Temporal Fractional Derivatives Research Articles

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.

Strong solution of a hydrodinamics problem with memory

In the last few years, the concepts of fractional calculus were frequently applied to other disciplines. Recently, this subject has been extended in various directions such as signal processing, applied mathematics, bio-engineering, viscoelasticity, fluid mechanics, and fluid dynamics. In fluid dynamics, the fractional derivative models were used widely in the past for the study of viscoelastic materials such as polymers in the glass transition and in the glassy state. Recently, it has increasingly been seen as an efficient tool through which a useful generalization of physical concepts can be obtained. The fractional derivatives used most are the Riemann--Liouville fractional derivative and the Caputo fractional derivative. It is well known that these operators exhibit difficulties in applications. For example, the Riemann--Liouville derivative of a constant is not zero. We deal with so called temporal fractional derivative as a prototype of general fractional derivative. We prove the global strong solvability of a linear and quasilinear initial-boundary value problems with a singular complete monotone kernels. Our main tool is a theory of evolutionary integral equations. An abstract fractional order differential equation is studied, which contains as particular case the Rayleigh–Stokes problem for a generalized second-grade fluid with a fractional derivative model. This paper concerns with an initial-boundary value problem for the Navier--Stokes--Voigt equations describing unsteady flows of an incompressible viscoelastic fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution in two-dimensional domain. We also establish an $L_2$ decay estimate for the velocity field under the assumption that the external forces field is conservative.

Efficient Exponential Time-Differencing Methods for the Optical Soliton Solutions to the Space-Time Fractional Coupled Nonlinear Schrödinger Equation

The coupled nonlinear Schrödinger equation is used in simulating the propagation of the optical soliton in a birefringent fiber. Hereditary properties and memory of various materials can be depicted more precisely using the temporal fractional derivatives, and the anomalous dispersion or diffusion effects are better described by the spatial fractional derivatives. In this paper, one-step and two-step exponential time-differencing methods are proposed as time integrators to solve the space-time fractional coupled nonlinear Schrödinger equation numerically to obtain the optical soliton solutions. During this procedure, we take advantage of the global Padé approximation to evaluate the Mittag-Leffler function more efficiently. The approximation error of the Padé approximation is analyzed. A centered difference method is used for the discretization of the space-fractional derivative. Extensive numerical examples are provided to demonstrate the efficiency and effectiveness of the modified exponential time-differencing methods.

Quasi-Analytical Model of the Transient Behavior Pressure in an Oil Reservoir Made Up of Three Porous Media Considering the Fractional Time Derivative

The present work proposes a new model to capture high heterogeneity of single phase flow in naturally fractured vuggy reservoirs. The model considers a three porous media reservoir; namely, fractured system, vugular system and matrix; the case of an infinite reservoir is considered in a full-penetrating wellbore. Furthermore, the model relaxes classic hypotheses considering that matrix permeability has a significant impact on the pressure deficit from the wellbore, reaching the triple permeability and triple porosity model wich allows the wellbore to be fed by all the porous media and not exclusively by the fractured system; where it is considered a pseudostable interporous flow. In addition, it is considered the anomalous flow phenomenon from the pressure of each independent porous medium and as a whole, through the temporal fractional derivative of Caputo type; the resulting phenomenon is studied for orders in the fractional derivatives in (0, 2), known as superdiffusive and subdiffusive phenomena. Synthetic results highlight the effect of anomalous flows throughout the entire transient behavior considering a significant permeability in the matrix and it is contrasted with the effect of an almost negligible matrix permeability. The model is solved analytically in the Laplace space, incorporating the Tartaglia–Cardano equations.

Traveling wave with beta derivative spatial-temporal evolution for describing the nonlinear directional couplers with metamaterials via two distinct methods

This work is reported the analytical solutions for describing the nonlinear directional couplers with metamaterials by including spatial–temporal fractional beta derivative evolution. The auxiliary ordinary differential equation method and the generalized Riccati method with the properties of beta derivative are implemented to secure such solutions. The solutions are obtained in the new forms by involving of some useful mathematical functions. The constraint conditions among the traveling wave parameters are evaluated. Some of the obtained solutions are presented graphically to illustrate the effectiveness of beta derivative parameter and mathematical techniques. It is investigated that the amplitudes of soliton are increased with the increase of fractional beta derivative parameter. The obtained results would be very useful to examine and understand the physical issues in nonlinear optics, especially in twin-core couplers with optical metamaterials.

DEVELOPMENT AND ANALYSIS OF NEW APPROXIMATION OF EXTENDED CUBIC B-SPLINE TO THE NONLINEAR TIME FRACTIONAL KLEIN–GORDON EQUATION

A new extended cubic B-spline (ECBS) approximation is formulated, analyzed and applied to obtain the numerical solution of the time fractional Klein–Gordon equation. The temporal fractional derivative is estimated using Caputo’s discretization and the space derivative is discretized by ECBS basis functions. A combination of Caputo’s fractional derivative and the new approximation of ECBS together with [Formula: see text]-weighted scheme is utilized to obtain the solution. The method is shown to be unconditionally stable and convergent. Numerical examples indicate that the obtained results compare well with other numerical results available in the literature.

Memory-dependent derivative versus fractional derivative (II): Remodelling diffusion process

The memory-dependent derivative (MDD) is a new substitution for the fractional derivative (FD). It reflects the memory effect in a more distinct way. As an application, the representative heat diffusion process is remodeled with it. In fact, due to the existence of heat-conduction paradox, the time-space evolution mechanisms of this process are challenges to the modelers. The paradox cann’t be ascribed to the classical Fourier law, and the results show that the newly-constructed temporal MDD model is more reasonable than the Maxwell-Cattaneo, the temporal FD, the spatial FD and the common ones. Moreover, different mediums may accord with different memory times and weighted functions. This freedom of choice reflects the flexibility of MDD in modelling. It can be borrowed for exploring other diffusion problems.

Higher-dimensional physical models with multimemory indices: analytic solution and convergence analysis

The purpose of this work is to analytically simulate the mutual impact for the existence of both temporal and spatial Caputo fractional derivative parameters in higher-dimensional physical models. For this purpose, we employ the γ̅-Maclaurin series along with an amendment of the power series technique. To supplement our idea, we present the necessary convergence analysis regarding the γ̅-Maclaurin series. As for the application side, we solved versions of the higher-dimensional heat and wave models with spatial and temporal Caputo fractional derivatives in terms of a rapidly convergent γ̅-Maclaurin series. The method performed extremely well, and the projections of the obtained solutions into the integer space are compatible with solutions available in the literature. Finally, the graphical analysis showed a possibility that the Caputo fractional derivatives reflect some memory characteristics.

Optimal H1 spatial convergence of a fully discrete finite element method for the time-fractional Allen-Cahn equation

A time-fractional Allen-Cahn problem is considered, where the spatial domain Ω is a bounded subset of $\mathbb {R}^{d}$ for some d ∈{1,2,3}. New bounds on certain derivatives of the solution are derived. These are used in the analysis of a numerical method (L1 discretization of the temporal fractional derivative on a graded mesh, with a standard finite element discretization of the spatial diffusion term, and Newton linearization of the nonlinear driving term), showing that the computed solution achieves the optimal rate of convergence in the Sobolev H1(Ω) norm. (Previous papers considered only convergence in L2(Ω).) Numerical results confirm our theoretical findings.

Error Analysis of a Finite Difference Method on Graded Meshes for a Multiterm Time-Fractional Initial-Boundary Value Problem

Abstract An initial-boundary value problem, whose differential equation contains a sum of fractional time derivatives with orders between 0 and 1, is considered. Its spatial domain is ( 0 , 1 ) d {(0,1)^{d}} for some d ∈ { 1 , 2 , 3 } {d\in\{1,2,3\}} . This problem is a generalisation of the problem considered by Stynes, O’Riordan and Gracia in SIAM J. Numer. Anal. 55 (2017), pp. 1057–1079, where d = 1 {d=1} and only one fractional time derivative was present. A priori bounds on the derivatives of the unknown solution are derived. A finite difference method, using the well-known L1 scheme for the discretisation of each temporal fractional derivative and classical finite differences for the spatial discretisation, is constructed on a mesh that is uniform in space and arbitrarily graded in time. Stability and consistency of the method and a sharp convergence result are proved; hence it is clear how to choose the temporal mesh grading in a optimal way. Numerical results supporting our theoretical results are provided.

On (2 + 1)-dimensional physical models endowed with decoupled spatial and temporal memory indices⋆

The current work concerns the development of an analytical scheme to handle (2 + 1) -dimensional partial differential equations endowed with decoupled spatial and temporal fractional derivatives (abbreviated by $ (\alpha,\beta)$ -models). For this purpose, a new bivariate fractional power series expansion has been integrated with the differential transform scheme. The mechanism of the submitted scheme depends mainly on converting the $ (\alpha,\beta)$ -model to a recurrence-differential equation that can be easily solved by virtue of an iterative procedure. This, in turn, reduces the computational cost of the Taylor power series method and consequently introduces a significant refinement for solving such hybrid models. To elucidate the novelty and efficiency of the proposed scheme, several $ (\alpha,\beta)$ -models are solved and the presence of remnant memory, due to the fractional derivatives, is graphically illustrated.

Effect of Leakage Delay on Hopf Bifurcation in a Fractional BAM Neural Network

Recently, experimental studies show that fractional calculus can depict the memory and hereditary attributes of neural networks more accurately. In this paper, we introduce temporal fractional derivatives into a six-neuron bidirectional associative memory (BAM) neural network with leakage delay. By selecting two different bifurcation parameters and analyzing corresponding characteristic equations, it is verified that the delayed fractional neural network generates Hopf bifurcation when the bifurcation parameters pass through some critical values. In order to measure how much is the impact of leakage delay on Hopf bifurcation, sensitivity analysis methods, such as scatter plots and partial rank correlation coefficients (PRCCs), are introduced to assess the sensitivity of bifurcation amplitudes to leakage delay. Numerical examples are carried out to illustrate the theoretical results and help us gain an insight into the effect of leakage delay.

A variety of soliton solutions for the fractional Wazwaz-Benjamin-Bona-Mahony equations

Abstract In the present paper, the new three-dimensional modified Benjamin-Bona-Mahony equations recently introduced are analyzed with the introduction of the spatial and temporal fractional order derivatives using conformable fractional derivative. Various solutions including hyperbolic and periodic function solutions are constructed using simple ansatzs with the aid of the Wolfram Mathematica software. Some graphical representations of the solutions are depicted.

A fractally fractional diffusion model of composite dual-porosity for multiple fractured horizontal wells with stimulated reservoir volume in tight gas reservoirs

Based on fractal theory (FT) and fractional calculus (FC), a new fractally fractional diffusion model (FFDM) of composite dual-porosity has been developed to evaluate performance of multiple fractured horizontal wells (MFHWs) with stimulated reservoir volume (SRV) in tight gas reservoirs (TGRs). More specifically, FT is used to characterize the complex and heterogeneous fracture network (FN) both inside and outside of SRV, while anomalous behavior of diffusion processes both inside and outside of SRV is quantified by applying the temporal fractional derivatives. The FFDM is then solved by the Laplace transformation, line source function, the numerical discrete method, and superposition principle. The transient pressure responses are then inversely converted from Laplace domain into real time domain with the Stehfest algorithm, and the FFDM is also validated, and type curves are generated as well. Flow stages are subsequently identified together with analysis on characteristics of the type curves, especially the anomalous features different with those generated from the conventional Euclidean model. Sensitivity analyses of some related parameters have also been discussed as well. And the FFDM is then also matched with the real field well-testing data of a MFHW with SRV in a TGR. The proposed FFDM provides a new understanding of the performance of MFHWs with SRV in TGRs, which can be used to interpret the field pressure data more accurately and appropriately.

New solitary wave solutions of the time-fractional Cahn-Allen equation via the improved ( $\frac{G'}{G}$ G G )-expansion method

An improved $(\frac{G'}{G})$ -expansion method is proposed for extracting more general solitary wave solutions of the nonlinear fractional Cahn-Allen equation. The temporal fractional derivative is taken in the sense of Jumarie’s fractional derivative. The results of this article are generalized and extended version of previously reported solutions.

Pattern Formation through Temporal Fractional Derivatives

It is well known that temporal first-derivative reaction-diffusion systems can produce various fascinating Turing patterns. However, it has been found that many physical, chemical and biological systems are well described by temporal fractional-derivative reaction-diffusion equations. Naturally arises an issue whether and how spatial patterns form for such a kind of systems. To address this issue clearly, we consider a classical prey-predator diffusive model with the Holling II functional response, where temporal fractional derivatives are introduced according to the memory character of prey’s and predator’s behaviors. In this paper, we show that this fractional-derivative system can form steadily spatial patterns even though its first-derivative counterpart can’t exhibit any steady pattern. This result implies that the temporal fractional derivatives can induce spatial patterns, which enriches the current mechanisms of pattern formation.

Unstructured mesh finite difference/finite element method for the 2D time-space Riesz fractional diffusion equation on irregular convex domains

• Establish and prove some new definitions and lemmas of fractional derivative space on convex domains. • An unstructured mesh finite element method is presented. • The stability and convergence of the method are discussed on different irregular convex domains. • Extend the theory and develop a computational model for the case of a multiply-connected domain. • Apply to solve the coupled 2D fractional Bloch–Torrey equation on human brain-like domains. Fractional differential equations are powerful tools to model the non-locality and spatial heterogeneity evident in many real-world problems. Although numerous numerical methods have been proposed, most of them are limited to regular domains and uniform meshes. For irregular convex domains, the treatment of the space fractional derivative becomes more challenging and the general methods are no longer feasible. In this work, we propose a novel numerical technique based on the Galerkin finite element method (FEM) with an unstructured mesh to deal with the space fractional derivative on arbitrarily shaped convex and non-convex domains, which is the most original and significant contribution of this paper. Moreover, we present a second order finite difference scheme for the temporal fractional derivative. In addition, the stability and convergence of the method are discussed and numerical examples on different irregular convex domains and non-convex domains illustrate the reliability of the method. We also extend the theory and develop a computational model for the case of a multiply-connected domain. Finally, to demonstrate the versatility and applicability of our method, we solve the coupled two-dimensional fractional Bloch–Torrey equation on a human brain-like domain and exhibit the effects of the time and space fractional indices on the behaviour of the transverse magnetization.

Fractional Time Fluctuations in Viscoelasticity: A Comparative Study of Correlations and Elastic Moduli.

We calculate the transverse velocity fluctuations correlation function of a linear and homogeneous viscoelastic liquid by using a generalized Langevin equation (GLE) approach. We consider a long-ranged (power-law) viscoelastic memory and a noise with a long-range (power-law) auto-correlation. We first evaluate the transverse velocity fluctuations correlation function for conventional time derivatives and then introduce time fractional derivatives in their equations of motion and calculate the corresponding fractional correlation function. We find that the magnitude of the fractional correlation is always lower than the non-fractional one and decays more rapidly. The relationship between the fractional loss modulus and is also calculated analytically. The difference between the values of for two specific viscoelastic fluids is quantified. Our model calculation shows that the fractional effects on this measurable quantity may be three times as large as compared with its non-fractional value. The fact that the dynamic shear modulus is related to the light scattering spectrum suggests that the measurement of this property might be used as a suitable test to assess the effects of temporal fractional derivatives on a measurable property. Finally, we summarize the main results of our approach and emphasize that the eventual validity of our model calculations can only come from experimentation.

Existence and exact controllability of fractional evolution inclusions with damping

The aim of this paper is to deal with the existence of mild solutions and exact controllability for a class of fractional evolution inclusions with damping (FEID, for short) in Banach spaces. Firstly, we provide the representation of mild solutions for FEID by applying the method of Laplace transform and the theory of (α,κ)‐regularized families of operators. Next, we are concerned with the existence and exact controllability of FEID under some suitable sufficient conditions by using the method of measure of noncompactness and an appropraite fixed point theorem. Finally, an application to nonlinear partial differential equations with temporal fractional derivatives is presented to illustrate the effectiveness of our main results. Copyright © 2017 John Wiley &amp; Sons, Ltd.