System Of Fractional Differential Equations Research Articles

Terminal value problem for the system of fractional differential equations with additional restrictions

This paper deals with the study of terminal value problem for the system of fractional differential equations with Caputo derivative. Additional conditions are imposed on the solutions of this problem in the form of a linear vector functional. Using the theory of pseudo-inverse matrices, we obtain the necessary and sufficient conditions for the solvability and the general form of the solution of this boundary-value problem. In the one-dimensional case, the obtained results are generalized to the case of a multi-point boundary-value problem. The issue of obtaining similar results for the terminal value problem for the system of fractional differential equations with tempered and Ψ–tempered fractional derivatives of Caputo type is considered.

On the Extended Simple Equations Method (SEsM) and Its Application for Finding Exact Solutions of the Time-Fractional Diffusive Predator–Prey System Incorporating an Allee Effect

In this paper, I extend the Simple Equations Method (SEsM) and adapt it to obtain exact solutions of systems of fractional nonlinear partial differential equations (FNPDEs). The novelty in the extended SEsM algorithm is that, in addition to introducing more simple equations in the construction of the solutions of the studied FNPDEs, it is assumed that the selected simple equations have different independent variables (i.e., different coordinates moving with the wave). As a consequence, nonlinear waves propagating with different wave velocities will be observed. Several scenarios of the extended SEsM are applied to the time-fractional predator–prey model under the Allee effect. Based on this, new analytical solutions are derived. Numerical simulations of some of these solutions are presented, adequately capturing the expected diverse wave dynamics of predator–prey interactions.

Well-Posedness of the Mild Solutions for Incommensurate Systems of Delay Fractional Differential Equations

Well-Posedness of the Mild Solutions for Incommensurate Systems of Delay Fractional Differential Equations

On Symmetrically Stochastic System of Fractional Differential Equations and Variational Inequalities

In this work, we are devoted to discussing a system of fractional stochastic differential variational inequalities with Lévy jumps (SFSDVI with Lévy jumps), that comprises both parts, that is, a system of stochastic variational inequalities (SSVI) and a system of fractional stochastic differential equations(SFSDE) with Lévy jumps. Here it is noteworthy that the SFSDVI with Lévy jumps consists of both sections that possess a mutual symmetry structure. Invoking Picard’s successive iteration process and projection technique, we obtain the existence of only a solution to the SFSDVI with Lévy jumps via some appropriate restrictions. In addition, the major outcomes are invoked to deduce that there is only a solution to the spatial-price equilibria system in stochastic circumstances. The main contributions of the article are listed as follows: (a) putting forward the SFSDVI with Lévy jumps that could be applied for handling different real matters arising from varied domains; (b) deriving the unique existence of solutions to the SFSDVI with Lévy jumps under a few mild assumptions; (c) providing an applicable instance for spatial-price equilibria system in stochastic circumstances affected with Lévy jumps and memory.

Lyapunov Inequalities for Systems of Tempered Fractional Differential Equations with Multi-Point Coupled Boundary Conditions via a Fix Point Approach

In this paper, we study a system of nonlinear tempered fractional differential equations with multi-point coupled boundary conditions. By applying the properties of Green’s function and the operator and combining the method of matrix analysis, we obtain the corresponding Lyapunov inequalities under two Banach spaces. And, we have compared two Lyapunov inequalities under certain conditions. An example is given to verify our results.

On generalized asymmetric harmonic oscillator with quadratic nonlinearity within fractional variational principles

This work studies the nonlinear fractional dynamics of asymmetric harmonic oscillators. The classical description of the physical system is generalized using the principles of fractional variational analysis. As a system of two-coupled fractional differential equations with a quadratic nonlinear component, the fractional Euler–Lagrange equations of the motion of the corresponding system are obtained. The Adams–Bashforth predictor-corrector numerical approach is used to approximate the system’s outcomes, which are then simulated comparatively with respect to various model parameter values, including mass, linear and quadratic nonlinear stiffness, and the order of the fractional derivative. The simulations provided the possibility of investigating various dynamical behaviours within the same physical model that is generalized by the use of fractional operators.

EXISTENCE RESULT FOR A COUPLED SYSTEM OF HYBRID FRACTIONAL DIFFERENTIAL EQUATIONS IN A BANACH ALGEBRA

In this work, we investigate the existence result for a coupled system of hybrid fractional differential equations in a Banach algebra. Our main result is based on a generalization of Darbo’s fixed point theorem in Banach algebra. We apply in our approach the technique of measure of non-compactness, we prove that the Kuratowski measure of noncompactness satisfies a condition (m) which will be useful in our considerations. An example is given to illustrate the feasibility of our main result. An example is provided to illustrate our result.

Existence and Uniqueness Results for a Coupled System of Nonlinear Hadamard Fractional Differential Equations

In this paper, we examine the existence and uniqueness of solutions to a system of four-point non-separated Hadamard fractional differential equations. Applying Leray-Schauder’s alternative yields the existence of solutions; Banach’s contraction principle establishes the uniqueness of the solution. Lastly, we provide two examples to demonstrate our results.

Existence results for coupled systems of fractional stochastic differential equations involving Hilfer derivatives

Abstract In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 < α < 1 \frac{1}{2}<\alpha<1 . Under some assumptions, we prove the existence of mild solutions for our system based on Perov’s and Schaefer’s fixed point theorems. An example illustrating our results is also provided.

Existence and stability of solution for a coupled system of Caputo–Hadamard fractional differential equations

In our present work, we study a coupled system of Caputo–Hadamard fractional differential equations supplemented with a novel set of initial value conditions involving the η=(tddt)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\eta =(t\\frac{d}{dt})$\\end{document} derivatives. We provided sufficient criteria for the existence and stability of the solutions for a coupled system of fractional differential equations by applying the Hyers–Ulam stability theory, the fixed point theorems of Banach, Krasnoselskii, and the Leray–Schauder nonlinear alternative. When computing priori bounds in Leray–Schauder nonlinear alternative and stability of the solutions, a novel Gronwall type inequality related to Hadamard integral is employed. This study investigates the properties of a solution, such as existence, uniqueness, and stability, to a given problem without attempting to solve the exact solution, and its theoretical applications are illustrated by providing an example.

Fixed Point and Stability Analysis of a Tripled System of Nonlinear Fractional Differential Equations with n-Nonlinear Terms

This research article investigates a tripled system of nonlinear fractional differential equations with n terms. The study explores this novel class of differential equations to establish existence and stability results. Utilizing Schaefer’s and Banach’s fixed point theorems, we derive sufficient conditions for the existence of at least one solution, as well as a unique solution. Furthermore, we apply Hyers–Ulam stability analysis to establish criteria for the stability of the system. To demonstrate the applicability of the main results, a detailed example is provided.

Dynamical behavior of whooping cough SVEIQRP model via system of fractal fractional differential equations

In this work, the application of the system of fractal fractional differential equations is exploited. Infectious diseases modeling of non-integer order attracts many mathematicians and biological scientists. The presentation of an advanced fractal fractional model for studying the exact dynamical behavior of infectious diseases in epidemiology is necessary. In this study, a new version of a mathematical model for whooping cough is presented. Whooping cough is a disease and can be transmitted from humans to humans through various means, i.e., touch, cough, and air droplets. In this study, we considered the whooping cough model with a new permanent compartment. Our modified model after incorporating the permanent recovered compartment explains better regarding the individuals who have developed long term immunity against whooping cough. This addition to the model enhances the model’s accuracy towards real world scenarios. The considered model was analyzed by using a system of fractal fractional differential equations, and numerical simulation was established for the findings of this study. The fixed point theorem was used to determine the existence, uniqueness, and Hyers–Ulam stability of the model. Numerical results of the dynamical behavior of the model are also recorded in tables.

Reliable Computational Method for Systems of Fractional Differential Equations Endowed with ψ-Caputo Fractional Derivative

This study develops a highly convergent computational method, the ψ-Laplace Adomian Decomposition Method (ψ-LADM), for solving coupled systems ofψ-Caputo Fractional Differential Equations (FDEs). The effectiveness of the proposed method has been assessed using various numerical test examples, including a real-world application for atmospheric convection models utilizing Lorenz chaotic dynamical systems. Notably, the method consistently produced solutions that matched the true solutions of the governing models. In the case of the Lorenz chaotic system, the obtained solutions accurately portrayed the characteristic phase portraits of a true chaotic system.

Chlamydia infection with vaccination asymptotic for qualitative and chaotic analysis using the generalized fractal fractional operator

In this work, we solve a system of fractional differential equations utilizing a Mittag-Leffler type kernel through a fractal fractional operator with two fractal and fractional orders. A six-chamber model with a single source of chlamydia is studied using the concept of fractal fractional derivatives with nonsingular and nonlocal fading memory. The fractal fractional model of the Chlamydia system can be solved by using the characteristics of a non-decreasing and compact mapping. A suggested model with the Lipschitz criteria and linear growth is studied both qualitatively and quantitatively, taking into account boundedness, uniqueness, and positive solutions at equilibrium points with Leray-Schauder results under time scale concepts. We examined the framework of local and global stability and insight into Lyapunov function properties for the infectious disease model. Chaos Control will employ the regulate for linear responses approach to stabilize the system following its equilibrium points. This will take into consideration a fractional order framework with a managed design, where solutions are bounded in the feasible domain and have a greater impact at the lower minimum infectious rate. To illustrate the implications of fractional and fractal dimensions with varying interest rate values through simulations with Newton’s polynomial method under the Mittag-Lefller kernel. Additionally, a comparative analysis of results is also derived by employing power and exponential decay kernels at various fractional orders.

Application of fractional differential transform method and Bell polynomial for solving system of fractional delay differential equations

In this article, a new numerical technique is presented to obtain numerical solution of a system of fractional delay differential equations (FDDE’s) involving proportional and time dependent delay terms. The fractional derivative is used in Caputo sense. The proposed technique is the combination of fractional differential transform and Bell polynomial. The existence and uniqueness results are discussed for FDDE’s. Three numerical problems are discussed to show reliability and efficiency of the method. Numerical results are compared with exact and Matlab DDENSD solution. The main advantage of the present method is handing effectively the nonlinear terms present in the FDDEs by using Bell polynomial. The present method can deal with both linear and nonlinear FDDEs. The convergence result is discussed, and error analysis is presented in detail.

Solution of Coupled System of Caputo Fractional Differential Equations withMulti-Point Boundary Conditions

This article relies on the Caputo fractional derivative for the objective of is to examine an interconnected system of fractional differential equations. The problem under consideration involves four fractional-Caputo-derivatives under neath initial conditions. We state and prove the existence and uniqueness theorem of solution by application of Banach fixed point theorem. Then, another result that deals with the existence of at least one solution is delivered and some sufficient conditions for this result are established by means of the fixed-point theorem of Schaefer. We end the paper by presenting to the reader by illustrative example.

NUMERICAL ASSESSMENT OF SOME SEMI-ANALYTICAL TECHNIQUES FOR SOLVING A FRACTIONAL-ORDER LEPTOSPIROSIS MODEL

This research aims to apply and compare two semi-analytical techniques, the Variational Iterative Method (VIM) and the New Iterative Method (NIM), for solving a pre-formulated mathematical model of Fractional-order Leptospirosis. Leptospirosis is a significant bacterial infection affecting humans and animals. By implementing the VIM and NIM algorithms, numerical experiments are conducted to solve the leptospirosis model. Comparing the obtained findings demonstrates that VIM and NIM are effective semi-analytical methods for solving systems of fractional differential equations. Notably, our study unveils a crucial dynamic in the disease's spread. The application of VIM and NIM offers a refined depiction of the biological dynamics, highlighting that the susceptible human population gradually decreases, the infectious human population declines, the recovered human population increases, and a significant rise in the infected vector population is observed over time. This nuanced portrayal of the disease's dynamics is crucial for understanding the intricate interplay of Leptospirosis among human and vector populations. The study's outcomes contribute valuable insights into the applicability and performance of the methods in solving the Fractional Leptospirosis model. Results indicate rapid convergence and comparable outcomes for both methods.

Fixed-point methodologies and new investments for fuzzy fractional differential equations with approximation results

The application of Caputo-Katugampola gH−differentiability to solve systems of fractional partial differential equations is investigated in this work. The existence and uniqueness of two types of gH−weak solutions for fuzzy fractional coupled partial differential equations are established. Lipschitz conditions are employed, and the Banach fixed-point theorem and mathematical induction are used in our approach. To address the challenges of the initial value problems, a matrix-form Cornwall’s inequality is developed. Additionally, a novel analysis of the continuous dependence of the coupled system’s solutions on the given conditions and approximate solutions is provided. An illustrative example is presented to validate the findings.

Existence results for some fractional order coupled systems with impulses and nonlocal conditions on the half line

Abstract. In this paper, we deal with initial value problems for coupled systems of nonlinear fractional differential equations, subject to coupled nonlocal initial and impulsive conditions on the half line. Global existence-uniqueness results are obtained under weak conditions allowing the reaction part of the problem to increase indefinitely with time. Our approach relies mainly to some fixed point theorem of Perov’s type in generalized gauge spaces. The obtained results improve, generalize and complement many existing results in the literature. An example illustrating our main finding is also given. Mathematics Subject Classification (2010): 26A33, 93C23, 35E15, 47H10. Keywords: Fractional differential equation, coupled systems, generalized spaces in Perov’s sens, nonlocal initial conditions, impulses.

On the inverse problem of time dependent coefficient in a time fractional diffusion problem by sinc wavelet collocation method

The object of this study is to establish the unknown function in a time fractional diffusion problem and the solution as well by utilizing Sinc wavelet collocation method (SWCM) and residual power series method (RPSM) together. SWCM enables us to convert time fractional diffusion problem into a system of fractional ordinary differential and algebraic equations. At this stage, the unknown function and the solution are constructed in the series form by employing RPSM. The novelty of this study is that the combination of SWCM and RPSM is utilized to establish the solution of inverse coefficient problem for the first time. Demonstrative examples are presented to articulate the implementation and importance of the proposed method.