Nonlinear Time-fractional Diffusion Equations Research Articles

Numerical analysis for optimal quadratic spline collocation method in two space dimensions with application to nonlinear time-fractional diffusion equation

Numerical analysis for optimal quadratic spline collocation method in two space dimensions with application to nonlinear time-fractional diffusion equation

A fast linearized virtual element method on graded meshes for nonlinear time-fractional diffusion equations

A fast linearized virtual element method on graded meshes for nonlinear time-fractional diffusion equations

A numerical approach for nonlinear time-fractional diffusion equation with generalized memory kernel

A numerical approach for nonlinear time-fractional diffusion equation with generalized memory kernel

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−.

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.

An inverse problem of identifying the coefficient in a nonlinear time-fractional diffusion equation

An inverse problem of identifying the coefficient in a nonlinear time-fractional diffusion equation

Solutions of certain class of non-linear time-fractional diffusion equations via the fractional differential transform method

Solutions of certain class of non-linear time-fractional diffusion equations via the fractional differential transform method

Superconvergence analysis of a two-grid finite element method for nonlinear time-fractional diffusion equations

Superconvergence analysis of a two-grid finite element method for nonlinear time-fractional diffusion equations

An Efficient Local Meshfree Method for Signal Smoothing by a Time–Fractional Nonlinear Diffusion Equation

An Efficient Local Meshfree Method for Signal Smoothing by a Time–Fractional Nonlinear Diffusion Equation

A new and efficient approach for solving linear and nonlinear time-fractional diffusion equations of distributed order

A new and efficient approach for solving linear and nonlinear time-fractional diffusion equations of distributed order

Partially explicit time discretization for nonlinear time fractional diffusion equations

Nonlinear time fractional partial differential equations are widely used in modeling and simulations. In many applications, there are high contrast changes in media properties. For solving these problems, one often uses a coarse spatial grid for spatial resolution. For temporal discretization, implicit methods are often used. For implicit methods, though the time step can be relatively large, the equations are difficult to compute due to the nonlinearity and the fact that one deals with large-scale systems. On the other hand, the discrete systems in explicit methods are easier to compute but it requires small time steps. In this work, we propose a partially explicit scheme following earlier works on developing partially explicit methods for nonlinear diffusion equations. In these schemes, the diffusion term is treated partially explicitly and the reaction term is treated fully explicitly. With the appropriate construction of spaces and stability analysis, we find that the required time step in our proposed scheme scales as the coarse mesh size, which creates a computational saving. The main novelty of this work is the extension of our earlier works for diffusion equations to time fractional diffusion equations. For the case of fractional diffusion equations, the constraints on time steps are more severe and the proposed methods alleviate this problem since the time step in the partially explicit method scales as the coarse mesh size. We present stability analysis. Numerical results are presented where we compare our proposed partially explicit methods with a fully implicit approach. We show that our proposed approach provides similar results while treating many degrees of freedom in nonlinear terms explicitly.

An efficient technique for solving fractional-order diffusion equations arising in oil pollution

In this article, non-linear time-fractional diffusion equations are considered to describe oil pollution in the water. The latest technique, fractional reduced differential transform method (FRDTM), is used to acquire approximate solutions of the time fractional-order diffusion equation and two cases of Allen–Cahn equations. The acquired results are collated with the exact solutions and other results from literature for integer-order α, which reveal that the proposed method is effective. Hence, FRDTM can be employed to obtain solutions for different types of nonlinear fractional-order IVPs arising in engineering and science.

Variational Regularization For The Inverse Problem of a Nonlinear Time-Fractional Diffusion Equations

The nonlinear time-fractional diffusion equations were considered in the 2D domain, and the physical information in initial state <inlineformula><texmath id="M2">\begin{document}$ u(0,y,t) $\end{document}</texmath><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="amm-42-0168_M2.jpg"></graphic><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="amm-42-0168_M2.png"></graphic></alternatives></inlineformula> of the material was recovered from the measured data in the final state. This problem is seriously ill-posed, that is, the solution to this problem does not continuously depend on the measured data. Therefore, a variational regularization method was proposed to construct the approximate solution to the problem, and the convergence error estimates of the exact and approximate solutions were obtained under the assumption of the priori bounds on the exact solutions. Finally, a numerical example was given to verify the effectiveness of the proposed method.

Implementation of the ADMM approach to constrained optimal control problem with a nonlinear time-fractional diffusion equation

In this paper, we study the inverse problem of identifying the parameters in a nonlinear subdiffusion model from an observation defined in the given $ \Omega_1 $ subset of $ \Omega $. The nonlinear subdiffusion model involves a Caputo fractional derivative of order $ \alpha\in (0,1) $ in time. To address our model, we first examine the regularity of the solution for the direct problem using the Mittag-Leffler function. To investigate our inverse parameter problem, we reformulate first it in to Least-Squares optimization problem. Then, we establish the existence of the optimal solution and prove the convexity of the considered cost function by using its first derivative. To solve this problem numerically, we adapt a recent method in the literature known as the alternating direction method of multiplier (ADMM) which we establish its convergence. In order to show the effectiveness of the proposed method we present some numerical experiments.

Correction to: A numerical scheme to solve a class of two-dimensional nonlinear time-fractional diffusion equations of distributed order

Correction to: A numerical scheme to solve a class of two-dimensional nonlinear time-fractional diffusion equations of distributed order

Two-grid methods for nonlinear time fractional diffusion equations by [formula omitted]-Galerkin FEM

In this paper, two efficient two-grid algorithms with L1 scheme are presented for solving two-dimensional nonlinear time fractional diffusion equations. The classical L1 scheme is considered in the time direction, and the two-grid FE method is used to approximate spatial direction. To linearize the discrete equations, the Newton iteration approach and correction technique are applied. The two-grid algorithms reduce the solution of the nonlinear fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithms save total computational cost. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Moreover, the numerical experiment presented further confirms the theoretical results.

A numerical scheme to solve a class of two-dimensional nonlinear time-fractional diffusion equations of distributed order

This article is devoted to obtain the numerical solution for a class of nonlinear two-dimensional distributed-order time-fractional diffusion equations. We discretize the problem by using a finite difference scheme in the time direction. Then, we solve the discretized nonlinear problem by a collocation approach based on the Legendre polynomials. The numerical algorithm is fully described and convergence analysis of the scheme is evaluated. Finally, few numerical implementations are presented to highlight the flexibility and the convergence rate of this method.

Anisotropic non-linear time-fractional diffusion equation with a source term: Classification via Lie point symmetries, analytic solutions and numerical simulation

The non-linear anomalous diffusion in anisotropic media is using in various fields, e.g. in molecular dynamics, hydrology, financial systems, porous media analysis, etc. The Lie group method is developed to study the time-fractional diffusion equation with a source term in anisotropic media. As an application, the complete Lie group classification is performed up to the equivalence transformations for all special cases of the coefficients. Some similarity reductions are obtained for implicit cases. The analytical invariant subspace method is used in order to find some exact solutions. The work is concluded by the fractional Chebyshev pseudo-spectral (FCPS) method for constructing a numerical simulation for some of the reduced equations in the symmetry analysis section.

A fast linearized numerical method for nonlinear time-fractional diffusion equations

In this paper, we study a fast linearized numerical method for solving nonlinear time-fractional diffusion equations. A new weighted method is proposed to construct linearized approximation, which enables the unconditional convergence to be established when the nonlinearity f(u) is only locally Lipschitz continuous. In order to reduce the computational cost, the sum-of-exponentials (SOE) technique is employed to evaluate the kernel function in the Caputo derivative. By using the complementary discrete kernels for the coefficients of the refined fast weighted discretization, the proposed method is shown to be unconditionally convergent with respect to the discrete H1-norm. The fast linearized method can also be extended to nonlinear multi-term and distributed-order time-fractional diffusion equations. Numerical examples with different types of nonlinear functions are provided to demonstrate the behavior of proposed methods for both smooth and weakly singular solutions.