Caputo Fractional Differential Equations Research Articles

Krawtchouk wavelets method for solving Caputo and Caputo–Hadamard fractional differential equations

PurposeThe purpose of the present work is to develop a new wavelet method, named as Krawtchouk wavelets method, for solving both Caputo fractional and Caputo–Hadamard fractional differential equations on a semi‐infinite domain.Design/methodology/approachWe have utilized the discrete Krawtchouk orthogonal polynomial for the construction of Krawtchouk wavelets method. The supporting analysis of the method such as construction of operational matrices, procedure of implementation, and convergence analysis of the method are being provided. We have also proposed a method by combining the Krawtchouk wavelets method with the method of step for the solution of Caputo–Hadamard fractional delay differential equations.FindingsWe have provided the orthogonality condition for the Krawtchouk wavelets. We have derived and constructed the Krawtchouk wavelets matrix, Krawtchouk wavelets operational matrix of Riemann–Liouville and Hadamard‐type fractional‐order integration, and Krawtchouk wavelets operational matrix of Riemann–Liouville and Hadamard‐type fractional‐order integration for boundary value problems. These matrices are successfully utilized for the solution of Caputo and Caputo–Hadamard fractional differential equations. Operational matrices contains many zero entries, which makes the present method more efficient.Furthermore, we workout on the procedure of implementation of the method for the Caputo fractional differential equations as well as for the Caputo–Hadamard fractional differential equations. We also derived the convergence analysis of the Krawtchouk wavelets method, which completes the theoretical analysis of the proposed method.We have applied the Krawtchouk wavelets method for the numerical solutions of several Caputo fractional differential equations and Caputo–Hadamard fractional differential equations and compare the obtained results with the analytical solutions. The comparison shows the effectiveness of the present numerical method. In this paper, we have considered initial value problem, boundary value problem, and delay problem. According to the numerical results, the present method is more efficient and accurate.Originality/valueSince fractional differential equation is a latest and emerging field, many engineers and scientists can utilize the present method for solving their fractional models. To the best of the author's knowledge, the present wavelets method has never been introduced and implemented for Caputo fractional and Caputo–Hadamard fractional differential equations.

Application of piecewise fractional differential equation to COVID-19 infection dynamics

We proposed a new mathematical model to study the COVID-19 infection in piecewise fractional differential equations. The model was initially designed using the classical differential equations and later we extend it to the fractional case. We consider the infected cases generated at health care and formulate the model first in integer order. We extend the model into Caputo fractional differential equation and study its background mathematical results. We show that the fractional model is locally asymptotically stable when R0<1 at the disease-free case. For R0≤1, we show the global asymptotical stability of the model. We consider the infected cases in Saudi Arabia and determine the parameters of the model. We show that for the real cases, the basic reproduction is R0≈1.7372. We further extend the Caputo model into piecewise stochastic fractional differential equations and discuss the procedure for its numerical simulation. Numerical simulations for the Caputo case and piecewise models are shown in detail.

On boundary value problems of Caputo fractional differential equation of variable order via Kuratowski MNC technique

In this manuscript, we examine both the existence and the stability of solutions to the boundary value problem of Caputo fractional differential equations of variable order by converting it into an equivalent standard Caputo boundary value problem of the fractional constant order with the help of the generalized intervals and the piece-wise constant functions. All results in this study are established using Darbo’s fixed point theorem combined with the Kuratowski measure of noncompactness. 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.

Fractional Euler numbers and generalized proportional fractional logistic differential equation

We solve a logistic differential equation for generalized proportional Caputo fractional derivative. The solution is found as a fractional power series. The coefficients of that power series are related to the Euler polynomials and Euler numbers as well as to the sequence of Euler’s fractional numbers recently introduced. Some numerical approximations are presented to show the good approximations obtained by truncating the fractional power series. This generalizes previous cases including the Caputo fractional logistic differential equation and Euler’s numbers.

Generalized Proportional Caputo Fractional Differential Equations with Delay and Practical Stability by the Razumikhin Method

Practical stability properties of generalized proportional Caputo fractional differential equations with bounded delay are studied in this paper. Two types of stability, practical stability and exponential practical stability, are defined and considered, and also some sufficient conditions to guarantee stability are presented. The study is based on the application of Lyapunov like functions and their generalized proportional Caputo fractional derivatives among solutions of the studied system where appropriate Razumikhin like conditions are applied (appropriately modified in connection with the fractional derivative considered). The theory is illustrated with several nonlinear examples.

Attractors of Caputo fractional differential equations with triangular vector fields

It is shown that the attractor of an autonomous Caputo fractional differential equation of order \(\alpha \in (0,1)\) in \({\mathbb {R}}^d\) whose vector field has a certain triangular structure and satisfies a smooth condition and dissipativity condition is essentially the same as that of the ordinary differential equation with the same vector field. As an application, we establish several one-parameter bifurcations for scalar fractional differential equations including the saddle-node and the pichfork bifurcations. The proof uses a result of Cong & Tuan [2] which shows that no two solutions of such a Caputo FDE can intersect in finite time.

Boyd-Wong contractions in F-metric spaces and applications

The main aim of this paper is to study the Boyd-Wong type fixed point result in the F-metric context and apply it to obtain some existence and uniqueness criteria of solution(s) to a second order initial value problem and a Caputo fractional differential equation. We substantiate our obtained result by finding a suitable non-trivial example.

Existence, uniqueness, continuous dependence and Ulam stability of mild solutions for an iterative fractional differential equation

In this work, we study the existence, uniqueness, continuous dependence and Ulam stability of mild solutions for an iterative Caputo fractional differential equation by first inverting it as an integral equation. Then we construct an appropriate mapping and employ the Schauder fixed point theorem to prove our new results. At the end we give an example to illustrate our obtained results.

Coupled system of $\psi$--Caputo fractional differential equations without and with delay in generalized Banach spaces

The main objective of this research manuscript is to establish various existence and uniqueness results as well as the Ulam--Hyers stability of solutions to a Coupled system of $\psi$--Caputo fractional differential equations without and with delay in generalized Banach spaces. Existence and uniqueness results are obtained by applying Krasnoselskii's type fixed point theorem, Schauder's fixed point theorem in generalized Banach spaces, and Perov's fixed point theorem combined with the Bielecki norm. While Urs's approach is used to analyze the Ulam--Hyers stability of solutions for the proposed problem. Finally, Some examples are given to illustrate the obtained results.

Global exponential stability of discrete-time almost automorphic Caputo–Fabrizio BAM fuzzy neural networks via exponential Euler technique

Exponential Euler discrete schemes have been widely employed in the studies of Caputo fractional order differential equations, but almost no literature concerns the Caputo–Fabrizio case. In current work, exponential discrete form has been set up to study Caputo–Fabrizio fuzzy BAM neural networks(CF-FBAMNNs). The research findings tell someone that (1) the exponential discrete form characterizes the continuous CF-FBAMNNs superior to the classical Euler discrete technique; (2) this type discrete technique pertains to the implicit Euler form, which can be calculated by PECE algorithm. Furthermore, the existence of a unique bounded asymptotically almost automorphic sequence solution and global exponential stability of the proposed discrete-time models are investigated. More importantly, the current works make up for the lacks in the existing literatures and build a set of new theories and methods in studying discrete-time Caputo–Fabrizio models in the fields of science and engineering.

New Relation-Theoretic Fixed Point Theorems in Fuzzy Metric Spaces with an Application to Fractional Differential Equations

In this paper, we introduce the notion of fuzzy R−ψ−contractive mappings and prove some relevant results on the existence and uniqueness of fixed points for this type of mappings in the setting of non-Archimedean fuzzy metric spaces. Several illustrative examples are also given to support our newly proven results. Furthermore, we apply our main results to prove the existence and uniqueness of a solution for Caputo fractional differential equations.

Stability of Gene Regulatory Networks Modeled by Generalized Proportional Caputo Fractional Differential Equations

A model of gene regulatory networks with generalized proportional Caputo fractional derivatives is set up, and stability properties are studied. Initially, some properties of absolute value Lyapunov functions and quadratic Lyapunov functions are discussed, and also, their application to fractional order systems and the advantage of quadratic functions are pointed out. The equilibrium of the generalized proportional Caputo fractional model and its generalized exponential stability are defined, and sufficient conditions for the generalized exponential stability and asymptotic stability of the equilibrium are obtained. As a special case, the stability of the equilibrium of the Caputo fractional model is discussed. Several examples are provided to illustrate our theoretical results and the influence of the type of fractional derivative on the stability behavior of the equilibrium.

A variation of constant formula for Caputo fractional stochastic differential equations with jump–diffusion

In this paper, the existence and uniqueness of solutions for the Caputo fractional stochastic differential equations with jump–diffusion is discussed. Then, a variation of constant formula for the equation is established. The main ingredient of the proof is to use Itoˆ’s isometry of Poisson jumps and martingale representation theorem.

A note on a class of Caputo fractional differential equations with respect to another function

The mathematical analysis and solutions for a class of ψ-Caputo fractional differential equations are discussed. Assuming that ψ(t) is strictly monotone and armed by the possibility of converting the ψ-Caputo fractional differential equations with respect to another function ψ to its Caputo counterpart by a mapping transformation, the solutions of the ψ-Caputo fractional differential equations can be deduced from the solution representation for the Caputo version via an inverse transformation. We show that the mapping transformation for such derivatives is extremely useful in practical applications. The representation of solutions for constant order time ψ-Caputo fractional diffusion equation and variable order ψ-Caputo fractional mobile-immobile diffusion equation is investigated and the regularity estimates are deduced accordingly.

Analysis of Multi Term Fractional Differential Equations using Variational Iteration Method

In this paper, we have obtained an approximate solution of multi-term Caputo fractional differential equations (MFDEs) using the Variational iteration method (VIM). Further, we have obtained the convergence criteria and error approximation of VIM for solving generalized multi term fractional differential equations. The obtained results are shown using plots to demonstrated the efficiency and accuracy of the VIM

Stability of Generalized Proportional Caputo Fractional Differential Equations by Lyapunov Functions

In this paper, nonlinear nonautonomous equations with the generalized proportional Caputo fractional derivative (GPFD) are considered. Some stability properties are studied by the help of the Lyapunov functions and their GPFDs. A scalar nonlinear fractional differential equation with the GPFD is considered as a comparison equation, and some comparison results are proven. Sufficient conditions for stability and asymptotic stability were obtained. Examples illustrating the results and ideas in this paper are also provided.

A high-order numerical scheme for right Caputo fractional differential equations with uniform accuracy

&lt;abstract&gt;&lt;p&gt;In this paper, we mainly study the high-order numerical scheme of right Caputo time fractional differential equations with uniform accuracy. Firstly, we construct the high-order finite difference method for the right Caputo fractional ordinary differential equations (FODEs) based on piecewise quadratic interpolation. The local truncation error of right Caputo FODEs is given, and the stability analysis of the right Caputo FODEs is proved in detail. Secondly, the time fractional partial differential equations (FPDEs) with right Caputo fractional derivative is studied by coupling the time-dependent high-order finite difference method and the spatial central second-order difference scheme. Finally, three numerical examples are used to verify that the convergence order of high-order numerical scheme is $ 3-\lambda $ in time with uniform accuracy.&lt;/p&gt;&lt;/abstract&gt;

On a study of the representation of solutions of a $ \Psi $-Caputo fractional differential equations with a single delay

&lt;abstract&gt;&lt;p&gt;We give a representation of solutions to linear nonhomogeneous $ \Psi $-fractional delayed differential equations with noncommutative matrices. We newly define $ \Psi $-delay perturbation of Mittag-Leffler type matrix function with two parameters and apply the method of variation of constants to obtain the representation of the solutions. We investigate the existence and uniqueness of solutions for a class of $ \Psi $-fractional delayed semilinear differential equations by using Banach Fixed Point Theorem. Further, we establish the Ulam-Hyers stability result for the analyzed problem. Finally, we provide some examples to illustrate the applicability of our results.&lt;/p&gt;&lt;/abstract&gt;

Analysis of nonlinear coupled Caputo fractional differential equations with boundary conditions in terms of sum and difference of the governing functions

&lt;abstract&gt;&lt;p&gt;In this paper, we introduce a new class of nonlocal multipoint-integral boundary conditions with respect to the sum and difference of the governing functions and analyze a coupled system of nonlinear Caputo fractional differential equations equipped with these conditions. The existence and uniqueness results for the given problem are proved via the tools of the fixed point theory. We also discuss the case of nonlinear Riemann-Liouville integral boundary conditions. The obtained results are well-illustrated with examples.&lt;/p&gt;&lt;/abstract&gt;

Existence results for nonlinear sequential Caputo and Caputo-Hadamard fractional differential equations with three-point boundary conditions in Banach spaces

In this paper, we study the existence of solutions for nonlinear sequential Caputo and Caputo-Hadamard fractional differential equations with three-point boundary conditions by using measure of noncompactness combined with fixed point theorem of M?nch. An example illustrating the effectiveness of the theoretical results is presented.