Multi-term Fractional Differential Equations Research Articles

Modeling of implicit multi term fractional delay differential equation: Application in pollutant dispersion problem

This work explores a new abstract model using multi-term fractional differential operator and delay effect. The model is defined by the existence of a delay parameter and insertion of multi term fractional differential operators in the input function. For the solution of the newly formulated problem, we utilized fixed-point theorems, which provide a powerful analytical tool for establishing the existence and uniqueness of solutions in functional spaces. Additionally, we investigate the stability properties of the implicit multi-term fractional delay differential equation (IMTFDDE) and employ the concept of Ulam-Hyers stability to assess their behaviour. The Ulam-Hyers stability framework is a contemporary approach that characterizes the sensitivity of solutions in functional spaces. Furthermore, we use an example to show how to apply our results. Lastly, for the real implementation of this research endeavour, we provided a real model devoted to modeling the dispersion of pollutants in a river while accounting for delayed effects and strip boundary circumstances.

An RBF-FD Method for Numerical Solutions of 2D Diffusion-Wave and Diffusion Equations of Distributed Fractional Order

AbstractThe subject of this paper is to propose a numerical algorithm for solving 2D diffusion and diffusion-wave equations of distributed order fractional derivatives. Such equations arise in modelling complex systems and have many important applications. Existence of integral term over the order of fractional derivative causes the high complexity of these equations and so their numerical solutions needs special cares. Using Gauss quadrature approach for discretizing the integral term of fractional derivative converts the distributed equation into a multi-term fractional differential equation. Then, the time variable is discretized with a suitable finite difference approach. The resultant semi-discretized equations are fully discretized by a radial basis function-generated finite difference based method. Convergence of the method are studied numerically. Various kind of test problems are considered for a comprehensive numerical study and the results confirm the efficiency of the method.

Existence of mild solutions for a multi-term fractional differential equation via ψ-(γ,σ)-resolvent operators

In this paper, we consider a multi-term fractional differential equation involving the [Formula: see text]-Caputo fractional derivative. Under some suitable hypotheses, we prove that our proposed problem has at least one mild solution expressed in terms of the [Formula: see text]-[Formula: see text]-resolvent. As an application, we provide a suitable example at the end of this work to illustrate our theoretical results.

Solving existence results in multi-term fractional differential equations via fixed points

Many researchers are interested in the existence theory of solutions to fractional differential equations. In the literature, existence results have been obtained by using a variety of fixed point problems, including the fixed-point problems of Lefschetz, Kleene, Tychonoff, and Banach. In this article, we propose a generalized version of the contraction principle in the context of controlled rectangular metric space. With this result, we address the existence and uniqueness results for the following fractional-order differential equations.1. The nonlinear multi-term fractional delay differential equation L(D)ζ(ϖ)=σ(ϖ,ζ(ϖ),ζ(ϖ−τ)),ϖ∈J=[0,T],T>0;ζ(ϖ)=σ̄(ϖ),ϖ∈[−τ,0]. where, L(D)=γwcDδw+γw−1cDδw−1+...+γ1cDδ1+γ0cDδ0,γ♭∈R(♭=0,1,2,3....w),γw≠0,0≤δ0<δ1<δ2....δw−1<δw<1, and cDδ denotes the Caputo fractional derivative of order δ.2. The Caputo type fractional differential equation cDδζ(ϖ)=σ(ϖ,ζ(ϖ)),5<δ≤6,ϖ∈[0,T], with ζ(0)=−ζ(T),ζ′(0)=−ζ′(T),ζ′′(0)=−ζ′′(T),ζ′′′(0)=−ζ′′′(T),ζ4(0)=−ζ4(T),ζ5(0)=−ζ5(T)

Cauchy–Dirichlet Problem to Semilinear Multi-Term Fractional Differential Equations

In this paper, we analyze the well-posedness of the Cauchy–Dirichlet problem to an integro-differential equation on a multidimensional domain Ω⊂Rn in the unknown u=u(x,t), Dtν0(ϱ0u)−Dtν1(ϱ1u)−L1u−∫0tK(t−s)L2u(x,s)ds=f(x,t)+g(u),0&lt;ν1&lt;ν0&lt;1, where Dtνi are the Caputo fractional derivatives, ϱi=ϱi(x,t) with ϱ0≥μ0&gt;0, and Li are uniform elliptic operators with time-dependent smooth coefficients. The principal feature of this equation is related to the integro-differential operator Dtν0(ϱ0u)−Dtν1(ϱ1u), which (under certain assumption on the coefficients) can be rewritten in the form of a generalized fractional derivative with a non-positive kernel. A particular case of this equation describes oxygen delivery through capillaries to tissue. First, under proper requirements on the given data in the linear model and certain relations between ν0 and ν1, we derive a priori estimates of a solution in Sobolev–Slobodeckii spaces that gives rise to providing the Hölder regularity of the solution. Exploiting these estimates and constructing appropriate approximate solutions, we prove the global strong solvability to the corresponding linear initial-boundary value problem. Finally, obtaining a priori estimates in the fractional Hölder classes and assuming additional conditions on the coefficients ϱ0 and ϱ1 and the nonlinearity g(u), the global one-valued classical solvability to the nonlinear model is claimed with the continuation argument method.

Chaotic attractors and fixed point methods in piecewise fractional derivatives and multi-term fractional delay differential equations

Using generalized cyclic contractions, we establish some fixed point results in controlled rectangular metric spaces. Some subsequent outcomes are obtained. Moreover, some necessary conditions to demonstrate the existence of solutions for the multi-term fractional delay differential equations with wth order and the piecewise equations under the setting of non-singular type derivative are established in this paper. In order to demonstrate the effectiveness of our results, we provided some numerical examples.

Stability Properties of Multi-Term Fractional-Differential Equations

Necessary and sufficient stability and instability conditions are reviewed and extended for multi-term homogeneous linear fractional differential equations with Caputo derivatives and constant coefficients. A comprehensive review of the state of the art regarding the stability analysis of two-term and three-term fractional-order differential equations is provided, which is then extended to the case of four-term fractional-order differential equations. The stability and instability properties are characterized with respect to the coefficients of the multi-term fractional differential equations, leading to both fractional-order-dependent and fractional-order-independent characterizations. In the general case, fractional-order-independent stability and instability properties are described for fractional-order differential equations with an arbitrary number of fractional derivatives.

Numerical Treatment of Multi-Term Fractional Differential Equations via New Kind of Generalized Chebyshev Polynomials

The main aim of this paper is to introduce a new class of orthogonal polynomials that generalizes the class of Chebyshev polynomials of the first kind. Some basic properties of the generalized Chebyshev polynomials and their shifted ones are established. Additionally, some new formulas concerned with these generalized polynomials are established. These generalized orthogonal polynomials are employed to treat the multi-term linear fractional differential equations (FDEs) that include some specific problems that arise in many applications. The basic idea behind the derivation of our proposed algorithm is built on utilizing a new power form representation of the shifted generalized Chebyshev polynomials along with the application of the spectral Galerkin method to transform the FDE governed by its initial conditions into a system of linear equations that can be efficiently solved via a suitable numerical solver. Some illustrative examples accompanied by comparisons with some other methods are presented to show that the presented algorithm is useful and effective.

BOUNDARY-VALUE PROBLEM FOR THE MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATION WITH CAPUTO DERIVATIVE

The extensive application of fractional differential equations and boundary-value problems for these equations promotes the development of the theory and the appearance of numerous publications in this field. One of the types of such equations are equations containing more than one differential operator of fractional order. This paper deals with the study of linear boundary-value problem for the multi-term fractional differential equation with the Caputo derivative. We considered the left fractional Caputo derivative, which is convenient for the description of systems with memory. The boundaryvalue problem is specified by linear vector functional such that the number of it components does not coincide with the number of the orders of the derivative. Assume that the coefficients of the equation are continuous functions and the orders of the derivative are commensurate. A multi-term fractional differential equation is reduced to an equivalent system of differential equations containing only one fractional operator. The general solution of the system of fractional differential equations consisting of a general solution of the associated homogeneous system and the arbitrary particular solution of the inhomogeneous system is considered. The particular solution we found, which is also a solution of the system of linear Volterra integral equations of the second kind with square summable kernels. The question of the solvability of the boundary-value problem for the multi-term fractional differential equations was studied. We considered the critical case, i.e. case when the homogeneous problem has nontrivial solutions. By using the theory of pseudo-inverse matrices, the necessary and sufficient conditions for solvability of the given problem are established. Moreover, a family of linearly independent solutions of this boundary-value problem is constructed.

Initial-boundary value problems to semilinear multi-term fractional differential equations

For $ \nu, \nu_i, \mu_j\in(0, 1) $, we analyze the semilinear integro-differential equation on the one-dimensional domain $ \Omega = (a, b) $ in the unknown $ u = u(x, t) $$ {\mathbf{D}}_{t}^{\nu}(\varrho_{0}u)+\sum\limits_{i = 1}^{M}{\mathbf{D}}_{t}^{\nu_{i}}(\varrho_{i}u) -\sum\limits_{j = 1}^{N}{\mathbf{D}}_{t}^{\mu_{j}}(\gamma_{j}u) -\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u+f(u) = g(x, t), $where $ {\mathbf{D}}_{t}^{\nu}, {\mathbf{D}}_{t}^{\nu_{i}}, {\mathbf{D}}_{t}^{\mu_{j}} $ are Caputo fractional derivatives, $ \varrho_0 = \varrho_0(t)&gt;0, $ $ \varrho_{i} = \varrho_{i}(t) $, $ \gamma_{j} = \gamma_{j}(t) $, $ \mathcal{L}_{k} $ are uniform elliptic operators with time-dependent smooth coefficients, $ \mathcal{K} $ is a summable convolution kernel. Particular cases of this equation are the recently proposed advanced models of oxygen transport through capillaries. Under certain structural conditions on the nonlinearity $ f $ and orders $ \nu, \nu_i, \mu_j $, the global existence and uniqueness of classical and strong solutions to the related initial-boundary value problems are established via the so-called continuation arguments method. The crucial point is searching suitable a priori estimates of the solution in the fractional Hölder and Sobolev spaces. The problems are also studied from the numerical point of view.

A stability analysis for multi-term fractional delay differential equations with higher order

As a widely used tool modeling some processes and systems in a variety of fields, fractional delay differential equations (FDDEs) with higher order have attracted much attention of the scientific community for years. Motivated by Yao et al. (2022), in which a single term has been done, we are much more interested in the stability analysis for multi-term FDDEs. In addition to the widely used Laplace transform method and decoupling technique for the characteristic equation, a region embedding technique is first introduced to handle the multiple fractional exponents. The existing results are generalized to multi-term FDDEs with higher order and the damping term of the classical integer-order delay differential equation is extended to fractional calculus. Numerical simulations for FDDEs and time-fractional telegraph equations with time delay are presented to illustrate the efficiency and validity of our results.

Necessary and sufficient condition for the existence of solutions to two-point boundary value problem of fuzzy linear multi-term fractional differential equations

Necessary and sufficient condition for the existence of solutions to two-point boundary value problem of fuzzy linear multi-term fractional differential equations

Existence and numerical analysis using Haar wavelet for fourth-order multi-term fractional differential equations

Existence and numerical analysis using Haar wavelet for fourth-order multi-term fractional differential equations

An Euler–Maruyama method and its fast implementation for multiterm fractional stochastic differential equations

In this paper, we derive an Euler–Maruyama (EM) method for a class of multiterm fractional stochastic nonlinear differential equations and prove its strong convergence. The strong convergence order of this EM method is , where is the order of Caputo fractional derivative satisfying that , and . Then, a fast implementation of this proposed EM method is also presented based on the sum‐of‐exponentials approximation technique. Finally, some numerical experiments are given to verify the theoretical results and computational efficiency of our EM method.

Self-similarity and response of fractional differential equations under white noise input

Self-similarity, fractal behaviour and long-range dependence are observed in various branches of physical, biological, geological, socioeconomics and mechanical systems. Self-similarity, also termed self-affinity, is a concept that links the properties of a phenomenon at a certain scale with the same properties at different time scales as it happens in fractal geometry. The fractional Brownian motion (fBm), i.e. the Riemann–Liouville fractional integral of the Gaussian white noise, is self-similar; in fact by changing the temporal scale t→at (a>0), the statistics in the new time axis (at) remain proportional to those calculated in the previous axis (t). The proportionality coefficient is a2H being H>0 the Hurst index. In the practical applications, the phenomena are usually ruled by fractional differential equations involving more terms. In this paper it is shown that the response of a multi-term fractional differential equation is a linear combination of self-similar processes with increasing order of Hurst exponent. The consequences of self-similarity are discussed in detail, closed forms of correlation and variance are presented for the general case and particularized for the cases of engineering interest.

Formulation, Solution’s Existence, and Stability Analysis for Multi-Term System of Fractional-Order Differential Equations

In the recent past, multi-term fractional equations have been studied using symmetry methods. In some cases, many practical test problems with some symmetries are provided to demonstrate the authenticity and utility of the used techniques. Fractional-order differential equations can be formulated by using two types of differential operators: single-term and multi-term differential operators. Boundary value problems with single- as well as multi-term differential operators have been extensively studied, but several multi-term fractional differential equations still need to be formulated, and examination should be done with symmetry or any other feasible techniques. Therefore, the purpose of the present research work is the formulation and study of a new couple system of multi-term fractional differential equations with delay, as well as supplementation with nonlocal boundary conditions. After model formulation, the existence of a solution and the uniqueness conditions will be developed, utilizing fixed point theory and functional analysis. Moreover, results related to Ulam’s and other types of functional stability will be explored, and an example is carried out to illustrate the findings of the work.

Galerkin operational approach for multi-dimensions fractional differential equations

The current manuscript introduces a novel numerical treatment for multi-term fractional differential equations with variable coefficients. The spectral Galerkin approach is developed with the operational matrix of fractional derivatives to get a system of algebraic equations that can be solved using a suitable technique. The operational matrix is introduced based on a combination of the shifted Jacobi polynomials. The proposed approach is applied also for 1+1 and 2+1 multi-term time-fractional diffusion and diffusion-wave equations. In addition, the convergence analysis for the presented approaches is investigated. Some specific numerical examples are given to ascertain the wide applicability and the good efficiency of the suggested algorithms, and to confirm that nonlocal numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account.

Uniqueness and Ulam–Hyers–Mittag–Leffler stability results for the delayed fractional multiterm differential equation involving the Φ-Caputo fractional derivative

The principal aim of this paper is to establish the uniqueness and Ulam–Hyers–Mittag–Leffler (UHML) stability of solutions for a new class of multiterm fractional time-delay differential equations in the context of the Φ-Caputo fractional derivative. To achieve this purpose, the generalized Laplace transform method alongside facet with properties of the Mittag–Leffler functions (M–LFs) are utilized to give a new representation formula of the solutions for the aforementioned problem. Besides that, the uniqueness of the solutions of the considered problem is also proved by applying the well-known Banach contraction principle coupled with the Φ-fractional Bielecki-type norm, while the Φ-fractional Gronwall type inequality and the Picard operator (PO) technique, combined with the abstract Gronwall lemma, are used to prove the UHML stability results for the proposed problem. Lastly, an example is offered to assure the validity of the obtained theoretical results.

A Generalized NPCM for Solving Multi-Term Fractional Differential Equations

A Generalized NPCM for Solving Multi-Term Fractional Differential Equations

Existence and Hyers–Ulam Stability for a Multi-Term Fractional Differential Equation with Infinite Delay

This paper is devoted to investigating one type of nonlinear two-term fractional order delayed differential equations involving Caputo fractional derivatives. The Leray–Schauder alternative fixed-point theorem and Banach contraction principle are applied to analyze the existence and uniqueness of solutions to the problem with infinite delay. Additionally, the Hyers–Ulam stability of fractional differential equations is considered for the delay conditions.