System Of Fractional Differential Equations Research Articles

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.

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.

A high order multi step method for solving system of fractional differential equations

The primary objective of this research paper is to present an advanced method utilizing quadratic interpolation to address the challenges encountered in solving a system of fractional initial value problems. By employing this high-order technique, we aim to improve the accuracy and efficiency of numerical computations associated with fractional differential equations. In our proposed methodology, we strive to enhance the conventional block-by-block approach by mitigating the interdependence of unknown solutions at each block iteration, except for the initial two steps. This modification aims to maintain the favorable stability characteristic inherent in block-by-block schemes, while effectively reducing coupling effects. We meticulously examine the convergence properties of the proposed method and conduct a rigorous analysis of the associated errors that prove that the numerical solution convergence to the true solution with an order of 3 + β when the parameter 0 < β ≤ 1. To substantiate the theoretical claims put forth in our research, we present a series of numerical examples. These examples serve as empirical evidence to demonstrate the effectiveness and practical applicability of the proposed method. The presented method superiority over alternative approaches is highlighted through a comprehensive comparison of the numerical results.

Numerical Comparisons Between Some Recent Modifications of Fractional Euler Methods

For dealing with systems of Fractional Differential Equations (FDEs), the primary goal of this work is to offer several graphical comparisons performed with the use of specific recent adjustments of the so-called Fractional Euler Method (FEM). These numerical adjustments are the Modified FEM (or MFEM) and Improved Modified FEM (or IMFEM) coupled with the FEM itself. This would reveal the best approximate solution to the fractional-order system consisting of FDEs in comparison with its exact solution.

Study of Caputo fractional derivative and Riemann–Liouville integral with different orders and its application in multi‐term differential equations

In this article, we initially provided the relationship between the RL fractional integral and the Caputo fractional derivative of different orders. Additionally, it is clear from the literature that studies into boundary value problems involving multi‐term operators have been conducted recently, and the aforementioned idea is used in the formulation of several novel models. We offer a unique coupled system of fractional delay differential equations with proper respect for the role that multi‐term operators play in the research of fractional differential equations, taking into account the newly established solution for fractional integral and derivative. We also made the assumptions that connected integral boundary conditions would be added on top of ‐fractional differential derivatives. The requirements for the existence and uniqueness of solutions are also developed using fixed‐point theorems. While analyzing various sorts of Ulam's stability results, the qualitative elements of the underlying model will also be examined. In the paper's final section, an example is given for purposes of demonstration.

Effective Method for Solving Higher-Order Fractional Differential Equations Using Spline Functions

In this paper, appropriate spline functions in polynomial form are outlined and applied to solve higher-order linear fractional differential equations (H-OLFDEs). Several techniques have been proposed to solve this type of equations. A description of the projected methodology is first introduced. The method involves transforming the H-OLFDE of order α, with given initial conditions, into a system of fractional differential equations with , denoting the order of the Caputo fractional derivative for each equation i in the transformed system, where the number of equations in the resulting system is equal to that of initial conditions. The numerical results obtained using this strategy demonstrate substantial agreement with the precise analytical solutions. The results in this paper affirm the robustness and ease of application of the proposed technique.

Existence results for a coupled system of fractional stochastic differential equations involving Hilfer derivative

Abstract In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 &lt; α &lt; 1 {\frac{1}{2}&lt;\alpha&lt;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 result is provided.

Fractional Jacobi–Picard iteration method using Gauss–Seidel technique for solving a system of nonlinear fractional differential equations

The main objective of this study is to introduce an improvement of Picard’s method, a technique commonly used to effectively solve a set of nonlinear fractional differential equations based on Caputo’s fractional derivative. Using the Picard’s method to solve fractional differential equations is straightforward. However, dealing with the integral in each Picard’s iteration becomes tough or even impossible for nonlinear problems. Thus, we propose an iterative strategy called the fractional Jacobi–Picard iteration method, which combines Picard’s iteration method with the shifted Jacobi polynomial. The computation of the fractional integrals of the shifted Jacobi polynomials is easily achieved at each step by utilizing properties of the fractional integral and shifted Jacobi polynomial. Furthermore, this approach not only transforms the system of equations into a reversible form but also solves it using the Gauss–Seidel technique. The convergence analysis of the method has been carefully performed. We performed detailed numerical simulations to show how well our method performs compared to other methods. Our results demonstrate the effectiveness and accuracy of our approach, especially in handling problems with non-smooth solutions.

Numerical solution of FDE-IVPs by using fractional HBVMs: the fhbvm code

AbstractIn this paper we describe the efficient numerical implementation of Fractional HBVMs, a class of methods recently introduced for solving systems of fractional differential equations. The reported arguments are implemented in the Matlab$$^{\copyright } $$ © code , which is made available on the web. An extensive experimentation of the code is reported, to give evidence of its effectiveness.

Multivalued nonlinear dominated mappings on a closed ball and associated numerical illustrations with applications to nonlinear integral and fractional operators

In the context of complete strong b-metric-like spaces, we prove new multi-fixed point solutions for the pair of multivalued, dominated operators that fulfill the generalized nonlinear contractions on a closed ball. We employ a mix of two different types of mappings in our approach: one is a class of multi-dominated mappings, while the other is a weaker class of strictly increasing mappings. Additionally, some new fixed-point results concerning the multi-graph-dominated structure in graph contraction are presented. To validate the hypothesis of acquired results, a few sample cases are presented. A numerical experiment has been carried out to approximate the fixed point. In addition, to demonstrate the originality of our findings, we proposed simple and efficient solutions to the system of fractional differential equations and nonlinear Volterra-type integral equations.

Coupled systems of conformable fractional differential equations

This paper deals with some existence of solutions for some classes of coupled systems of conformable fractional differential equations with initial and boundary conditions in Banach and Fréchet spaces. Our results are based on some fixed point theorems. Some illustrative examples are presented in the last section.

A Study on Linear Prabhakar Fractional Systems with Variable Coefficients

The focus of this paper is on addressing the initial value problem related to linear systems of fractional differential equations characterized by variable coefficients, incorporating Prabhakar fractional derivatives of Riemann–Liouville and Caputo types. Utilizing the generalized Peano–Baker series technique, the state-transition matrix is acquired. The paper presents closed form solutions for both homogeneous and inhomogeneous cases, substantiated by illustrative examples.

Existence Results for the System of Fractional-Order Sequential Integrodifferential Equations via Liouville–Caputo Sense

We investigate the conditions for the existence and uniqueness of solutions in a nonlinear system of sequential fractional differential equations using the Liouville–Caputo type with varying orders. This system is enriched by nonlocal coupled integral boundary conditions. The desired outcomes are attained by employing traditional fixed-point theorems. It is essential to emphasize that the fixed-point approach proves to be an effective method for establishing the existence of solutions in boundary value problems. Furthermore, we provide constructed examples to illustrate the obtained results.

Efficient Modified Adomian Decomposition Method for Solving Nonlinear Fractional Differential Equations

Based on the application of the standard Adomian method, the current paper proposes two modification approaches for the classes of the fractional differential equations and the system of fractional differential equations, which are featured through initial-value problems. Certainly, the constructed iterative schemes for the two classes are shown to be reliable, considering a number of test problems for demonstration, and upon deploying other existing numerical approaches for contrasting. In fact, the proposed schemes are found to portray less error, rapidity, accuracy and consume less computational time among others.

Wave propagation in three-dimensional fractional viscoelastic infinite solid body

In order to analyze the wave propagation in three-dimensional isotropic and viscoelastic body, the Cauchy initial value problem on unbounded domain is considered for the wave equation written as a system of fractional partial differential equations consisting of equation of motion of three-dimensional solid body, equation of strain, as well as of the constitutive equation, obtained by generalizing the classical Hooke’s law of three-dimensional isotropic and elastic body by replacing Lamé coefficients with the relaxation moduli to account for different memory kernels corresponding to the propagation of compressive and shear waves. By the use of the method of integral transforms, namely Laplace and Fourier transforms, the displacement field, as a solution of the Cauchy initial value problem, is expressed through Green’s functions corresponding to both compressive and shear wave propagation. Two approaches are adopted in solving the Cauchy problem: in the first one the displacement field is expressed through the scalar and vector fields, obtained as solutions of wave equations which are consequences of decoupling the wave equation for displacement field, while in the second approach the displacement field is obtained by the action of the resolvent tensor on the initial conditions. Fractional anti-Zener and Zener model I+ID.ID, as well as fractional Burgers model VII, as representatives of models yielding infinite and finite wave propagation speed, are used in numerical calculations and graphical representations of Green’s functions and its short and large time asymptotic behavior.