p$-Laplacian through variational method

Fractional Differential Equations with Impulsive Effects

This paper discusses impulsive effects on fractional differential equations. Two approaches are taken to obtain our results: either with fixed or changing lower limits in Caputo fractional derivatives. First, we derive an existence result for periodic solutions of fractional differential equations with periodically changing lower limits. Then, the impulsive effects are modeled for fractional differential equations regarding the nonlinearities rather than the initial value conditions. The proposed impulsive model differs from common discontinuous and nonsmooth dynamical systems.

Optimizing Fractional Differential Equation Solutions with Novel Müntz Space Basis Functions

This paper introduces innovative basis functions derived from Müntz spaces, aimed at addressing the computational challenges of Fractional Differential Equations (FDEs). Our primary focus is the creation of these functions using singular indices linked to the solutions of FDEs. We thoroughly investigate the properties of these fundamental functions to understand their operational potential. These functions are particularly adept at capturing initial singular indices, making them highly suitable for solving FDEs. The proposed numerical method is distinguished by its rapid convergence rates, showcasing its efficiency in computational evaluations. We validate our approach by presenting numerical examples that highlight its accuracy and reliability. These examples confirm the effectiveness and efficiency of the new basis functions from Müntz spaces in accurately solving FDEs. This research advances numerical methods for FDEs and serves as a valuable resource for researchers seeking robust and reliable techniques

Solutions of fractional differential models by using Sumudu transform method and its hybrid

This paper presents the Sumudu transform method and its hybrid for the construction of solutions of differential equations, both with integer-order and fractional derivatives. The paper discusses the construction of solutions of fractional differential equations with varying delay proportional to the independent variable by using two methods. The paper considers the mathematical model for the decay of Iodine 135 as an application of fractional differential equations in nuclear physics. The application indicates that fractional differential equations with variable delay proportional to the independent variable are a useful tool for the modeling of many anomalous phenomena in nature and in the theory of complex systems.

Pathway to Fractional Integrals, Fractional Differential Equations, and Role of the H-Function

In this paper, the pathway model for the real scalar variable case is re-explored and its connections to fractional integrals, solutions of fractional differential equations, Tsallis statistics and superstatistics in statistical mechanics, the reaction-rate probability integral, Krätzel transform, pathway transform, etc., are explored. It is shown that the common thread in these connections is their H-function representations. The pathway parameter is shown to be connected to the fractional order in fractional integrals and fractional differential equations.

Solvability of a boundary value problem with integrable conditions for partial differential equations of fractional order

Differential equations have a remarkable ability to predict the world around us. They are used in a wide variety of disciplines, from biology, economics, physics, chemistry and engineering. The study of existence and uniqueness, periodicity, asymptotic behavior, stability, and methods of analytic and numerical solutions of fractional differential equations have been studied extensively in a large cycle works; Especially, The study of existence and uniqueness of solution of fractional partial differential equations are then proved by mani methods as: the well-known Lax–Milgram theorem, Energy estimate and fixed point theorem... Among them, we only mention here the apriori estimate method (energy inequality method). The current piece of work concentrates on exploring the existence and uniqueness of a solution for a non-linear boundary value problem that has integrable conditions in the case of fractional partial differential equations. For this we split the proof into two sections: linear and non-linear problem; for the associated linear problem, we derive the a priori bound and demonstrate the density of the operator generated by the problem posed; we solve the non-linear problem by introducing a iterative process. The results show the efficiency of energy inequality method in the case of time fractional order differential equations with integrable conditions our results illustrate the existence and uniqueness of the continuous dependence of solution on fractional order. The work of the authors could be considered as a contribution to the development of the functional analysis method. And to conclude we argue that the research carried out in this article could serve as a contribution for possible studies in the field of applied mathematics.

An Intelligent Parameter Identification Algorithm of Linear Fuzzy Fractional Differential Equation

Abstract The traditional linear generation fuzzy fractional differential equation parameter identification algorithm lacks the update of parameter identification process, has large amount of calculation, slow convergence speed of parameter identification and strong dependence on initial values. In this paper, a new intelligent recognition algorithm for linearly generated fuzzy fractional differential equations is proposed. The parameters of the equation are re expressed by the constant level set. The piecewise constant level set algorithm based on equation parameter intelligent identification is used to solve the steady-state solution of fractional differential equation, and the non convergence problem caused by too much calculation is solved. A new algorithm scheme for linearly generating fuzzy fractional differential equations is established, the constraints of the level set of the differential equations are calculated, and the updated algorithm for parameter identification of the equation is obtained. The evolutionary algorithm is used to solve the updating algorithm to realize the intelligent identification algorithm of linear fractional fuzzy differential equations. Experimental results show that the algorithm has the advantages of fast convergence speed, high calculation accuracy and low initial value.

A New Fixed Point Approach for Approximating Solutions of Fractional Differential Equations in Banach Spaces

A New Fixed Point Approach for Approximating Solutions of Fractional Differential Equations in Banach Spaces

Soliton solutions and sensitive analysis to nonlinear wave model arising in optics

In this study, we use analytical algorithms, specifically the auxiliary equation (AE) approach, the improved F-expansion method, and the modified Sardar sub-equation (MSSE) method to investigate complex wave structures for plentiful solutions associated with the fractional perturbed Gerdjikov-Ivanov (PGI) model with the M-fractional operator. The investigated model is a well-established mathematical model used to represent a variety of physical events in nonlinear dynamics and mathematical physics. By using the aforementioned techniques, we scrutinize some new optical wave solutions in the framework of dark, bright, periodic, combo, W-shaped, M-shape, V-shape, kink type, singular rational, exponential, trigonometric, and hyperbolic solutions. The acquired solutions address a wide range of optical solutions in the form of 3D plots, contour plots, and 2D plots, declaring the free parameters of such optical soliton solutions and comprehending their dynamic behavior. Also, the sensitive analysis of the selected model is analyzed. The main contribution of this study is to extract diverse solitary wave solutions of the adopted model. Some of the solutions are similar and some diverge from the previous solutions which justifies the novelty of the study. Finally, we discovered that the current technique provides a reliable instrument for investigating the analytic solutions of fractional differential equations. The proposed PGI model can be used to transmit ultra-fast pulses across optical fibers. This research goes beyond to the advancement of mathematical techniques for solving fractional differential equations and broadens their application to a wide range of real-world scientific and engineering problems.

Fixed Point Method for Nonlinear Fractional Differential Equations with Integral Boundary Conditions on Tetramethyl-Butane Graph

Until now, little investigation has been done to examine the existence and uniqueness of solutions for fractional differential equations on star graphs. In the published articles on the subject, the authors used a star graph with one junction node that has edges with the other nodes, although there are no edges between them. These graph structures do not cover more generic non-star graph structures; they are specific examples. The purpose of this study is to prove the existence and uniqueness of solutions to a new family of fractional boundary value problems on the tetramethylbutane graph that have more than one junction node after presenting a labeling mechanism for graph vertices. The chemical compound tetramethylbutane has a highly symmetrical structure, due to which it has a very high melting point and a short liquid range; in fact, it is the smallest saturated acyclic hydrocarbon that appears as a solid at a room temperature of 25 °C. With vertices designated by 0 or 1, we propose a fractional-order differential equation on each edge of tetramethylbutane graph. Employing the fixed-point theorems of Schaefer and Banach, we demonstrate the existence and uniqueness of solutions for the suggested fractional differential equation satisfying the integral boundary conditions. In addition, we examine the stability of the system. Lastly, we present examples that illustrate our findings.

Numerical simulations for fractional differential equations of higher order and a wright-type transformation

In this work, a new relationship is established between the solutions of higher order fractional differential equations and a Wright-type transformation. Solutions could be interpreted as expected values of functions in a random time process. As applications, we solve the fractional beam equation, fractional electric circuits with special functions as external sources, derive d’Alembert’s formula and show the existence of explicit solutions for a general fractional wave equation with variable coefficients. Due to this relationship, we present two methods for simulating solutions of fractional differential equations. The two approaches use the interpretation of the Caputo derivative of a function as a Wright-type transformation of the higher derivative of the function. In the first approach, we use the Runge–Kutta method of hybrid orders 4 and 5 to solve ordinary differential equations combined with the Monte Carlo integration to conduct the Wright-type transformation. The second method uses a feedforward neural network to simulate the fractional differential equation.

Analytical investigation of vesicle dynamics via the modified Riemann-Liouville fractional derivative: Mittag-Leffler function solution and comparative analysis with Caputo's derivative.

The solution of fractional differential equations is a significant focus of current research, given their prevalence in various fields of application. This paper introduces an innovative exploration of vesicle dynamics using Jumarie's modified Riemann-Liouville fractional derivative within a five-dimensional fractional rigid sphere model. The study reveals an exact solution through the Mittag-Leffler function, providing a deep understanding of intricate vesicle dynamics, including alternative motions, such as tank-treading with over-damped and under-damped vesicle oscillations, respectively, TT-OD and TT-UD. A comparative analysis with Caputo's derivative emphasizes the effectiveness of these fractional derivatives, contributing not only to theoretical insights but also practical implications in applied mathematics and biophysical systems. The findings advance our understanding of complex vesicle behaviors, particularly in mimicking real cell-like behaviors, and pave the way for further research and applications in the field.

Employing the Akbari Ganji Method (AGM) to conduct a semi-analytical analysis of transient Eyring-Powell compressible flow in a tensile Surface under the influence of a magnetic field

This study explores the transfer of mass and heat within unstable two-dimensional flows of non-Newtonian material under conditions involving radiation generation, absorption, and thermal radiation. Additionally, it investigates the impact of magnetic hydromagnetic joule (MHD) heating on these processes. The researchers converted the partial differential equations into ordinary ones through appropriate transformations. Subsequently, a new idea was considered, involving coupling fractional differential equations using the AGM method, with an order of 0.5 < a <0.8 and the initial condition x (0) = x0. A new technique is introduced to find the exact solution of fractional differential equations by solving the correct order differential equations. The primary aim of this paper is to explore the impact of parameter variations on velocity, temperature, local skin friction coefficient, and local Nusselt and Sherwood numbers. This article investigates the effect of multi-parameter changes on local skin friction coefficient and Schmidt number. In most fluid heat transfer problems, especially in non-Newtonian fluids, fractional differential equations are widely used in liquids. The obtained results indicate that the Lorentz force, influenced by the magnetic field parameter (Ha), diminishes the velocity distribution. Additionally, it is observed that the temperature profile decreases as the radiation parameter (R) increases.

Modified special functions: properties, integral transforms and applications to fractional differential equations

In this paper, we defined the modified gamma and beta functions, which involving the generalized M-series at their kernels, and then defined the modified Gauss and confluent hypergeometric functions by using the modified beta function. Furthermore, we presented some of their properties such that, integral representations, summation formulas and derivative formulas. Also, we applied beta, Mellin, Laplace, Sumudu, Elzaki and general integral transforms to newly defined the modified special functions. Then, we obtained solution of fractional differential equations involving the modified special functions, as examples. Finally, we gave the relationships between the modified functions with some of the generalized special functions, which can be found in the literature.

Existence and multiplicity of solutions for fractional differential equations with p-Laplacian at resonance

In this article, we investigate the existence and multiplicity of solutions for a fractional differential equations with p-Laplacian equation at resonance in the \(\psi\)-fractional space \(H^{\alpha, \beta; \psi}_p\). In addition, we show that the energy functional satisfies the Palais-Smale condition. For more information see https://ejde.math.txstate.edu/Volumes/2024/34/abstr.html

Asymptotically periodic solutions of fractional order systems with applications to population models

Motivated by applications in population models, we consider S-asymptotically periodic solution of fractional differential equations with periodic environment forces or asymptotically periodic ones. The system is quasi-monotone, and the existence of positive S-asymptotically periodic solution is established by using upper and lower solutions. The sufficient conditions that ensure the uniqueness of positive S-asymptotically periodic solution are also established on the basis of theory of sublinear operator. The applications of the general conclusions to classical population models yield the global convergence of positive S-asymptotically periodic solution in logistic equation with or without weak Allee effect, and the model of two cooperative populations.

Solution of Fractional Differential Equations Involving Hilfer-Hadamard Fractional Derivatives

Objectives: The aim is to establish prerequisite properties for the Hilfer-Hadamard fractional derivatives and address boundary value problems related to fractional polar Laplace and fractional Sturm-Liouville equations involving Hilfer-Hadamard fractional derivatives. Methods: Existing definitions and findings are utilized to obtain the properties for fractional derivatives, and the Adomian decomposition method is employed to solve the fractional differential equations. Findings: Validity conditions for the law of exponents are determined, and the study investigates the fractional differential equations and their corresponding solutions, possessing the capacity to replace the traditional polar Laplace and Sturm-Liouville boundary value problems to effectively represent real-world phenomena. Novelty: The study introduces the substitution of two consecutively operated Hilfer-Hadamard fractional derivatives with a corresponding single Hilfer-Hadamard fractional derivative using the law of exponents. Additionally, the polar Laplace and Sturm-Liouville boundary value problems are extended to their respective fractional counterparts, expressed in a concise format using HilferHadamard fractional derivatives. Keywords: Adomian decomposition method, Hilfer-Hadamard fractional derivative, Fractional polar Laplace equation, Fractional Sturm-Liouville boundary value problem