Temporal Fractional Derivatives Research Articles

α-Robust Error Analysis of L2-1σ Scheme on Graded Mesh for Time-Fractional Nonlocal Diffusion Equation

Abstract In this work, a time-fractional nonlocal diffusion equation is considered. Based on the L2-1σ scheme on a graded mesh in time and the standard finite element method (FEM) in space, the fully-discrete L2-1σ finite element method is investigated for a time-fractional nonlocal diffusion problem. We prove the existence and uniqueness of fully-discrete solution. The α-robust error bounds are derived, i.e., bounds remain valid as α→1−, where α ∈(0,1) is the order of a temporal fractional derivative. The numerical experiments are presented to justify the theoretical findings.

The dimension reduction method of two-grid Crank–Nicolson mixed finite element solution coefficient vectors for nonlinear fourth-order reaction diffusion equation with temporal fractional derivative

Herein, we mainly resort to a proper orthogonal decomposition (POD) to study the dimension reduction of unknown solution coefficient vectors in the two-grid Crank–Nicolson mixed finite element (CNMFE) (TGCNMFE) method for the nonlinear fourth-order reaction diffusion equation with temporal fractional derivative and establish a new reduced-dimension extrapolated TGCNMFE (RDETGCNMFE) method. For this purpose, we firstly retrospect the TGCNMFE method for the nonlinear fourth-order reaction diffusion equation and provide the existence, unconditional stability, and error estimates of the TGCNMFE solutions. Thereafter, we use the POD method to reduce the dimensionality of the unknown TGCNMFE solution coefficient vectors of the nonlinear fourth-order reaction diffusion equation and develop the new RDETGCNMFE method, and use matrix analysis to analyze the existence, unconditional stability, and errors of the RDETGCNMFE solutions. Finally, we perform numerical experiments to verify our theoretical results and show the advantages of the RDETGCNMFE method. It is worth noting that the RDETGCNMFE method herein is completely distinguished from the existing ones. Hence, the work in this paper is brand-new.

Innovative Method for Computing Approximate Solutions of Non-Homogeneous Wave Equations with Generalized Fractional Derivatives

In this work, a well-known non-homogeneous wave equation with temporal fractional derivative is approximately investigated. A recently defined generalized non-local fractional derivative is utilized as the fractional operator. A novel technique is proposed to approximate the solutions of wave equation with generalized fractional derivative. The proposed method is based on the shifted Chebyshev polynomials and a combination of collocation and residual function methods. Theoretical analysis of the convergence of the proposed method is performed. Approximate solutions are derived in both rectangular and non-rectangular (general) domains.

An efficient spline technique for solving time-fractional integro-differential equations

Spline curves are very prominent in the mathematics due to their simple construction, accuracy of assessment and ability to approximate complicated structures into interactive curved designs. A spline is a smooth piece-wise polynomial function. The primary goal of this study is to use extended cubic B-spline (ExCuBS) functions with a new second order derivative approximation to obtain the numerical solution of the weakly singular kernel (SK) non-linear fractional partial integro-differential equation (FPIDE). The spatial and temporal fractional derivatives are discretized by ExCuBS and the Caputo finite difference scheme, respectively. The present study found that it is stable and convergent. The validity of the current approach is examined on a few test problems, and the obtained outcomes are compared with those that have previously been reported in the literature.

Analysis of anomalous transport with temporal fractional transport equations in a bounded domain

Anomalous transport in magnetically confined plasmas is investigated using temporal fractional transport equations. The use of temporal fractional transport equations means that the order of the partial derivative with respect to time is a fraction. In this case, the Caputo fractional derivative relative to time is utilized, because it preserves the form of the initial conditions. A numerical calculation reveals that the fractional order of the temporal derivative α (α ∈ (0,1), sub-diffusive regime) controls the diffusion rate. The temporal fractional derivative is related to the fact that the evolution of a physical quantity is affected by its past history, depending on what are termed memory effects. The magnitude of α is a measure of such memory effects. When α decreases, so does the rate of particle diffusion due to memory effects. As a result, if a system initially has a density profile without a source, then the smaller the α is, the more slowly the density profile approaches zero. When a source is added, due to the balance of the diffusion and fueling processes, the system reaches a steady state and the density profile does not evolve. As α decreases, the time required for the system to reach a steady state increases. In magnetically confined plasmas, the temporal fractional transport model can be applied to off-axis heating processes. Moreover, it is found that the memory effects reduce the rate of energy conduction and hollow temperature profiles can be sustained for a longer time in sub-diffusion processes than in ordinary diffusion processes.

Novel analytical technique to find diversity of solitary wave solutions for Wazwaz-Benjamin-Bona Mahony equations of fractional order

The newly modified Benjamin-Bona-Mahony equations in three dimensions are examined in the current work through the use of conformable fractional derivatives to incorporate spatial and temporal fractional order derivatives. With the aid of the Wolfram Mathematica software, a variety of solutions, including hyperbolic and periodic function solutions, are created. The information was gathered to evaluate how well the proposed novel (G’/G2)-expansion method could compute exact solutions of the WBBM equations that could be applied to the implementation of the nonlinear water model in the ocean and coastal engineering. Table 1 shows a comparison between the recently discovered solutions and those that have previously been reported in the literature. There are some visual representations of the solutions.

An efficient numerical scheme and its analysis for the multiterm time‐fractional convection‐diffusion‐reaction equation

This article aims at developing a robust numerical method based on graded mesh for solving the multiterm time‐fractional convection‐diffusion‐reaction (TFCDR) equation whose solution very likely exhibits a weak singularity at the initial time. The time‐fractional derivative in the model problem is described in terms of Liouville–Caputo. In order to handle the weak singularity at the initial time, we use a graded mesh technique for discretization of multiterm temporal fractional derivatives. The space derivatives are approximated by means of a compact finite difference (CFD) method. The proposed method is analyzed for its stability and convergence. Two numerical examples are considered to demonstrate the applicability and accuracy of the method. It is shown that the proposed graded mesh technique provides an optimal rate of convergence in time direction for the problem with nonsmooth exact solution, while the method on the uniform mesh yields a nonoptimal rate of convergence.

A CONVERGENCE ANALYSIS OF THE MOBILE–IMMOBILE ADVECTION–DISPERSION MODEL OF TEMPORAL FRACTIONAL ORDER ARISING IN WATERSHED CATCHMENTS AND RIVERS

An efficient high-order computational procedure is going to be created in this paper to determine the solution to the mobile–immobile advection–dispersion model (MIAD) of temporal fractional order [Formula: see text], which can be employed to model the solute forwarding in watershed catchments and floods. To do it, the temporal-first derivative of MIAD is discretized by using the finite-difference technique’s first-order precision and the linear interpolation’s temporal-fractional derivative. On either side, the space derivative is simulated using a collocation approach based on the Legendre basis to generate the full-discrete method. The order of MIAD-convergence for the implicit numerical structure is explained. Additionally, a basic conceptual discussion of the temporal-discretized stability mechanism is included in this paper. Finally, two models are provided to show the reliability and excellence of the organized approach.

Investigation and analysis of the numerical approach to solve the multi-term time-fractional advection-diffusion model

<abstract><p>In this paper, a methodical approach is presented to approximate the multi-term fractional advection-diffusion model (MT-FAD). The Lagrange squared interpolation is used to discretize temporal fractional derivatives, and Legendre polynomials are shifted as an operator to discretize the spatial fractional derivatives. The advantage of these numerical techniques lies in the orthogonality of Legendre polynomials and its matrix operations. A quadratic implicit design as well as its stability and convergence analysis are evaluated. It should be noted that the theoretical proof obtained from this study represents the first results for these numerical schemes. Finally, we provide three numerical examples to verify the validity of the proposed methods and demonstrate their accuracy and effectiveness in comparison with previous studies shown in [W. P. Bu, X. T. Liu, Y. F. Tang, J. Y. Yang, Finite element multigrid method for multi-term time fractional advection diffusion equations, <italic>Int. J. Model. Simul. Sci. Comput.</italic>, <bold>6</bold> (2015), 1540001].</p></abstract>

A hybrid collocation method for the computational study of multi-term time fractional partial differential equations

In this article, we focus on the numerical study of one and two dimensional higher order multi-term time fractional partial differential equations. In the adopted strategy, the temporal fractional derivative is replaced via well known L 1 formula and the integer order space derivatives are approximated by truncated one and two dimensional wavelet series. The fascinating nature of the scheme is to use collocation approach which convert the governing equations to the system of algebraic equations from which the wavelet coefficients can be calculated. Next, stability of the proposed scheme is investigated theoretically which is also the fundamental subject of the current work. Further computational convergence rate is computed which predicts that the order is approximately two. The scheme is applied to solve one dimensional beam models (fourth order partial differential equations) and two dimensional fissured rock models (Sobolev equations). Efficiency of the scheme is examined with the help of various error norms such as I ∞ , I r m s and I 2 . Simulations indicate pretty much good results from which we can say that the scheme is suitable and robust for both one and two dimensional problems.

Sensitive Demonstration of the Twin-Core Couplers including Kerr Law Non-Linearity via Beta Derivative Evolution

To obtain new solitary wave solutions for non-linear directional couplers using optical meta-materials, a new extended direct algebraic technique (EDAT) is used. This model investigates solitary wave propagation inside a fiber. As a result, twin couplers are the subject of this study. Kerr law is the sort of non-linearity addressed there. Because it offers solutions to problems with large tails or infinite fluctuations, the resulting solution set is more generalized than the current solution because it is turned into a fractional-order derivative. Furthermore, the found solutions are fractional solitons with spatial–temporal fractional beta derivative evolution. In intensity-dependent switches, these nonlinear directional couplers also serve as limiters. Non-linearity alters the transmission constants of a system’s modes. The significance of the beta derivative parameter and mathematical approach is demonstrated graphically, with a few of the extracted solutions. A parametric analysis revealed that the fractional beta derivative parameter has a significant impact on the soliton amplitudes. With the aid of the advanced software tools for numerical computations, the categories of semi-dark solitons, singular dark-pitch solitons, single solitons of Type-1 along with 2, intermingled hyperbolically, trigonometric, and rational solitons were established and evaluated. We also discussed sensitivity analysis, which is an inquiry that determines how sensitive our system is. A comparative investigation via different fractional derivatives was also studied in this paper so that one can easily understand the correlation with other fractional derivatives. The findings demonstrate that the approach is simple and efficient and that it yields generalized analytical results. The findings will be extremely beneficial in examining and comprehending physical issues in nonlinear optics, specifically in twin-core couplers with optical metamaterials.

A high-order numerical scheme based on graded mesh and its analysis for the two-dimensional time-fractional convection-diffusion equation

This work proposes a numerical method for solving the two-dimensional time-fractional convection-diffusion (TFCD) equation. The time-fractional derivative is considered in the sense of Caputo. The considered problem with Caputo temporal fractional derivative generally shows a weak singularity at the initial time. In order to handle the weak singularity at t = 0 , the time-fractional derivative is discretized on a non-uniform graded mesh. A high-order compact operator is employed for approximating the derivatives in space directions. The resulting system of equations is then solved using the Alternating Direction Implicit (ADI) approach. We prove the stability and convergence of this scheme. Numerical experiment is performed to demonstrate the applicability and accuracy of the method. The considered numerical example subjected to nonsmooth solution confirms that the method has an order min ⁡ ( 2 − α , r α , 2 α + 1 ) in the temporal direction, where α ∈ ( 0 , 1 ) is the order of the fractional derivative and r is a parameter used in construction of the graded mesh and has a rate of convergence of order four in spatial direction. The proposed method on graded mesh has an advantage in terms of numerical accuracy over the method on the uniform mesh. The CPU time (in seconds) for the present method is also provided.

An efficient numerical scheme and its stability analysis for a time-fractional reaction diffusion model

An efficient numerical scheme is presented to solve a time-fractional reaction–diffusion (TFRD) model with Caputo temporal–fractional derivative. The L1 scheme is employed to approximate the Caputo temporal–fractional derivative and a collocation method based on sextic B-spline (SBS) basis functions is used for discretization of the spatial variable. The stability of the method is investigated. Three numerical examples are provided to illustrate the performance and accuracy of the method. The convergence order of proposed method is O(Δt2−α+Δx6), where α is the order of fractional derivative (FD) (0<α<1) and Δt and Δx represent, respectively, the temporal step size and the spatial mesh size. The influence of the order of fractional derivative α on the solution profile is examined. The computed results and CPU times are compared with those obtained by the standard second-order finite difference method.

Collocation Method for Optimal Control of a Fractional Distributed System

In this paper, a collocation method based on the Jacobi polynomial is proposed for a class of optimal-control problems of a fractional distributed system. By using the Lagrange multiplier technique and fractional variational principle, the stated problem is reduced to a system of fractional partial differential equations about control and state functions. The uniqueness of this fractional coupled system is discussed. For spatial second-order derivatives, the proposed method takes advantage of Jacobi polynomials with different parameters to approximate solutions. For a temporal fractional derivative in the Caputo sense, choosing appropriate basis functions allows the collocation method to be implemented easily and efficiently. Exponential convergence is verified numerically under continuous initial conditions. As a particular example, the relation between the state function and the order of the fractional derivative is analyzed with a discontinuous initial condition. Moreover, the numerical results show that the integration of the state function will decay as the order of the fractional derivative decreases.

Turing instability and pattern formation of a fractional Hopfield reaction–diffusion neural network with transmission delay

It is well known that integer-order neural networks with diffusion have rich spatial and temporal dynamical behaviors, including Turing pattern and Hopf bifurcation. Recently, some studies indicate that fractional calculus can depict the memory and hereditary attributes of neural networks more accurately. In this paper, we mainly investigate the Turing pattern in a delayed reaction–diffusion neural network with Caputo-type fractional derivative. In particular, we find that this fractional neural network can form steadily spatial patterns even if its first-derivative counterpart cannot develop any steady pattern, which implies that temporal fractional derivative contributes to pattern formation. Numerical simulations show that both fractional derivative and time delay have influence on the shape of Turing patterns.

On a discrete model that dissipates the free energy of a time-space fractional generalized nonlinear parabolic equation

The present work is the first manuscript of the literature in which a numerical model that preserves the dissipation of free energy is proposed to solve a time-space fractional generalized nonlinear parabolic system. More precisely, we investigate an extension of the multidimensional heat equation with nonlinear reaction and fractional derivatives in space and time. The model considers homogeneous Dirichlet boundary conditions and initial data. The temporal fractional derivative is understood in the Caputo sense, and Riesz fractional derivatives are employed in the spatial variables. The system possesses a free energy functional and we prove mathematically that it is a decreasing function of time. Motivated by these facts, we propose a discretization of the space-time-fractional model which dissipates the discrete free energy. The discrete model is obtained using fractional-order centered differences to approximate the Riesz derivatives, and the L1 scheme to estimate the Caputo derivatives. The numerical model is a consistent approximation of the continuous system with quadratic order in space, and at most second order in time. Simulations confirm the capability of the numerical model to dissipate the free energy of the continuous fractional system.

On the nonlinear wave transmission in a nonlinear continuous hyperbolic regime with Caputo-type temporal fractional derivative

This present manuscript studies a nonlinear hyperbolic model in fractional form which generalizes the nonlinear Klein–Gordon system. The equation under investigation includes the presence of a time-fractional operator of the Caputo type. A space-fractional form of that equation with integer-order temporal derivative has been previously investigated to elucidate the existence of localized wave transmission in relativistic wave equations. Here, we employ numerical techniques to estimate the solution of the fractional equation. The method has consistency of fourth order in space. Meanwhile, the temporal order of consistency is equal to 3−α. We considered herein a sinusoidal perturbation of the medium. The simulations show the existence of the transmission of localized nonlinear modes in some complex fractional media governed by hyperbolic models. Physically, the present work investigates the phenomenon of nonlinear supratransmission in a continuous generalization of linear chains of harmonic oscillators with memory effects. In particular, the present work corroborates the presence of this nonlinear phenomenon in chains of pendula with memory and arrays of Josephson junctions attached through superconducting wires and memory effects.

Dynamics and pattern formation of a diffusive predator - prey model in the subdiffusive regime in presence of toxicity

Motivated by the fact that the restrictive conditions for a Turing instability are relaxed in subdiffusive regime, we investigate the effects of subdiffusion in the predator - prey model with toxins under the homogeneous Neumann boundary condition. First, the stability analysis of the corresponding ordinary differential equation is carried out. From this analysis, it follows that stability is closely related to the coefficient of toxicity. In addition, the temporal fractional derivative does not systematically widen the range of parameters to maintain a point in the stability domain. Furthermore, we derive the condition which links the Turing instability to the coefficient of toxicity in the subdiffusive regime. System parameters are varied in order to test our mathematical predictions while comparing them to ecological literature. It turns out that the memory effects, linked to the transport process can, depending on the parameters, either stabilize an ecosystem or make a completely different configuration.

A general framework for the numerical analysis of high-order finite difference solvers for nonlinear multi-term time-space fractional partial differential equations with time delay

This paper is devoted to introducing a novel methodology to prove the convergence and stability of a Crank–Nicolson difference approximation for a class of multi-term time-fractional diffusion equations with nonlinear delay and space fractional derivatives in case of sufficient smooth solutions. The temporal fractional derivatives are approximated by a specific form of L1 scheme at tk+1/2. A fourth-order difference approximation for the spatial fractional derivatives is employed by using the weighted average of the shifted Grünwald formulae. This methodology is based on a class of discrete fractional Grönwall inequalities convenient with the quadrature formula used to approximate the Caputo derivative at tk+1/2. In the present work, the method of energy inequalities is utilized to show that the used difference scheme is stable and converges to the exact solution with order O(τ2−αJ+h4), in the case that 0<αJ<1 satisfies 3αJ≥32, which means that 0.369≤αJ<1, such that αJ is the maximum α-th order in the multi-order fractional operators.

A numerical technique based on B‐spline for a class of time‐fractional diffusion equation

AbstractThis paper presents an efficient numerical technique for solving a class of time‐fractional diffusion equation. The time‐fractional derivative is described in the Caputo form. The L1 scheme is used for discretization of Caputo fractional derivative and a collocation approach based on sextic B‐spline basis function is employed for discretization of space variable. The unconditional stability of the fully‐discrete scheme is analyzed. Two numerical examples are considered to demonstrate the accuracy and applicability of our scheme. The proposed scheme is shown to be sixth order accuracy with respect to space variable and (2 − α)‐th order accuracy with respect to time variable, where α is the order of temporal fractional derivative. The numerical results obtained are compared with other existing numerical methods to justify the advantage of present method. The CPU time for the proposed scheme is provided.