Space Fractional Derivatives Research Articles

SIMULTANEOUS DETERMINATION OF A SOURCE TERM AND DIFFUSION CONCENTRATION FOR A MULTI-TERM SPACE-TIME FRACTIONAL DIFFUSION EQUATION

An inverse problem of determining a time dependent source term along with diffusion/temperature concentration from a non-local over-specified condition for a space-time fractional diffusion equation is considered. The space-time fractional diffusion equation involve Caputo fractional derivative in space and Hilfer fractional derivatives in time of different orders between 0 and 1. Under certain conditions on the given data we proved that the inverse problem is locally well-posed in the sense of Hadamard. Our method of proof based on eigenfunction expansion for which the eigenfunctions (which are Mittag-Leffler functions) of fractional order spectral problem and its adjoint problem are considered. Several properties of multinomial Mittag-Leffler functions are proved.

Fourth-order numerical method for the Riesz space fractional diffusion equation with a nonlinear source term

‎This paper aims to propose a high-order and accurate numerical scheme for the solution of the nonlinear diffusion equation with Riesz space fractional derivative. To this end, we first discretize the Riesz fractional derivative with a fourth-order finite difference method, then we apply a boundary value method (BVM) of fourth-order for the time integration of the resulting system of ordinary differential equations. The proposed method has a fourth-order of accuracy in both space and time components and is unconditionally stable due to the favorable stability property of BVM. The numerical results are compared with analytical solutions and with those provided by other methods in the literature. Numerical experiments obtained from solving several problems including fractional Fisher and fractional parabolic-type sine-Gordon equations show that the proposed method is an efficient algorithm for solving such problems and can arrive at the high-precision.

Prospecting black hole thermodynamics with fractional quantum mechanics

This paper investigates whether the framework of fractional quantum mechanics can broaden our perspective of black hole thermodynamics. Concretely, we employ a space-fractional derivative (Riesz in Acta Math 81:1, 1949) as our main tool. Moreover, we restrict our analysis to the case of a Schwarzschild configuration. From a subsequently modified Wheeler–DeWitt equation, we retrieve the corresponding expressions for specific observables. Namely, the black hole mass spectrum, M, its temperature T, and entropy, S. We find that these bear consequential alterations conveyed through a fractional parameter, alpha . In particular, the standard results are recovered in the specific limit alpha =2. Furthermore, we elaborate how generalizations of the entropy-area relation suggested by Tsallis and Cirto (Eur Phys J C 73:2487, 2013) and Barrow (Phys Lett B 808:135643, 2020) acquire a complementary interpretation in terms of a fractional point of view. A thorough discussion of our results is presented.

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.

Approximation Solution of the Fractional Parabolic Partial Differential Equation by the Half-Sweep and Preconditioned Relaxation

In this study, the numerical solution of a space-fractional parabolic partial differential equation was considered. The investigation of the solution was made by focusing on the space-fractional diffusion equation (SFDE) problem. Note that the symmetry of an efficient approximation to the SFDE based on a numerical method is related to the compatibility of a discretization scheme and a linear system solver. The application of the one-dimensional, linear, unconditionally stable, and implicit finite difference approximation to SFDE was studied. The general differential equation of the SFDE was discretized using the space-fractional derivative of Caputo with a half-sweep finite difference scheme. The implicit approximation to the SFDE was formulated, and the formation of a linear system with a coefficient matrix, which was large and sparse, is shown. The construction of a general preconditioned system of equation is also presented. This study’s contribution is the introduction of a half-sweep preconditioned successive over relaxation (HSPSOR) method for the solution of the SFDE-based system of equation. This work extended the use of the HSPSOR as an efficient numerical method for the time-fractional diffusion equation, which has been presented in the 5th North American International Conference on industrial engineering and operations management in Detroit, Michigan, USA, 10–14 August 2020. The current work proposed several SFDE examples to validate the performance of the HSPSOR iterative method in solving the fractional diffusion equation. The outcome of the numerical investigation illustrated the competence of the HSPSOR to solve the SFDE and proved that the HSPSOR is superior to the standard approximation, which is the full-sweep preconditioned SOR (FSPSOR), in terms of computational complexity.

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.

On the solutions for generalised multiorder fractional partial differential equations arising in physics

In this article, we have studied solutions of a generalised multiorder fractional partial differential equations involving the Caputo time‐fractional derivative and the Riemann–Liouville space fractional derivatives using Laplace–Fourier transform technique. Proposed generalised multiorder fractional partial differential equation is reducible to Schrödinger equation, wave equation and diffusion equation in a more general sense, and hence, solutions of these equations are specifically noted. Not only this, solutions of equation proposed in the stochastic resetting theory in the context of Brownian motion can also be found in a general regime.

Effects of non-locality on unsteady nonequilibrium sediment transport in turbulent flows: A study using space fractional ADE with fractional divergence

This study aims to develop a mathematical model that can describe the vertical distribution of suspended sediment particles in an open channel turbulent flow under unsteady nonequilibrium condition with non-local mixing effect. A space non-local transport of sediment is considered that arises due to turbulent bursting where the hopping height of a particle is not limited to a small distance as described by classical theory of turbulence, rather it can jump upto any height. To take into account this fact unlike previous researchers who used Rouse equation, Hunt equation or traditional advection–diffusion equation as the governing equation, the present study uses fractional advection–diffusion equation (fADE) where space fractional derivative for the local diffusive flux has been used that counts the effect of non-locality. The fractional derivative is considered in Caputo sense and the order α of the fractional derivative corresponds to α-order Le´vy stable distribution with value 1<α≤2. The non-local effect has been considered in the boundary conditions as well as in the non-linear models of sediment diffusivity. Four types of sediment diffusivity are considered which are generalized non-linear models for broad applicability of the study. The fADE together with the boundary and initial conditions, is solved by Chebyshev collocation method and Euler backward method. This proposed method is unconditionally convergent and converges more rapidly than other previous methods used in similar studies. Effects of non-local mixing have been investigated for transient and bottom concentration distributions. Proposed models have been validated for sediment distribution under steady and unsteady condition with existing experimental data and satisfactory results are obtained for all choices of sediment diffusivity. The variation of transient and bottom concentration with non-local effects are physically justified. Validation results show that the model can be applied to describe vertical concentration distribution in unsteady and steady turbulent flows in practical situation.

Parsimonious modeling of skeletal muscle perfusion: Connecting the stretched exponential and fractional Fickian diffusion.

To develop an anomalous (non-Gaussian) diffusion model for characterizing skeletal muscle perfusion using multi-b-value DWI. Fick's first law was extended for describing tissue perfusion as anomalous superdiffusion, which is non-Gaussian diffusion exhibiting greater particle spread than that of the Gaussian case. This was accomplished using a space-fractional derivative that gives rise to a power-law relationship between mean squared displacement and time, and produces a stretched exponential signal decay as a function of b-value. Numerical simulations were used to estimate parameter errors under in vivo conditions, and examine the effect of limited SNR and residual fat signal. Stretched exponential DWI parameters, α and , were measured in thigh muscles of 4 healthy volunteers at rest and following in-magnet exercise. These parameters were related to a stable distribution of jump-length probabilities and used to estimate microvascular volume fractions. Numerical simulations showed low dispersion in parameter estimates within 1.5% and 1%, and bias errors within 3% and 10%, for α and , respectively. Superdiffusion was observed in resting muscle, and to a greater degree following exercise. Resting microvascular volume fraction was between 0.0067 and 0.0139 and increased between 2.2-fold and 4.7-fold following exercise. This model captures superdiffusive molecular motions consistent with perfusion, using a parsimonious representation of the DWI signal, providing approximations of microvascular volume fraction comparable with histological estimates. This signal model demonstrates low parameter-estimation errors, and therefore holds potential for a wide range of applications in skeletal muscle and elsewhere in the body.

On a nonlinear energy-conserving scalar auxiliary variable (SAV) model for Riesz space-fractional hyperbolic equations

In this work, we consider a fractional extension of the classical nonlinear wave equation, subjected to initial conditions and homogeneous Dirichlet boundary data. We consider space-fractional derivatives of the Riesz type in a bounded real interval. It is known that the problem has an associated energy which is preserved through time. The mathematical model is presented equivalently using the scalar auxiliary variable (SAV) technique, and the expression of the energy is obtained using the new scalar variable. The new differential system is discretized then following the SAV approach. The proposed scheme is a nonlinear implicit method which has an associated discrete energy, and we prove that the discrete model is also conservative. The present work is the first report in which the SAV method is used to design nonlinear conservative numerical method to solve a Hamiltonian space-fractional wave equations.

Some inverse problems for wave equations with fractional derivative attenuation

The damped wave equation with the attenuation proportional to velocity is ubiquitous in science and engineering and a common situation is when the attenuation depends on frequency. The usual way to incorporate this effect is to introduce fractional order derivatives either as a replacement for u t or as modifier through a spatial component with space fractional derivatives. Models for these are very well developed and the effort in this paper is towards the analysis of the inverse problem of recovering critical coefficients or initial states although we also develop constructive methods for these and analyse their degree of ill-conditioning.

An efficient multigrid solver for two-dimensional spatial fractional diffusion equations with variable coefficients

Extrapolation cascadic multigrid (EXCMG) method with the conjugate gradient smoother is shown to be an efficient solver for large sparse symmetric positive definite systems resulting from linear finite element discretization of second-order elliptic boundary value problems [Pan et al. J. Comput. Phys. 344 (2017) 499–515]. In this paper, we generalize the EXCMG method to solve a class of spatial fractional diffusion equations (SFDEs) with variable coefficients. Both steady-state and time-dependent problems are considered. First of all, space-fractional derivatives defined in Riemann–Liouville sense are discretized by using the weighted average of shifted Grünwald formula, which results in a fully dense and nonsymmetric linear system for the steady-state problem, or a semi-discretized ODE system for the time-dependent problem. Then, to solve the former problem, we propose the EXCMG method with the biconjugate gradient stabilized smoother to deal with the dense and nonsymmetric linear system. Next, such technique is extended to solve the latter problem since it becomes fully discrete when the Crank-Nicolson scheme is introduced to handle the temporal derivative. Finally, several numerical examples are reported to show that the EXCMG method is an efficient solver for both steady-state and time-dependent SFDEs, and performs much better than the V-cycle multigrid method with banded-splitting smoother for time-dependent SFDEs [Lin et al. J. Comput. Phys. 336 (2017) 69–86].

Combined Galerkin spectral/finite difference method over graded meshes for the generalized nonlinear fractional Schrödinger equation

The main aim of this paper is to construct an efficient Galerkin–Legendre spectral approximation combined with a finite difference formula of L1 type to numerically solve the generalized nonlinear fractional Schrodinger equation with both space- and time-fractional derivatives. We discretize the Riesz space-fractional derivative using the Legendre–Galerkin spectral method and the time-fractional derivative using the L1 scheme on nonuniform meshes. The stability and convergence analyses of the numerical scheme are studied in detail. The scheme is unconditionally stable and convergent of $$\min \{\kappa \theta ,2-\theta \}$$ order convergence in time and of spectral accuracy in space, where $$\theta $$ is the order of fractional derivative and $$\kappa $$ is the grading mesh parameter. To verify the efficiency of the proposed algorithm, two numerical test problems are performed with convergence and error analysis.

ON THE NUMERICAL SIMULATION OF TIME-SPACE FRACTIONAL COUPLED NONLINEAR SCHRÖDINGER EQUATIONS UTILIZING WENDLAND’S COMPACTLY SUPPORTED FUNCTION COLLOCATION METHOD

This research describes an efficient numerical method based on Wendland’s compactly supported functions to simulate the time-space fractional coupled nonlinear Schrödinger (TSFCNLS) equations. Here, the time and space fractional derivatives are considered in terms of Caputo and Conformable derivatives, respectively. The present numerical discussion is based on the following ways: we first approximate the Caputo fractional derivative of the proposed equation by a scheme order O(∆t2−α), 0 &lt; α &lt; 1 and then the Crank-Nicolson scheme is employed in the mentioned equation to discretize the equations. Second, applying a linear difference scheme to avoid solving nonlinear systems. In this way, we have a linear, suitable calculation scheme. Then, the conformable fractional derivatives of the Wendland’s compactly supported functions are established for the scheme. The stability analysis of the suggested scheme is also examined in a similar way to the classic Von-Neumann technique for the governing equations. The efficiency and accuracy of the present method are verified by solving two examples.

Banded Preconditioners for Riesz Space Fractional Diffusion Equations

In this paper, we consider numerical methods for Toeplitz-like linear systems arising from the one- and two-dimensional Riesz space fractional diffusion equations. We apply the Crank–Nicolson technique to discretize the temporal derivative and apply certain difference operator to discretize the space fractional derivatives. For the one-dimensional problem, the corresponding coefficient matrix is the sum of an identity matrix and a product of a diagonal matrix and a symmetric Toeplitz matrix. We transform the linear systems to symmetric linear systems and introduce symmetric banded preconditioners. We prove that under mild assumptions, the eigenvalues of the preconditioned matrix are bounded above and below by positive constants. In particular, the lower bound of the eigenvalues is equal to 1 when the banded preconditioner with diagonal compensation is applied. The preconditioned conjugate gradient method is applied to solve relevant linear systems. Numerical results are presented to verify the theoretical results about the preconditioned matrices and to illustrate the efficiency of the proposed preconditioners.

A space-time finite element method for solving linear Riesz space fractional partial differential equations

In this paper, numerical solutions for linear Riesz space fractional partial differential equations with a second-order time derivative are considered. A space-time finite element method is proposed to solve these equations numerically. In the time direction, the C0-continuous Galerkin method is used to approximate the second-order time derivative. In the space direction, the usual linear finite element method is developed to approximate the space fractional derivative. The matrix equivalent form of this numerical method is deduced. The stability of the discrete solution is established and the optimal error estimates are investigated. Some numerical tests are given to validate the theoretical results.

Numerical approach to simulate diffusion model of a fluid-flow in a porous media

When a particle distributes at a rate that deviates from the classical Brownian motion model, fractional space derivatives have been used to simulate anomalous diffusion or dispersion. When a fractional derivative substitutes the second-order derivative in a diffusion or dispersion model, amplified diffusion occurs (named super-diffusion). The proposed approach in this paper allows seeing the physical background of the newly defined Caputo space-time-fractional derivative and indicates that the order of convergence to approximate such equations has increased.

New types of soliton solutions for space-time fractional cubic nonlinear Schrodinger equation

New definitions for traveling wave transformation and using of new conformable fractional derivative for converting fractional nonlinear evolution equations into the ordinary differential equations are presented in this study. For this aim we consider the time and space fractional derivatives cubic nonlinear Schrodinger equation. Then by using of the efficient and powerful method the exact traveling wave solutions of this equation are obtained. The new definition introduces a promising tool for solving many space-time fractional partial differential equations.

Space-time fractional diffusion: transient flow to a line source

Nonlocal diffusion to a line source well is addressed by space-time fractional diffusion to model transients governed by both long-range connectivity and distorted flow paths that result in interruptions in the geological medium as a consequence of intercalations, dead ends, etc.The former, superdiffusion, results in long-distance runs and the latter, subdiffusion, in pauses. Both phenomena are quantified through fractional constitutive laws, and two exponentsαandβare used to model subdiffusion and superdiffusion, respectively. Consequently, we employ both time and space fractional derivatives. The spatiotemporal evolution of transients in 2D is evaluated numerically and insights on the structure of solutions described through asymptotic solutions are confirmed numerically. Pressure distributions may be classified through two situations (i) wherein 2α = β + 1 in which case solutions may be grouped on the basis of the classical Theis solution, and (ii) wherein 2α ≠ β + 1 in which case conventional expectations do not hold; regardless, at long enough times for the combined case, power-law responses are similar to those for pure subdiffusive flows. Pure superdiffusion on the other hand, although we consider a system that is infinite in its areal extent, interestingly, results in behaviors similar to steady-state flow. To our knowledge, documented behaviors are yet to be reported.

Explicit Finite Difference Approximation Method for Solving the Riesz Space Fractional Percolation Equation

ABSTRACT This paper presents an efficient approximation method for solving the Riesz space fractional percolation equation mixed fractional derivative (RSFPEMFD) with the initial condition (IC) and derivative boundary conditions (DBC) using explicit finite difference method (EFDM). The shifted Grunwald estimate was used for analysis of the mixed fractional derivatives. In addition, the analysis method of consistency, stability, and convergence was used for analyzing the EFDM. To demonstrate the efficiency and validity of the proposed algorithm, some illustrative examples are given and comparing results with the exact solution. The results are displayed when necessary in tables using the MathCAD and MATLAB software package. The EFDM for solving RSFPEMFD has been demonstrated to be effective and reliable.