Fractional Diffusion Equation Research Articles

A fast modified $$\overline{L1}$$ finite difference method for time fractional diffusion equations with weakly singular solution

A fast modified $$\overline{L1}$$ finite difference method for time fractional diffusion equations with weakly singular solution

An efficient approach for solving a class of fractional anomalous diffusion equation with convergence

Abstract This article presents a study on Fractional Anomalous Diffusion (FAD) and proposes a novel numerical algorithm for solving Cupoto’s type fractional sub-diffusion equations. to convert the fractional model into a set of nonlinear algebraic equations. These equations are efficiently solved using the Levenberg-Marquardt algorithm. The study provides the error analysis to validate the proposed method. The effectiveness and accuracy of the method are demonstrated through several test problems, and its performance and reliability are compared with other existing methods in the literature. The results indicate that the proposed method is a reliable and efficient technique for solving fractional sub-diffusion equations, with better accuracy and computational efficiency than other existing methods. The study’s findings have important implications for researchers working in the field of fractional calculus. They could provide a valuable tool for solving sub-diffusion equations in various applications, including physics, chemistry, biology, and engineering.

LAPLACE DECOMPOSITION METHOD FOR SOLVING THE TWO-DIMENSIONAL DIFFUSION PROBLEM IN FRACTAL HEAT TRANSFER

In this paper, the Local Fractional Laplace Decomposition Method (LFLDM) is used for solving a type of Two-Dimensional Fractional Diffusion Equation (TDFDE). In this method, first we apply the Laplace transform and its inverse to the main equation, and then the Adomian decomposition is used to obtain approximate/analytical solution. The accuracy and applicability of the LFLDM is shown through two examples. The LFLDM results are in good agreement with the exact solution of the problems.

High Order Numerical Scheme for Generalized Fractional Diffusion Equations

High Order Numerical Scheme for Generalized Fractional Diffusion Equations

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.

Conformal and Non-Minimal Couplings in Fractional Cosmology

Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered “anomalous”. It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus must reflect non-local, frequency- and history-dependent properties of power-law phenomena. This tool has various important applications, such as fractional mass conservation, electrochemical analysis, groundwater flow problems, and fractional spatiotemporal diffusion equations. It can also be used in cosmology to explain late-time cosmic acceleration without the need for dark energy. We review some models using fractional differential equations. We look at the Einstein–Hilbert action, which is based on a fractional derivative action, and add a scalar field, ϕ, to create a non-minimal interaction theory with the coupling, ξRϕ2, between gravity and the scalar field, where ξ is the interaction constant. By employing various mathematical approaches, we can offer precise schemes to find analytical and numerical approximations of the solutions. Moreover, we comprehensively study the modified cosmological equations and analyze the solution space using the theory of dynamical systems and asymptotic expansion methods. This enables us to provide a qualitative description of cosmologies with a scalar field based on fractional calculus formalism.

Numerical algorithm for a general fractional diffusion equation

In this manuscript, the analytical solution to the general time fractional diffusion equation followed by the regularity analysis of its solution is presented. Further, a hybrid efficient numerical algorithm for a general fractional derivative of order α∈(0,1) is designed. The physical application of the developed approximation formula is employed to design a hybrid numerical algorithm to determine the numerical solution of the corresponding general time fractional diffusion equation. The proposed numerical techniques are subjected to solvability, numerical stability, and convergence analysis. Numerical example is utilized to verify that (3−α)-th order convergence is attained in time which is higher than the existing scheme (Xu et al., 2013). In the spatial direction, the fourth order convergence is obtained utilizing the compact finite difference method. Besides, such a hybrid numerical algorithm is also used in the stochastic model.

A fast algorithm for multi-term time-space fractional diffusion equation with fractional boundary condition

A fast algorithm for multi-term time-space fractional diffusion equation with fractional boundary condition

Unique recovery of the initial state of distributed order time fractional diffusion equation

We use Phragm\'en-Lindel\"of-Liouville argument to prove the uniqueness for the determining the initial state of solution for the time fractional diffusion equation with distributed order derivative. Several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

Identifying the source term and the initial value simultaneously for Caputo–Hadamard fractional diffusion equation on spherically symmetric domain

Identifying the source term and the initial value simultaneously for Caputo–Hadamard fractional diffusion equation on spherically symmetric domain

Double fast algorithm for solving time-space fractional diffusion problems with spectral fractional Laplacian

This paper presents an efficient and concise double fast algorithm to solve high dimensional time-space fractional diffusion problems with spectral fractional Laplacian. We first establish semi-discrete scheme of time-space fractional diffusion equation, which uses linear finite element or fourth-order compact difference method combining with matrix transfer technique to approximate spectral fractional Laplacian. Then we introduce a fast time-stepping L1 scheme for time discretization. The proposed scheme can exactly evaluate fractional power of matrix and perform matrix-vector multiplication at per time level by using discrete sine transform, which doesn't need to resort to any iteration method and can significantly reduce computation cost and memory requirement. Further, we address stability and convergence analyses of full discrete scheme based on fast time-stepping L1 scheme on graded time mesh. Our error analysis shows that the choice of graded mesh factor ω=(2−α)/α shall give an optimal temporal convergence O(N−(2−α)) with N denoting the number of time mesh. Finally, ample numerical examples are delivered to illustrate our theoretical analysis and the efficiency of the suggested scheme.

An equation for complex fractional diffusion created by the Struve function with a T-symmetric univalent solution

Abstract A T T -symmetric univalent function is a complex valued function that is conformally mapping the unit disk onto itself and satisfies the symmetry condition ϕ [ T ] ( ζ ) = [ ϕ ( ζ T ) ] 1 ∕ T {\phi }^{\left[T]}\left(\zeta )={\left[\phi \left({\zeta }^{T})]}^{1/T} for all ζ \zeta in the unit disk. In other words, it is a complex function that preserves the unit disk’s shape and orientation and is symmetric about the unit circle. They are used in the study of geometric function theory and the theory of univalent functions. In recent effort, we extend the class of fractional anomalous diffusion equations in a symmetric complex domain. we aim to present the analytic univalent solution for such a class using special functions technique. Our analysis and comparative findings are further supported by the geometric simulations for the univalent solution such as the convexity and starlikeness of the diffusion. As a consequence of illustration of a list of conditions yielding the univalent solutions (normalize analytic function in the open unit disk), the normalization of diffusion shape is achieved.

The first boundary value problem for the fractional diffusion equation in a degenerate angular domain

This article addresses the problems observed in branching fractal structures, where super-slow transport processes can occur, a phenomenon described by diffusion equations with a fractional time derivative. The characteristic feature of these processes is their extremely slow relaxation rate, where a physical quantity changes more gradually than its first derivative. Such phenomena are sometimes categorized as processes with “residual memory”. The study presents a solution to the first boundary problem in an angular domain degenerating into a point at the initial moment of time for a fractional diffusion equation with the RiemannLiouville fractional differentiation operator with respect to time. It establishes the existence theorem of the problem under investigation and constructs a solution representation. The need for understanding these super-slow processes and their impact on fractal structures is identified and justified. The paper demonstrates how these processes contribute to the broader understanding of fractional diffusion equations, proving the theorem’s existence and formulating a solution representation.

IDENTIFICATION OF A TIME-DEPENDENT SOURCE TERM IN A NONLOCAL PROBLEM FOR TIME FRACTIONAL DIFFUSION EQUATION

This paper is concerned with the inverse problem of recovering the time dependent source term in a time fractional diffusion equation, in the case of nonlocal boundary condition and integral overdetermination condition. The boundary conditions of this problem are regular but not strongly regular. The existence and uniqueness of the solution are established by applying generalized Fourier method based on the expansion in terms of root functions of a spectral problem, weakly singular Volterra integral equation and fractional type Gronwall’s inequality. Moreover, we show its continuous dependence on the data.

SIMULTANEOUS INVERSION OF THE SOURCE TERM AND INITIAL VALUE OF THE TIME FRACTIONAL DIFFUSION EQUATION

In this paper, the problem we investigate is to simultaneously identify the source term and initial value of the time fractional diffusion equation. This problem is ill-posed, i.e., the solution (if exists) does not depend on the measurable data. We give the conditional stability result under the a-priori bound assumption for the exact solution. The modified Tikhonov regularization method is used to solve this problem, and under the a-priori and the a-posteriori selection rule for the regularization parameter, the convergence error estimations for this method are obtained. Finally, numerical example is given to prove the effectiveness of this regularization method.

INITIAL-BOUNDARY VALUE PROBLEMS TO THE TIME-NONLOCAL DIFFUSION EQUATION

This article investigates a fractional diffusion equation involving Caputo fractional derivative and Riemann-Liouville fractional integral. The equation is supplemented by initial and boundary conditions in the domain defined by the interval by space 0<x<1 and interval by time 0<t<T. The fractional operators are defined rigorously, utilizing the Caputo fractional derivative of order β and the Riemann-Liouville fractional integral of order α, where 0<α<β≤1. The main results include the presentation of well-known properties associated with fractional operators and the establishment of the unique solution to the given problem. The key findings are summarized through a theorem that provides the explicit form of the solution. The solution is expressed as a series involving the two-parameter Mittag-Leffler function and orthonormal eigenfunctions of the Sturm-Liouville operator. The uniqueness of the solution is proven, ensuring that the problem has a single, well-defined solution under specific conditions on the initial function. Furthermore, the article introduces and proves estimates related to the Mittag-Leffler function, providing bounds crucial for the convergence analysis. The convergence of the series is investigated, and conditions for the solution to belong to a specific function space are established. The uniqueness of the solution is demonstrated, emphasizing its singularity within the given problem. Finally, the continuity of the solution in the specified domain is confirmed through the uniform convergence of the series.

Elzaki residual power series method to solve fractional diffusion equation.

The time-fractional order differential equations are used in many different contexts to analyse the integrated scientific phenomenon. Hence these equations are the point of interest of the researchers. In this work, the diffusion equation for a one-dimensional time-fractional order is solved using a combination of residual power series method with Elzaki transforms. The residual power series approach is a useful technique for finding approximate analytical solutions of fractional differential equations that needs the residual function's (n-1)α derivative. Since it is challenging to determine a function's fractional-order derivative, the traditional residual power series method's application is somewhat constrained. The Elzaki transform with residual power series method is an attempt to get over the limitations of the residual power series method. The obtained numerical solutions are compared with the exact solution of this equation to discuss the method's applicability and efficiency. The results are also graphically displayed to show how the fractional derivative influences the behaviour of the solutions to the suggested method.

Haar wavelet based numerical technique for the solutions of fractional advection diffusion equations

Haar wavelet based numerical technique for the solutions of fractional advection diffusion equations

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.

An $$\alpha $$-robust analysis of finite element method for space-time fractional diffusion equation

An $$\alpha $$-robust analysis of finite element method for space-time fractional diffusion equation