TT-M FE method for a 2D nonlinear time distributed-order and space fractional diffusion equation

Implicit Runge-Kutta and spectral Galerkin methods for the two-dimensional nonlinear Riesz space fractional diffusion equation

A fully discrete scheme is proposed for the nonlinear fractional delay diffusion equations with smooth solutions, where the fractional derivative is described in Caputo sense with the order $\\alpha$ ($0 \\lt \\alpha \\lt 1$). The scheme is constructed by combining finite difference method in time and Legendre spectral approximation in space. Stability and convergence are proved rigorously. Moreover, a modified scheme is proposed for the equation with nonsmooth solutions by adding correction terms to the approximations of fractional derivative operator and nonlinear term. Numerical examples are carried out to support the theoretical analysis.

ABSTRACTIn this paper a finite difference method is presented to solve time–space linear and nonlinear fractional diffusion equations. Specifically, the centred difference scheme is used to approximate the Riesz fractional derivative in space. A trapezoidal formula is used to solve a system of Volterra integral equations transformed from spatial discretization. Stability and convergence of the proposed scheme is discussed which shows second-order accuracy both in temporal and spatial directions. Finally, examples are presented to show the accuracy and effectiveness of the schemes.

Herein, we mainly focus on developing a new two-grid Crank-Nicolson (CN) mixed finite element (MFE) (TGCNMFE) method for the generalized nonlinear time fractional fourth-order reaction diffusion equation. To do so, by introducing an auxiliary function, the nonlinear time fractional fourth-order reaction diffusion equation is first split into two second-order nonlinear equations. Thereafter, a new time semi-discrete mixed CN (TSDMCN) scheme is constructed through discretizing the time derivative and time fractional derivative by the CN difference quotient, and the existence, steadiness, and errors of the TSDMCN solutions are analysed. Next, a new TGCNMFE method is developed through using two-grid MFE technique to discretize the spacial variables, and the existence, steadiness, and error estimations for the TGCNMFE solutions are discussed. Lastly, the correctness of theory results and the superiority of the TGCNMFE method are verified by some numerical experiments.

In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.

Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation

Background. The concept of fractal is one of the main paradigms of modern theoretical and experimental physics, radiophysics and radar, and fractional calculus is the mathematical basis of fractal physics, geothermal energy and space electrodynamics. We investigate the solvability of the Cauchy problem for linear and nonlinear inhomogeneous pseudodifferential diffusion equations. The equation contains a fractional derivative of a Riemann–Liouville time variable defined by Caputo and a pseudodifferential operator that acts on spatial variables and is constructed in a homogeneous, non-negative homogeneous order, a non-smooth character at the origin, smooth enough outside. The heterogeneity of the equation depends on the temporal and spatial variables and permits the Laplace transform of the temporal variable. The initial condition contains a restricted function. Objective. To show that the homotopy perturbation transform method (HPTM) is easily applied to linear and nonlinear inhomogeneous pseudodifferential diffusion equations. To prove the solvability and obtain the solution formula for the Cauchy problem series for the given linear and nonlinear diffusion equations. Methods. The problem is solved by the NPTM method, which combines a Laplace transform with a time variable and a homotopy perturbation method (HPM). After the Laplace transform, we obtain an integral equation which is solved as a series by degrees of the entered parameter with unknown coefficients. Substituting the input formula for the solution into the integral equation, we equate the expressions to equal parameter degrees and obtain formulas for unknown coefficients. When solving the nonlinear equation, we use a special polynomial which is included in the decomposition coefficients of the nonlinear function and allows the homotopy perturbation method to be applied as well for nonlinear non-uniform pseudodifferential diffusion equation. Results. The result is a solution of the Cauchy problem for the investigated diffusion equation, which is represented as a series of terms whose functions are found from the parametric series. Conclusions. In this paper we first prove the solvability and obtain the formula for solving the Cauchy problem as a series for linear and nonlinear inhomogeneous pseudodifferential equations.

Solving fractional diffusion and wave equations by modified homotopy perturbation method

In this paper, a wide view on the theory of the continuous time random walk (CTRW) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW. These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.

In this paper, we propose a class of variable coefficients fractional ordinary differential equations (FODEs). Using Mellin transform (MT), we have transformed this class into a functional equation which can’t be solved in general. So, we have selected many special cases of this functional equation that can be solved exactly. After solving these special cases of the functional equation and using the inverse MT, we obtained some exact solutions for the proposed class. The obtained solutions are given in the form of $$H$$ H -function and the Wright function. The results, as special cases, contain some special forms given in the literature. Also, the invariant subspace method (ISM) is utilized for solving a class of nonlinear fractional diffusion equations with variable coefficients. The solutions of this class of nonlinear fractional diffusion equations depend upon the solutions of the proposed class of FODEs.

We investigate the Cauchy problem for a nonlinear fractional diffusion equation, which is modified using the time-fractional hyper-Bessel derivative. The source function is a gradient source of Hamilton–Jacobi type. The main objective of our current work is to show the existence and uniqueness of mild solutions. Our desired goal is achieved using the Picard iteration method, and our analysis is based on properties of Mittag–Leffler functions and embeddings between Hilbert scales spaces and Lebesgue spaces.

Inverse problems for heat equation and space–time fractional diffusion equation with one measurement

Fractional nonlinear differential equations present an interplay between two common and important effective descriptions used to simplify high dimensional or more complicated theories: nonlinearity and fractional derivatives. These effective descriptions thus appear commonly in physical and mathematical modeling. We present a new series method providing systematic controlled accuracy for solutions of fractional nonlinear differential equations, including the fractional nonlinear Schrödinger equation and the fractional nonlinear diffusion equation. The method relies on spatially iterative use of power series expansions. Our approach permits an arbitrarily large radius of convergence and thus solves the typical divergence problem endemic to power series approaches. In the specific case of the fractional nonlinear Schrödinger equation we find fractional generalizations of cnoidal waves of Jacobi elliptic functions as well as a fractional bright soliton. For the fractional nonlinear diffusion equation we find the combination of fractional and nonlinear effects results in a more strongly localized solution which nevertheless still exhibits power law tails, albeit at a much lower density.

Solutions of fractional nonlinear diffusion equation and first passage time: Influence of initial condition and diffusion coefficient

Self-similarity is a ubiquitous concept in the physical sciences used to explain a wide range of spatial- or temporalstructures observed in a broad range of applications and natural phenomena. Indeed, they have been predicted or observed in the context of Raman scattering, spatial soliton fractals, propagation in the normal dispersion regime with strong nonlinearity, optical amplifiers, and mode-locked lasers. These self-similar structures are typically long-time transients formed by the interplay, often nonlinear, of the underlying dominant physical effects in the system. A theoretical model shows that in the context of the universal Ginzburg-Landau equation with rapidly-varying, mean-zero dispersion, stable and attracting self-similar pulses are formed with parabolic profiles: the zero-dispersion similariton. The zero-dispersion similariton is the final solution state of the system, not a long-time, intermediate asymptotic behavior. An averaging analysis shows the self-similarity to be governed by a nonlinear diffusion equation with a rapidly-varying, mean-zero diffusion coefficient. Indeed, the leadingorder behavior is shown to be governed by the porous media (nonlinear diffusion) equation whose solution is the well-known Barenblatt similarity solution which has a parabolic, self-similar profile. The alternating sign of the diffusion coefficient, which is driven by the dispersion fluctuations, is critical to supporting the zero-dispersion similariton which is, to leading-order, of the Barenblatt form. This is the first analytic model proposing a mechanism for generating physically realizable temporal parabolic pulses in the Ginzburg-Landau model. Although the results are of restricted analytic validity, the findings are suggestive of the underlying physical mechanism responsible for parabolic (self-similar) pulse formation in lightwave transmission and observed in mode-locked laser cavities.