Fractional Order Diffusion Equation Research Articles

On a Reconstruction of a Solely Time-Dependent Source in a Time-Fractional Diffusion Equation with Non-smooth Solutions

An inverse source problem for a non-automonous time fractional diffusion equation of order \((0<\beta <1)\) is considered in a bounded Lipschitz domain in \(\mathbb {R}^d\). The missing solely time-dependent source is recovered from an additional integral measurement. The existence, uniqueness and regularity of a weak solution is studied. We design two numerical algorithms based on Rothe’s method over uniform and graded grids, derive a priori estimates and prove convergence of iterates towards the exact solution. An essential feature of the fractional subdiffusion problem is that the solution lacks the smoothness near the initial time, although it would be smooth away from \(t = 0\). Rothe’s method on a uniform grid addresses the existence of a such a solution (non-smooth with \(t^\gamma \) term where \(1>\gamma > \beta \)) under low regularity assumptions, whilst Rothe’s method over graded grids has the advantage to cope better with the behaviour at \(t=0\) (also here \(t^\beta \) is included in the class of admissible solutions) for the considered problems. The theoretical obtained results are supported by numerical experiments and stay valid in case of smooth solutions to the problem.

NUMERICAL SIMULATIONS FOR THE VARIABLE ORDER TWO-DIMENSIONAL REACTION SUB-DIFFUSION EQUATION: LINEAR AND NONLINEAR

The applications and the fields that use the anomalous sub-diffusion equations cannot be easily listed due to their wide area. Sure, one of the main physical reasons for using and researching fractional order diffusion equations is to explain anomalous diffusion that occurs in transport processes through complex and/or disordered structures, such as fractal media. One of the important applications is their use in chemical reactions, where a single material continues to shift from a high concentration area to a low concentration area until the concentration across the space is equal. The mathematical model that describes these physical-chemical phenomena is called the reaction sub-diffusion equation. In our study, we try to solve the 2D variable order version of these equations (2DVORSE) (linear and nonlinear) by using an accurate numerical technique which is the variable weighted average finite difference method (WAFDM). We will analyze the stability of the resulting scheme by using a modified suitable version of the John Von Neumann procedure. Specific stability conditions that occur or some parameters in the resulting schemes are derived and checked. At the end of the study, numerical examples are simulated to check the stability and the accuracy of the proposed technique.

A high-order L2 type difference scheme for the time-fractional diffusion equation

The present paper is devoted to constructing L2 type difference analog of the Caputo fractional derivative. The fundamental features of this difference operator are studied and it is used to construct difference schemes generating approximations of the second and fourth order in space and the (3−α)th-order in time for the time fractional diffusion equation with variable coefficients. Difference schemes were also constructed for the variable-order diffusion equation and the generalized fractional-order diffusion equation of the Sobolev type. Stability of the schemes under consideration as well as their convergence with the rate equal to the order of the approximation error are proven. The received results are supported by the numerical computations performed for some test problems.

Numerical Analysis of Time-Fractional Diffusion Equations via a Novel Approach

The aim of this paper is a new semianalytical technique called the variational iteration transform method for solving fractional-order diffusion equations. In the variational iteration technique, identifying of the Lagrange multiplier is an essential rule, and variational theory is commonly used for this purpose. The current technique has the edge over other methods as it does not need extra parameters and polynomials. The validity of the proposed method is verified by considering some numerical problems. The solution achieved has shown that the better accuracy of the proposed technique. This paper proposes a simpler method to calculate the multiplier using the Shehu transformation, making a valuable technique to researchers dealing with various linear and nonlinear problems.

Numerical Solution of Fractional Order Advection Reaction Diffusion Equation with Fibonacci Neural Network

The authors have developed an efficient method with the Fibonacci neural network’s help to solve the fractional-order reaction-diffusion equation in the present article. The Fibonacci neural network consists of an input layer with one perceptron, a hidden layer with $$n\times m$$ perceptions, and an output layer with one perceptron. The authors have used various degrees of the Fibonacci polynomial as an activation function to the input in the hidden layer. The authors then convert the fractional order diffusion equation with initial and boundary conditions into a non-constrained optimization problem, which is then called the cost function. Marquardt’s method is used to update the values of weights to minimize the cost function. After that, the authors used the discussed method on four examples with an exact solution and showed that our method works more accurately than previously existing methods through comparison. In the last, authors have used the discussed method to solve the unsolved diffusion equation and observe the change in solute concentration for different fractional-order at different times.

Kernel functions‐based approach for distributed order diffusion equations

AbstractIn this work, we solve distributed order diffusion equations (DODEs) by applying the theory on reproducing kernel functions (RKFs). The classical numerical quadrature formulae is used to approximate the DODE to a multi‐term Caputo fractional order diffusion equation (FDE). The Mittag‐Leffler RKF is introduced to estimate fractional derivatives of Caputo. And a space–time RKFs collocation scheme is derived for the multi‐term Caputo time FDEs. The accuracy of the present numerical technique is indicated by employing several experiments.

Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme

This paper develops a numerical scheme for finding the approximate solution of space fractional order of the diffusion equation (SFODE). Firstly, the compact finite difference (CFD) with convergence order \begin{document}$ \mathcal{O}(\delta \tau ^{2}) $\end{document} is used for discretizing time derivative. Afterwards, the spatial fractional derivative is approximated by the Chebyshev collocation method of the fourth kind. Furthermore, time-discrete stability and convergence analysis are presented. Finally, two examples are numerically investigated by the proposed method. The examples illustrate the performance and accuracy of our method compared to existing methods presented in the literature.

Computation of solution to fractional order partial reaction diffusion equations

In this article, the considered problem of Cauchy reaction diffusion equation of fractional order is solved by using integral transform of Laplace coupled with decomposition technique due to Adomian scheme. This combination led us to a hybrid method which has been properly used to handle nonlinear and linear problems. The considered problem is used in modeling spatial effects in engineering, biology and ecology. The fractional derivative is considered in Caputo sense. The results are obtained in series form corresponding to the proposed problem of fractional order. To present the analytical procedure of the proposed method, some test examples are provided. An approximate solution of a fractional order diffusion equation were obtained. This solution was rapidly convergent to the exact solution with less computational cost. For the computation purposes, we used MATLAB.

A New Kind of Parallel Natural Difference Method for Multi-Term Time Fractional Diffusion Model

Multi-term time fractional diffusion model is not only an important physical subject, but also a practical problem commonly involved in engineering. In this paper, we apply the alternating segment technique to combine the classical explicit and implicit schemes, and propose a parallel nature difference method alternating segment pure explicit–implicit (PASE-I) and alternating segment pure implicit–explicit (PASI-E) difference schemes for multi-term time fractional order diffusion equations. The existence and uniqueness of the solutions are proved, and stability and convergence analysis of the two schemes are also given. Theoretical analyses and numerical experiments show that the PASE-I and PASI-E schemes are unconditionally stable and satisfy second-order accuracy in spatial precision and 2 − α order in time precision. When the computational accuracy is equivalent, the CPU time of the two schemes are reduced by up to 2 / 3 compared with the classical implicit difference method. It indicates that the PASE-I and PASI-E parallel difference methods are efficient and feasible for solving multi-term time fractional diffusion equations.

Numerical Solution of a Class of Time-Fractional Order Diffusion Equations in a New Reproducing Kernel Space

In this paper, we structure some new reproducing kernel spaces based on Jacobi polynomial and give a numerical solution of a class of time fractional order diffusion equations using piecewise reproducing kernel method (RKM). Compared with other methods, numerical results show the reliability of the present method.

The capture of particles on absorbing traps in the medium with anomalous diffusion: The effective fractional order diffusion equation and the slow temporal asymptotic of survival probability

The new asymptotic behavior of the survival probability of particles, which move in the anomalous sub-diffusion way, in the medium with absolutely absorbing traps was established. The new power temporal behavior of the survival probability of diffusing particles at long times was obtained. This result is caused by the anomalous character of diffusion.

Convergence analysis of the space fractional-order diffusion equation based on the compact finite difference scheme

This paper develops a numerical method for approximating the space fractional diffusion equation in Caputo derivative sense. In this discretization process, firstly, the compact finite difference with convergence order $${\mathcal {O}}(\delta \tau ^{2})$$ is used to obtain the semi-discrete in time derivative. Afterward, the spatial fractional derivative is discretized by using the Chebyshev collocation method of the third-kind. This collocation scheme is based on the operational matrix. In addition, time-discrete stability and convergence are theoretically proved in detail. We solve two examples by the proposed method and the obtained results are compared with other numerical methods. The numerical results show that our method is much more accurate than existing methods.

Homotopy Analysis Method for a Fractional Order Equation with Dirichlet and Non-Local Integral Conditions

The main purpose of this paper is to obtain some numerical results via the homotopy analysis method for an initial-boundary value problem for a fractional order diffusion equation with a non-local constraint of integral type. Some examples are provided to illustrate the efficiency of the homotopy analysis method (HAM) in solving non-local time-fractional order initial-boundary value problems. We also give some improvements for the proof of the existence and uniqueness of the solution in a fractional Sobolev space.

Numerical solution of the nonlinear diffusion equation by using non-standard/standard finite difference and Fibonacci collocation methods

In this article, the authors have introduced a new nonstandard/standard finite difference scheme with the Fibonacci polynomial. The considered nonlinear fractional diffusion equation is reduced into a system of linear ordinary differential equations which are solved by using nonstandard/standard finite difference method. The developed schemes are unconditionally stable and the effectiveness and efficiency of the method is confirmed by applying it in two linear and one nonlinear problems and through comparison of their numerical results with existing analytical results. After validation of the efficiency of the method, the proposed method is used to solve the highly nonlinear fractional order diffusion equation. The stability of the method is also discussed. The salient feature of the article is the graphical exhibitions of the effects on the solute profiles due to increasing in non-linearity of the model and of the order of the spatial derivative for different particular cases.

Numerical Approximations for the Variable Coefficient Fractional Diffusion Equations with Non-smooth Data

Abstract In this article, we study the numerical approximation of a variable coefficient fractional diffusion equation. Using a change of variable, the variable coefficient fractional diffusion equation is transformed into a constant coefficient fractional diffusion equation of the same order. The transformed equation retains the desirable stability property of being an elliptic equation. A spectral approximation scheme is proposed and analyzed for the transformed equation, with error estimates for the approximated solution derived. An approximation to the unknown of the variable coefficient fractional diffusion equation is then obtained by post-processing the computed approximation to the transformed equation. Error estimates are also presented for the approximation to the unknown of the variable coefficient equation with both smooth and non-smooth diffusivity coefficient and right-hand side. Numerical experiments are presented to test the performance of the proposed method.

New approach to solving the atmospheric pollutant dispersion equation using fractional derivatives

In atmospheric environments, traditional differential equations do not adequately describe the problem of turbulent diffusion because the usual derivatives are not well defined in the non-differentiable behaviour introduced by turbulence, where the fractional calculation has become a very useful tool for studying anomalous dispersion and other transportation processes. Considering a new direction, this paper presents an analytical series solution of a three-dimensional advection–diffusion equation of fractional order, in the Caputo sense, applied to the dispersion of atmospheric pollutants. The solution is obtained by applying the generalised integral transform technique (GITT), solving the transformed problem by the Laplace decomposition method (LDM), and considering the lateral and vertical turbulent diffusion dependence on the longitudinal distance from the source, as well as a fractional parameter. The fractional solution is more general than the traditional solution in the sense that consideration of the integer order of the fractional parameter yields the traditional solution. The solution considers the memory effect in eddy diffusivity and in the fractional derivative, and it is simple, easy to implement, and converges rapidly. Numerical simulations were conducted to compare the performance of the proposed fractional solution to the traditional solution using an experimental dataset and other models, which also made it possible to find a better parametrisation for use in Gaussian models. The best results are obtained with the fractional order of the derivative.

Analytical Solutions of Fractional-Order Diffusion Equations by Natural Transform Decomposition Method.

In the present article, fractional-order diffusion equations are solved using the Natural transform decomposition method. The series form solutions are obtained for fractional-order diffusion equations using the proposed method. Some numerical examples are presented to understand the procedure of the Natural transform decomposition method. The Natural transform decomposition method has shown the least volume of calculations and a high rate of convergence compared to other analytical techniques, the proposed method can also be easily applied to other non-linear problems. Therefore, the Natural transform decomposition method is considered to be one of the best analytical technique, to solve fractional-order linear and non-linear partial deferential equations, particularly fractional-order diffusion equation.

An accurate numerical technique for solving two-dimensional time fractional order diffusion equation

ABSTRACTIn this paper, we consider a numerical technique for solving the two-dimensional fractional order diffusion equation with a time fractional derivative. The proposed technique depends basically on the fact that an expansion of the required approximate solution in a series of shifted Chebyshev polynomials of the first kind in the time and the Sinc function in the space. Then, the expansion coefficients are determined by reducing the problem with the initial conditions to a system of algebraic equations. The fractional derivatives are expressed in the Caputo sense. The numerical results of the proposed technique are compared with other published results to show the efficiency of the presented technique.

Approximate Controllability of Sub-Diffusion Equation with Impulsive Condition

In this work, we study an impulsive sub-diffusion equation as a fractional diffusion equation of order &#x0D; &#x0D; &#x0D; &#x0D; α&#x0D; ∈&#x0D; (&#x0D; 0&#x0D; ,&#x0D; 1&#x0D; )&#x0D; &#x0D; &#x0D; &#x0D; . Existence, uniqueness and regularity of solution of the problem is established via eigenfunction expansion. Moreover, we establish the approximate controllability of the problem by applying a unique continuation property via internal control which acts on a sub-domain.

Finite element method for drifted space fractional tempered diffusion equation

Off-late many models in viscoelasticity, signal processing or anomalous diffusion equations are formulated in fractional calculus. Tempered fractional calculus is the generalization of fractional calculus and in the last few years several important partial differential equations occurring in different field of science have been reconsidered in this terms like diffusion wave equations, Schr\(\ddot{o}\)dinger equation and so on. In the present paper, a time dependent tempered fractional diffusion equation of order \(\gamma \in (0,1)\) with forcing function is considered. Existence, uniqueness, stability, and regularity of the solution has been proved. Crank–Nicolson discretization is used in the time direction. By implementing finite element approximation a priori space–time estimate has been derived and we proved that the convergent order is \(\mathcal {O}(h^2+\varDelta t ^2)\) where h is the space step size and \(\varDelta t\) is the time difference. A couple of numerical examples have been presented to confirm the accuracy of theoretical results. Finally, we conclude that the studied method is useful for solving tempered fractional diffusion equations.