Caputo Time-fractional Derivative Research Articles

A REACTION-DIFFUSION EQUATION ON STRATIFIED GROUPS

We study an extension of the classical Fisher-KPP equation to stratified groups. We show the positivity, boundedness, asymptotic time-behavior, and blow-up in finite time of solutions. These results are also discussed for the (time-fractional) Fisher-KPP equation by means of the Caputo time-fractional derivative.

First-passage time for the g-subdiffusion process of vanishing particles.

Subdiffusion equationand molecule survival equation, both with Caputo fractional time derivatives with respect to other functions g_{1} and g_{2}, respectively, are used to describe diffusion of a molecule that can disappear at any time with a constant probability. The process can be interpreted as an "ordinary" subdiffusion and "ordinary" molecule survival process in which timescales are changed by the functions g_{1} and g_{2}. We derive the first-passage time distribution for the process. The mutual influence of subdiffusion and molecule-vanishing processes can be included in the model when the functions g_{1} and g_{2} are related to each other. As an example, we consider the processes in which subdiffusion and molecule survival are highly related, which corresponds to the case of g_{1}≡g_{2}.

A numerical combined algorithm in cubic B-spline method and finite difference technique for the time-fractional nonlinear diffusion wave equation with reaction and damping terms

The applications of the diffusion wave model of a time-fractional kind with damping and reaction terms can occur within classical physics. This quantification of the activity can measure the diagnosis of mechanical waves and light waves. The goal of this work is to predict and construct numerical solutions for such a diffusion model based on the uniform cubic B-spline functions. The Caputo time-fractional derivative has been estimated using the standard finite difference technique, whilst, the uniform cubic B-spline functions have been employed to achieve spatial discretization. The convergence of the suggested blueprint is discussed in detail. To assert the efficiency and authenticity of the study, we compute the approximate solutions for a couple of applications of the diffusion model in electromagnetics and fluid dynamics. To show the mathematical simulation, several tables and graphs are shown, and it was found that the graphical representations and their physical explanations describe the behavior of the solutions lucidly. The key benefit of the resultant scheme is that the algorithm is straightforward and makes it simple to implement as utilized in the highlight and conclusion part.

A vertex-centred finite volume method for the 3D multi-term time and space fractional Bloch–Torrey equation with fractional Laplacian

We investigate the suitability of a fractional Laplacian in the construction of a three-dimensional (3D) multi-term time–space fractional Bloch–Torrey equation. Instead of using the Riesz fractional derivative as in most previous research, we consider the fractional Laplacian in space and multi-term Caputo fractional derivative in time. A vertex-centred finite volume method is firstly derived to discretise this 3D fractional dynamic system in combination with the matrix transfer technique and the weighted and shifted Grünwald–Letnikov formula. Since computing the fractional power of the matrix representation m(−Δ) of the Laplacian operator results in a highly dense matrix, we use matrix function approximations developed in a Krylov subspace framework and exploit the sparsity of m(−Δ) to reduce the computational overhead. Then the Lanczos method is employed to approximate vector products of matrix functions. A preconditioner is applied to accelerate the convergence process, which shows excellent performance and reduces the memory considerably. In particular, we introduce block matrix functions to deal with the coupled system. Furthermore, the analytical solution of the two-term time–space fractional diffusion equation is derived and expressed by the multinomial Mittag-Leffler function. The feasibility and effectiveness of the proposed numerical techniques are verified by some numerical examples. Compared with two-dimensional fractional models or single-term time fractional dynamic systems, this multi-term time–space fractional model provides more flexibility for the characterisation of anomalous diffusion in biological tissues.

Three-Point Boundary Value Problems of Coupled Nonlocal Laplacian Equations

This paper discusses a kind of coupled nonlocal Laplacian evolution equation with Caputo time-fractional derivatives and proportional delays. Green function and mild solution are firstly established by employing the approach of eigenvalues’ expansions and Fourier analysis technique. By the properties of eigenvalues and Mittag–Leffler functions, several vital estimations of Green functions are presented. In view of these estimations and some appropriate assumptions, the existence and uniqueness of the mild solution are studied by utilizing the Leray–Schauder fixed-point theorem and the Banach fixed-point theorem. Finally, an example is provided to illustrate the effectiveness of our main results.

Local Discontinuous Galerkin Method Coupled with Nonuniform Time Discretizations for Solving the Time-Fractional Allen-Cahn Equation

This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order α∈(0,1). Considering the weak singularity of the solution u(x,t) at the starting time, i.e., its first and/or second derivatives with respect to time blowing-up as t→0+ albeit the function itself being right continuous at t=0, two well-known difference formulas, including the nonuniform L1 formula and the nonuniform L2-1σ formula, which are used to approximate the Caputo time-fractional derivative, respectively, and the local discontinuous Galerkin (LDG) method is applied to discretize the spatial derivative. With the help of discrete fractional Gronwall-type inequalities, the stability and optimal error estimates of the fully discrete numerical schemes are demonstrated. Numerical experiments are presented to validate the theoretical results.

Recovery of the Order of Derivation for Fractional Diffusion Equations in an Unknown Medium

In this work, we investigate the recovery of a parameter in a diffusion process given by the order of derivation in time for a class of diffusion-type equations, including both classical and time-fractional diffusion equations, from the flux measurement observed at one point on the boundary. The mathematical model for time-fractional diffusion equations involves a Djrbashian--Caputo fractional derivative in time. We prove a uniqueness result in an unknown medium (e.g., diffusion coefficients, obstacle, initial condition, and source), i.e., the recovery of the order of derivation in a diffusion process having several pieces of unknown information. The proof relies on the analyticity of the solution at large time, asymptotic decay behavior, strong maximum principle of the elliptic problem, and suitable application of the Hopf lemma. Further we provide an easy-to-implement reconstruction algorithm based on a nonlinear least-squares formulation, and several numerical experiments are presented to complement the theoretical analysis.

Fractional order triple-phase-lag thermoelasticity in the context of two-temperature theory

A new mathematical model is designed through reconstructing the triple-phase-lag model proposed by Roy Choudhuri, where the integer ordered derivatives concerning the time variable are replaced by the Caputo time-fractional derivative of order δ∈(0,1]. Moreover, the three-phase-lag heat conduction equation is derived in the context of two-temperature theories to examine the joint effect of thermodynamical and conductive temperatures. The proposed formulation is applied to the finite dimensional solid slab, whose boundaries are subject to the thermal loading and free from mechanical forces. The integral transform technique is applied to converting the original boundary value problem into an eigenvalue problem. Analytical solutions of the unknown thermal parameters are obtained in the Laplace domain. Time-domain solutions of the analytical results are achieved numerically through the Gaver–Stehfest algorithm. The results are examined graphically for the fixed phase-lag variations admitting various existing thermoelasticity theories.

Numerical implementation and error analysis of nonlinear coupled fractional viscoelastic fluid model with variable heat flux

The studies conducted in the recent past reveals that the non-Newtonian are extensively applied in several engineering applications. The atypical Oldroyd-B fluid, a specific sub-category of non-Newtonian fluids, with fractional derivative constitutive equations, is very expedient to determine the approximation of many dilute polymer liquids. Since very few studies regarding the numerical solution of the aforesaid model are conducted, so the literature that can be referred to in this regard is scarce. The current article investigates the unsteady magneto-hydrodynamic flow of irregular Oldroyd-B fluid with a modified thermal flux restricted between two infinite parallel plates. Here, it is assumed that the thermal conductivity of the fluid is variable, and the impact of body forces like gravity is accordingly analyzed in terms of non-linear convection. The linear acceleration of the lower permeable plate in its plane prompts the flow. Further, we have investigated the heat transfer in the flow field through a modified fractional thermal gradient and internal heat source. The temperature gradient accelerates with time and exists among the flow boundaries. In order to determine the approximate solution unified with finite difference approximation for Caputo fractional time derivatives, we have utilized the standard Galerkin finite element method. Moreover, we have conducted a convergence analysis of the suggested numerical scheme and offered some valid error approximations. In the non-integer viscoelastic model, the local number and coefficient of skin friction are also deliberated. We have also performed some upright numerical simulations to highlight the ascendency of characteristic flow parameters of velocity and temperature fields in the proposed scheme. Here, we have analyzed that the resistive force increases by 40%, corresponding to an increment in α ranging from 0 to 0.4. Further, the heat transfer rate is diminish by 10.94% against an increase in variable thermal conductivity while it is enhanced by 37.78% corresponding to increased in Pr.

Numerical Analysis of Local Discontinuous Galerkin Method for the Time-Fractional Fourth-Order Equation with Initial Singularity

In this paper, efficient methods seeking the numerical solution of a time-fractional fourth-order differential equation with Caputo’s derivative are derived. The solution of such a problem has a weak singularity near the initial time t=0. The Caputo time-fractional derivative with derivative order α∈(0,1) is discretized by the well-known L1 formula on nonuniform meshes; for the spatial derivative, the local discontinuous Galerkin (LDG) finite element method is used. Based on the discrete fractional Gronwall’s inequality, we prove the stability of the proposed scheme and the optimal error estimate for the solution, i.e., (2−α)-order accurate in time and (k+1)-order accurate in space, when piece-wise polynomials of degree at most k are used. Moreover, a second-order and nonuniform time-stepping scheme is developed for the fractional model. The scheme uses the L2-1σ formula for the time fractional derivative and the LDG method for the space approximation. The stability and temporal optimal second-order convergence of the scheme are also shown. Finally, some numerical experiments are presented to confirm the theoretical results.

Abstract differential equations and Caputo fractional derivative

In this work we consider the abstract Cauchy problem with Caputo fractional time derivative of order $$\alpha \in (0,1]$$ , and discuss the continuity of the respective solutions regarding the parameter $$\alpha $$ . We also present a study about the continuity of the Mittag-Leffler families of operators (for $$\alpha \in (0,1]$$ ), when they are induced by sectorial operators.

Numerical treatments of the nonlinear coupled time‐fractional Schrödinger equations

In this article, we focused on solving numerically the coupled nonlinear Schrödinger equations (NLFSEs) with Caputo fractional derivative in time. The discrete schemes are constructed by using the trigonometric B‐spline collocation method and the non‐polynomial B‐spline method for space discretization respectively, while the ‐formula is applied in time discretization. The Von Neumann approach is applied to examine the stability of the proposed methods. For validating the accuracy and efficiency of the presented schemes, numerical tests are compared with the exact solution.

An inverse source problem of a semilinear time-fractional reaction–diffusion equation

In this paper, we investigate an inverse problem of determining the time-dependent source coefficient in a semilinear reaction–diffusion equation involving the Caputo fractional time derivative of order in the case of nonlocal boundary and integral overdetermination conditions. We prove the existence and uniqueness of the classical solution by Fourier analysis and the iteration method. Moreover, we show the continuous dependence of this solution upon the additional data of the inverse problem.

Numerical Solution of Diffusion Equation with Caputo Time Fractional Derivatives Using Finite-Difference Method with Neumann and Robin Boundary Conditions

Many problems in various branches of science, such as physics, chemistry, and engineering have been recently modeled as fractional ODEs and fractional PDEs. Thus, methods to solve such equations, especially in the nonlinear state, have drawn the attention of many researchers. The most important goal of researchers in solving such equations has been set to provide a solution with the possible minimum error. The fractional PDEs can be generally classified into two main types, spatial-fractional, and time-fractional differential equations. This study was designed to provide a numerical solution for the fractional-time diffusion equation using the finite-difference method with Neumann and Robin boundary conditions. The time fraction derivatives in the concept of Caputo were considered, also the stability and convergence of the proposed numerical scheme have been completely proven and a numerical test was also designed and conducted to assess the efficiency and precision of the proposed method. Eventually, it can be said that based on findings, the present technique can provide accurate results.

Fractional model of MHD blood flow in a cylindrical tube containing magnetic particles

In recent years, the use of magnetic particles for biomedicine and clinical therapies has gained considerable attention. Unique features of magnetic particles have made it possible to apply them in medical techniques. These techniques not only provide minimal invasive diagnostic tools but also transport medicine within the cell. In recent years, MRI, drug supply to infected tissue, Hyperthermia are more enhanced by the use of magnetic particles. The present study aims to observe heat and mass transport through blood flow containing magnetic particles in a cylindrical tube. Furthermore, the magnetic field is applied vertically to blood flow direction. The Caputo time fractional derivative is used to model the problem. The obtained partial fractional derivatives are solved using Laplace transform and finite Hankel transform. Furthermore, the effect of various physical parameters of our interest has also been observed through various graphs. It has been noticed that the motion of blood and magnetic particles is decelerated when the particle mass parameter and the magnetic parameter are increased. These findings are important for medicine delivery and blood pressure regulation.

Boundary stabilization and disturbance rejection for an unstable time fractional diffusion-wave equation

In this paper, we study boundary stabilization and disturbance rejection problem for an unstable time fractional diffusion-wave equation with Caputo time fractional derivative. For the case of no boundary external disturbance, both state feedback control and output feedback control via Neumann boundary actuation are proposed by the classical backstepping method. It is proved that the state feedback makes the closed-loop system Mittag-Leffler stable and the output feedback makes the closed-loop system asymptotically stable. When there is boundary external disturbance, we propose a disturbance estimator constructed by two infinite dimensional auxiliary systems to recover the external disturbance. A novel control law is then designed to compensate for the external disturbance in real time, and rigorous mathematical proofs are presented to show that the resulting closed-loop system is Mittag-Leffler stable and the states of all subsystems involved are uniformly bounded. As a result, we completely resolve, from a theoretical perspective, two long-standing unsolved mathematical control problems raised in Liang [Nonlinear Dyn. 38 (2004) 339–354] where all results were verified by simulations only.

Energy-dissipation for time-fractional phase-field equations

<p style='text-indent:20px;'>We consider a class of time-fractional phase field models including the Allen-Cahn and Cahn-Hilliard equations. We establish several weighted positivity results for functionals driven by the Caputo time-fractional derivative. Several novel criterions are examined for showing the positive-definiteness of the associated kernel functions. We deduce strict energy-dissipation for a number of non-local energy functionals, thereby proving fractional energy dissipation laws.</p>

Computational Approach via Half-Sweep and Preconditioned AOR for Fractional Diffusion

Solving time-fractional diffusion equation using a numerical method has become a research trend nowadays since analytical approaches are quite limited. There is increasing usage of the finite difference method, but the efficiency of the scheme still needs to be explored. A half-sweep finite difference scheme is well-known as a computational complexity reduction approach. Therefore, the present paper applied an unconditionally stable half-sweep finite difference scheme to solve the time-fractional diffusion equation in a one-dimensional model. Throughout this paper, a Caputo fractional operator is used to substitute the time-fractional derivative term approximately. Then, the stability of the difference scheme combining the half-sweep finite difference for spatial discretization and Caputo time-fractional derivative is analyzed for its compatibility. From the formulated half-sweep Caputo approximation to the time-fractional diffusion equation, a linear system corresponds to the equation contains a large and sparse coefficient matrix that needs to be solved efficiently. We construct a preconditioned matrix based on the first matrix and develop a preconditioned accelerated over relaxation (PAOR) algorithm to achieve a high convergence solution. The convergence of the developed method is analyzed. Finally, some numerical experiments from our research are given to illustrate the efficiency of our computational approach to solve the proposed problems of time-fractional diffusion. The combination of a half-sweep finite difference scheme and PAOR algorithm can be a good alternative computational approach to solve the time-fractional diffusion equation-based mathematical physics model.

Numerical Simulation of the Solitary Wave Collision Process in Time Fractional Orders Based on the Coupled Pure Meshless Method

In this paper, a coupled pure meshless finite pointset method (CFPM) is developed for the first time to numerically predict the inelastic collision process of solitary wave described by the time fractional coupled nonlinear Schrödinger (TF-CNLS) equation. Its construction process is as follows: 1) a high-precision difference scheme is used for the Caputo time fractional derivative; 2) FPM discrete scheme based on Taylor expansion and weighted least square method is adopted for spatial derivatives; 3) The region is locally refined and the double cosine kernel function with good stability is used to improve the numerical accuracy. In the numerical study, the one-dimensional TF-CNLS equations with analytical solutions are solved by CFPM, and the error and convergence rate are analyzed when the nodes are uniformly distributed or locally refined, showing that the proposed method has the approximate second-order accuracy and the flexibility of easy local refinement. Secondly, the inelastic collision process of solitary waves, which is described by the one-dimensional TF-CNLS equation without analytical solution, is numerically predicted by CFPM, and the wave collapse phenomenon depicted above is completely different from the multi-wave phenomenon under the integer order. Meanwhile, by comparing the result with that obtained by finite difference method, it suggests that the CFPM is reliable to predict the complex propagation of the inelastic collision process of the solitary waves under the time fractional order.

Blow-Up and Global Existence of Solutions for the Time Fractional Reaction–Diffusion Equation

In this paper, we investigate a reaction–diffusion equation with a Caputo fractional derivative in time and with boundary conditions. According to the principle of contraction mapping, we first prove the existence and uniqueness of local solutions. Then, under some conditions of the initial data, we obtain two sufficient conditions for the blow-up of the solutions in finite time. Moreover, the existence of global solutions is studied when the initial data is small enough. Finally, the long-time behavior of bounded solutions is analyzed.