Nonlinear Fractional Diffusion Equation Research Articles

Maximal $$L_p$$-regularity for x-dependent fractional heat equations with Dirichlet conditions

AbstractWe prove optimal regularity results in $$L_p$$ L p -based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator P of order 2a ($$0<a<1$$ 0 < a < 1 ) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian $$(-\Delta )^a$$ ( - Δ ) a , as well as elliptic operators $$(-\nabla \cdot A(x)\nabla +b(x))^a$$ ( - ∇ · A ( x ) ∇ + b ( x ) ) a . The proofs are based on general results on maximal $$L_p$$ L p -regularity and its relation to $$\mathcal {R}$$ R -boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.

Investigation of Well-Posedness for a Direct Problem for a Nonlinear Fractional Diffusion Equation and an Inverse Problem

Investigation of Well-Posedness for a Direct Problem for a Nonlinear Fractional Diffusion Equation and an Inverse Problem

A robust numerical scheme for solving Riesz-tempered fractional reaction–diffusion equations

The Fractional Diffusion Equation (FDE) is a mathematical model that describes anomalous transport phenomena characterized by non-local and long-range dependencies which deviate from the traditional behavior of diffusion. Solving this equation numerically is challenging due to the need to discretize complicated integral operators which increase the computational costs. These complexities are exacerbated by nonlinear source terms, nonsmooth data and irregular domains. In this study, we propose a second order Exponential Time Differencing Finite Element Method (ETD-RDP-FEM) to efficiently solve nonlinear FDE, posed in irregular domains. This approach discretizes matrix exponentials using a rational function with real and distinct poles, resulting in an L-stable scheme that damps spurious oscillations caused by non-smooth initial data. The method is shown to outperform existing second-order methods for FDEs with a higher accuracy and faster computational time.

Midpoint splitting methods for nonlinear space fractional diffusion equations

We propose a midpoint splitting method to solve nonlinear space fractional diffusion equations. We first discretize the equation in space using the weighted and shifted Grünwald difference (WSGD) operators. Then, we obtain the fully discrete scheme of the equation using the midpoint splitting method, which can alleviate computational costs and mitigate instability issues that may arise. We establish the stability result of the fully discrete scheme under certain conditions. Finally, we demonstrate its feasibility and efficiency through numerical examples.

Exact Solutions for a Class of Variable Coefficients Fractional Differential Equations Using Mellin Transform and the Invariant Subspace Method

AbstractIn 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.

Existence of solutions to a nonlinear fractional diffusion equation with exponential growth

In this paper, we study a Cauchy problem for a space–time fractional diffusion equation with exponential nonlinearity. Based on the standard Lp-Lq estimates of strongly continuous semigroup generated by fractional Laplace operator, we investigate the existence of global solutions for initial data with small norm in Orlicz space exp Lp(Rd) and a time weighted Lr(Rd) space. In the framework of the Hölder interpolation inequality, we also discuss the existence of local solutions without the Orlicz space.

Unconditionally optimal time two-mesh mixed finite element algorithm for a nonlinear fourth-order distributed-order time fractional diffusion equation

In this article, a fast second-order time two-mesh mixed finite element (TT-MMFE) algorithm is considered to numerically solve the nonlinear fourth-order distributed-order time fractional diffusion equation, where the generated formula for the fractional BDF2 and fractional trapezoidal rule through a free parameter θ (FBT-θ formula) combined with the composite trapezoid formula for the distributed-order derivative is used to discretize the time direction, and the MFE method is utilized to approximate the space direction, respectively. Further, to improve the computing efficiency of our numerical method, the TT-M algorithm is applied to deal with the nonlinear system. For the case of nonsmooth solution, the corrected TT-MMFE scheme by adding starting parts is designed. The existence and uniqueness of the numerical solution are proved in detail. Different from the traditional finite element analysis, the error between the exact solution and the finite element solution is split into temporal error and spatial error by introducing a so-called the error splitting technique, which helps us obtain the spatial error independent of the time step. Finally, numerical examples are given to validate that the TT-MMFE method can reduce computing time, and the corrected TT-MMFE scheme by adding starting parts can recover the convergence rate.

Local well-posedness results for the nonlinear fractional diffusion equation involving a Erdélyi-Kober operator

<p>In this paper, we study an initial boundary value problem of a nonlinear fractional diffusion equation with the Caputo-type modification of the Erdélyi-Kober fractional derivative. The main tools are the Picard-iteration method, fixed point principle, Mittag-Leffler function, and the embedding theorem between Hilbert scales spaces and Lebesgue spaces. Through careful analysis and precise calculations, the priori estimates of the solution and the smooth effects of the Erdélyi-Kober operator are demonstrated, and then the local existence, uniqueness, and stability of the solution of the nonlinear fractional diffusion equation are established, where the nonlinear source function satisfies the Lipschitz condition or has a gradient nonlinearity.</p>

A second-order numerical method for nonlinear variable-order fractional diffusion equation with time delay

In this paper, a linearized numerical scheme of nonlinear variable-order fractional diffusion equation with time delay is constructed. We apply the L2−1σ formula to discretize the temporal derivative and second-order central difference scheme to discretize the spatial derivative. The proposed method is unconditionally stable and convergent with Oτ2+h2, where τ and h are the time and space steps, respectively. Numerical experiment demonstrates the effectiveness and accuracy of the numerical scheme.

On the analysis and application of a spectral collocation scheme for the nonlinear two-dimensional fractional diffusion equation

In this paper, we propose and analyze a novel spectral scheme for the numerical solution of a two-dimensional time-fractional diffusion equation. The proposed scheme approximates the unknown function and its derivatives in space by finite Lucas and Fibonacci polynomial expansion, and time derivative/variable by L1 formula with θ-weighted difference scheme. Error estimates are derived and convergence is proved. Six test problems are then solved to verify the theory numerically. The obtained results confirm the low computational cost and better accuracy than known results in the literature.

Recovering solution of the Reverse nonlinear time Fractional diffusion equations with fluctuations data


 In this study, our focus is on obtaining an estimated solution for the nonlinear fractional time diffusion equation. Specifically, we have utilized the Riemann Liouville fractional derivative. Additionally, we have concerned Gaussian white noise in the input data. As we are aware, this problem is considered ill-posed according to Hadamard's definition. To tackle this problem, we have proposed a regularized solution and demonstrated the convergence between the mild solution and the regularized solution.

Global existence and convergence results for a class of nonlinear time fractional diffusion equation

This paper investigates Cauchy problems of nonlinear parabolic equation with a Caputo fractional derivative. When the initial datum is sufficiently small in some appropriate spaces, we demonstrate the existence in global time and uniqueness of a mild solution in fractional Sobolev spaces using some novel techniques. Under some suitable assumptions on the initial datum, we show that the mild solution of the time fractional parabolic equation converges to the mild solution of the classical problem when . Under some appropriate assumptions on the initial datum, we show that the mild solution of the time fractional diffusion equation converges to the mild solution of the classical problem when . Our theoretical results can be widely applied to many different equations such as the Hamilton–Jacobi equation, the Navier–Stokes equation in two cases: the fractional derivative and the classical derivative. Our paper also provides a completely new answer to the related open problem of convergence of solutions to fractional diffusion equations as the order of fractional derivative approaches 1−.

On Finite-Time Blow-Up Problem for Nonlinear Fractional Reaction Diffusion Equation: Analytical Results and Numerical Simulations

The study of the blow-up phenomenon for fractional reaction–diffusion problems is generally deemed of great importance in dealing with several situations that impact our daily lives, and it is applied in many areas such as finance and economics. In this article, we expand on some previous blow-up results for the explicit values and numerical simulation of finite-time blow-up solutions for a semilinear fractional partial differential problem involving a positive power of the solution. We show the behavior solution of the fractional problem, and the numerical solution of the finite-time blow-up solution is also considered. Finally, some illustrative examples and comparisons with the classical problem with integer order are presented, and the validity of the results is demonstrated.

Fractional centered difference schemes and banded preconditioners for nonlinear Riesz space variable-order fractional diffusion equations

Fractional centered difference schemes and banded preconditioners for nonlinear Riesz space variable-order fractional diffusion equations

Linear and nonlinear fractional order diffusion equations with initial and bowndary conditions

The usual differentiation and integration are expanded to any non-integer order in fractional calculus. The topic predates the development of differential calculus by Leibnitz and Newton and is therefore as old as classical theory. The concept of fractional calculus has generated interest not only among mathematicians but also among physicists and engineers. This concept is calculated in the same way as the classical methods of differential and integral calculus, and also dates back to the time when Leibniz and Newton invented differential calculus. The idea of calculating fractional order is of interest not only among individual mathematicians, but also among physicists and engineers. The method of upper and lower solutions has been extended to FDEs using these minimum-maximum principles, and various existence results have been established. In this paper, a one-dimensional subdiffusion equation is investigated using the principle of the maximum of the Riemann-Liouville derivative of fractional order. It is proved that for fractional order diffusion equations in linear and nonlinear time there is a unique classical solution to the initial boundary value problem and that the solution continuously depends on the initial and boundary conditions.

New results for convergence problem of fractional diffusion equations when order approach to $1^-$

This work studies the convergence problem for a class of fractional diffusion equations in which the time-derivative order approaches $1^-$. Up to now, few works have investigated this topic. The purpose of the article consists of three main contents. The first result is related to the convergence of the Caputo derivative and the Mittag-Leffler operators when $\alpha \to 1^-$. The second is to investigate the convergence problem for a linear fractional diffusion equation on $L^p$ spaces. And last result is concerned with the convergence problem for nonlinear fractional diffusion equations. The main analysis and techniques of the paper involve the evaluation related to Riemann-Liouville integration, Caputo derivative and Sobolev embeddings. Our analysis provides a complete and detailed answer to the convergence problem as fractional order tends to $1^-$.

Determination of a Nonlinear Coefficient in a Time-Fractional Diffusion Equation

In this paper, we study direct and inverse problems for a nonlinear time fractional diffusion equation. We prove that the direct problem has a unique weak solution and the solution depends continuously on the coefficient. Then we show that the inverse problem has a quasi-solution. The direct problem is solved by the method of lines using an operator approach. A quasi-Newton optimization method is used for the numerical solution to the inverse problem. The Tikhonov regularization is used to overcome the ill-posedness of the inverse problem. Numerical examples with noise-free and noisy data illustrate the applicability and accuracy of the proposed method to some extent.

The Quasi-Boundary Regularization Method for Recovering the Initial Value in a Nonlinear Time–Space Fractional Diffusion Equation

In this paper, we consider the inverse problem for identifying the initial value problem of the time–space fractional nonlinear diffusion equation. The uniqueness of the solution is proved by taking the fixed point theorem of Banach compression, and the ill-posedness of the problem is analyzed through the exact solution. The quasi-boundary regularization method is chosen to solve the ill-posed problem, and the error estimate between the regularization solution and the exact solution is given. Moreover, several numerical examples are chosen to prove the effectiveness of the quasi-boundary regularization method. Finally, our method can be used to solve high dimensional time–space fractional nonlinear diffusion equation, especially in cylindrical and spherical symmetric regions.

A novel numerical technique for solving time fractional nonlinear diffusion equations involving weak singularities

In this work, an efficient numerical approximation for the solution of the time fractional nonlinear diffuse interface model is studied. The solution to this problem has a weak singularity near the initial time . The fractional order nonlinear diffusion model is transformed into a system of nonlinear functional equations. The Daftardar–Gejji and Jafari method is employed to solve the corresponding nonlinear system. The L1 scheme is used to discretize the Caputo fractional derivative on a graded mesh in the time direction. In contrast, the spatial derivative is approximated by applying a classical central finite difference scheme to a uniform mesh. The convergence analysis and the error bounds are carried out. The analysis and the computational findings exhibit the effectiveness of the proposed method.

Iterative learning algorithms for boundary tracing problems of nonlinear fractional diffusion equations

<abstract><p>In this paper, the iterative learning control technique is extended to distributed parameter systems governed by nonlinear fractional diffusion equations. Based on $ P $-type and $ PI^{\theta} $-type iterative learning control methods, sufficient conditions for the convergences of systems are given. Finally, numerical examples are presented to illustrate the efficiency of the proposed iterative schemes. The numerical results show that the closed-loop iterative learning control scheme converges faster than the open-loop iterative learning control scheme and the $ PI^{\theta} $-type iterative learning control scheme converges faster than the $ P $-type and the $ PI $-type iterative learning control scheme.</p></abstract>