Fractional Diffusion-wave Equation Research Articles

Two linearized difference schemes on graded meshes for the time-space fractional nonlinear diffusion-wave equation with an initial singularity

Abstract In this article, two linearized difference schemes are considered to solve a class of time-space fractional nonlinear diffusion-wave equations with weak singularity near the initial time based on its equivalent integro-differential equation. The Crank-Nicolson method combined with the trapezoidal product integration rule with graded meshes is implemented to approximate the Riemann-Liouville integral. The fractional central difference formula and a novel compact difference method are used to construct fully discrete difference schemes. Using the energy method, the stability and convergence of the proposed schemes are demonstrated in terms of the L 2 norm with the accuracy of O N − min { r σ , 2 } + M − 2 and O N − min { r σ , 2 } + M − 4 , respectively. The theoretical analysis is complemented by numerical results.

MATHEMATICAL MODEL FOR NANOFIBERS’ DIFFUSION IN A NANOFLUID AND NUMERICAL SIMULATION

The diffusion process of the nanofibers in a nanofluid displays a combination of diffusion and wave characteristics. The wave-like behavior is attributed to the large aspect ratio of the nanofiber, which can be modeled as a bead-spring chain system, exhibiting wave-like properties. In order to describe this mixed diffusion-wave phenomenon, a fractional diffusion-wave equation is proposed, wherein multiple time fractional derivatives are employed. By appropriately regulating the fractional time terms, the model can be transformed into either the traditional diffusion equation or the traditional wave equation, as required. A numerical schedule is developed through the implementation of suitable time and space discretization, and a stability analysis is conducted. The numerical results substantiate the reliability of the numerical schedule and its applicability to other fractional differential equations.

A Novel and Accurate Algorithm for Solving Fractional Diffusion-Wave Equations

The main objective of this work is to apply a novel and accurate algorithm for solving the second-order and fourth-order fractional diffusion-wave equations (FDWEs). First, the desired equation is reduced to the corresponding Volterra integral equation (VIE). Then, the collocation method is applied, for which the Chebyshev cardinal functions (CCFs) have been considered as the bases. In this paper, the CCFs based on a Lobatto grid are introduced and used for the first time to solve these kinds of equations. To this end, the derivative and fractional integral operators are represented in CCFs. The main features of the method are simplicity, compliance with boundary conditions, and good accuracy. An exact analysis to show the convergence of the scheme is presented, and illustrative examples confirm our investigation.

Numerical simulation of an effective transform mechanism with convergence analysis of the fractional diffusion-wave equations

In the current study, we solve two very important mathematical models, such as the time fractional-order space-fractional telegraph and diffusion-wave equations using a reliable technique called the Adomian decomposition natural method (ADNM), which combines Adomian decomposition and natural transform. The diffusion wave equation describes the flood wave propagation, which is used in solving overland and open channel flow problems. For this reason, it is critical to fully understand and effectively solve the diffusion wave equations. Because telegraph equations are crucial for modeling and developing voltage or frequency transmission, they are widely used in physics and engineering. Furthermore, the designing process is greatly impacted by the uncertainty in the system parameters. For nonlinear ordinary differential equations based on the theorem of Banach fixed point, we provide existence and uniqueness theorem proofs. The present approach has been successfully used to explore exact solutions for time fractional-order and space fractional-order applications. The results show how effective and valuable the ADNM. This paper presents a methodology that will be used in future work to address similar nonlinear problems related to fractional calculus.

H3N3-2$$_\sigma $$-based difference schemes for time multi-term fractional diffusion-wave equation

H3N3-2$$_\sigma $$-based difference schemes for time multi-term fractional diffusion-wave equation

A New Perspective for Scientific Modelling: Sparse Reconstruction Based Approach for Learning Time-Space Fractional Differential Equations

Abstract This paper studies a sparse reconstruction based approach to learn time-space fractional differential equations, i.e. to identify parameter values and particularly the order of the fractional derivatives. The approach uses a generalized Taylor series expansion to generate, in every iteration, a feature matrix which is used to learn the fractional orders of both, temporal and spatial derivatives by minimizing the LASSO operator using Differential Evolution algorithm. To verify the robustness of the method, numerical results for time-space fractional diffusion equation, wave equation, and Burgers? equation at different noise levels in the data are presented. Finally, the methodology is applied to a realistic example where underlying fractional differential equation associated to published experimental data obtained from an in-vitro cell culture assay is learned.

Regularization of an inverse source problem for fractional diffusion-wave equations under a general noise assumption

Regularization of an inverse source problem for fractional diffusion-wave equations under a general noise assumption

Meshless analysis of fractional diffusion-wave equations by generalized finite difference method

In this paper, a meshless generalized finite difference method (GFDM) is proposed to solve the time fractional diffusion-wave (TFDW) equations. A second-order temporal discretization scheme is developed to tackle the Caputo fractional derivative, and then spatial discretization formulas are derived by the GFDM. Theoretical accuracy and convergence of the GFDM for TFDW equations are analyzed. Numerical results verify the theoretical results and the efficiency of the method.

A discrete spectral method for time fractional fourth-order 2D diffusion-wave equation involving [formula omitted]-Caputo fractional derivative

In this work, the ψ-Caputo fractional derivative, as a generalization of the classical Caputo fractional derivative in which the fractional derivative of a sufficiently differentiable function is defined with respect to another strictly increasing function, ψ, is utilized to define the time fractional fourth-order 2D diffusion-wave equation. A Chebyshev–Gauss–Lobatto scheme is developed to solve this equation. In this way, a new operational matrix for the ψ-Riemann–Liouville fractional integration of the Chebyshev polynomials is derived. The solution of the equation is obtained by determining the solution of the algebraic system extracted from approximation the ψ-Caputo fractional derivative term via a finite discrete Chebyshev series and employing the expressed operational matrix. The validity of the established approach is examined by solving two examples.

The Solution of Caputo Fractional Partial Differential Equations By Sumudu Transform

In this paper, we obtained an explicit solutions of the fractional diffusion-wave equations involving partial Caputo fractional derivative by using Sumudu transform. The solutions of the fractional diffusion-wave equations obtained in terms of Mittag-Leffler function and generalized Wright function.Some illustrative examples are also given.

On the critical exponents for a fractional diffusion-wave equation with a nonlinear memory term in a bounded domain

In this paper, we prove sharp blow-up and global existence results for a time fractional diffusion-wave equation with a nonlinear memory term in a bounded domain, where the fractional derivative in time is taken in the sense of the Caputo type. Moreover, we also give a result for nonexistence of global solutions to a wave equation with a nonlinear memory term in a bounded domain. The proof of blow-up results is based on the eigenfunction method and the asymptotic properties of solutions for an ordinary fractional differential inequality.

Error Analysis of the Nonuniform Alikhanov Scheme for the Fourth-Order Fractional Diffusion-Wave Equation

This paper considers the numerical approximation to the fourth-order fractional diffusion-wave equation. Using a separation of variables, we can construct the exact solution for such a problem and then analyze its regularity. The obtained regularity result indicates that the solution behaves as a weak singularity at the initial time. Using the order reduction method, the fourth-order fractional diffusion-wave equation can be rewritten as a coupled system of low order, which is approximated by the nonuniform Alikhanov scheme in time and the finite difference method in space. Furthermore, the H2-norm stability result is obtained. With the help of this result and a priori bounds of the solution, an α-robust error estimate with optimal convergence order is derived. In order to further verify the accuracy of our theoretical analysis, some numerical results are provided.

On the solution of multi-term time fractional diffusion-wave equation involving ultra-hyperbolic operator

A diffusion-wave equation with multi-term Hilfer fractional derivatives (HFDs) in time and ultra-hyperbolic operator (UHO) in space has been considered. Fundamental solution of the fractional diffusion-wave equation is obtained by using Laplace and Fourier transform with Mellin-Barnes integral representation. The solution obtained involved the Fox H-function. In addition, we provide some special cases of diffusion-wave equation.

An efficient spectral collocation method based on the generalized Laguerre polynomials to multi-term time fractional diffusion-wave equations

In this study, a spectral collocation method is proposed to solve a multi-term time fractional diffusion-wave equation. The solution is expanded by a series of generalized Laguerre polynomials, and then, by imposing the collocation nodes, the equation is reduced to a linear system of algebraic equations. The coefficients of the expansion can be determined by solving the resulting system. The convergence of the method is proved, and some numerical examples are presented to demonstrate the accuracy and efficiency of the scheme. Finally, conclusions are given.

Two Regularization Methods for Identifying the Spatial Source Term Problem for a Space-Time Fractional Diffusion-Wave Equation

In this paper, we delve into the challenge of identifying an unknown source in a space-time fractional diffusion-wave equation. Through an analysis of the exact solution, it becomes evident that the problem is ill-posed. To address this, we employ both the Tikhonov regularization method and the Quasi-boundary regularization method, aiming to restore the stability of the solution. By adhering to both a priori and a posteriori regularization parameter choice rules, we derive error estimates that quantify the discrepancies between the regularization solutions and the exact solution. Finally, we present numerical examples to illustrate the effectiveness and stability of the proposed methods.

Two-grid finite element method with an H2N2 interpolation for two-dimensional nonlinear fractional multi-term mixed sub-diffusion and diffusion wave equation

<abstract><p>In this paper, we studied the two-grid method (TGM) for two-dimensional nonlinear time fractional multi-term mixed sub-diffusion and diffusion wave equation. A fully discrete scheme with the quadratic Hermite and Newton interpolation (H2N2) method was considered in the temporal direction and the expanded finite element method is used to approximate the spatial direction. In order to reduce computational time, a dual grid method based on Newton iteration was constructed with order $ \alpha\in(0, 1) $ and $ \beta\in(1, 2) $. The global convergence order of the two-grid scheme reaches $ O(\tau^{3-\beta}+h^{r+1}+H^{2r+2}) $, where $ \tau $, $ H $ and $ h $ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. The error estimation and stability of the fully discrete scheme were derived. Theoretical analysis shows that the two grid algorithms maintain asymptotic optimal accuracy while saving computational costs. In addition, numerical experiments further confirmed the theoretical results.</p></abstract>

Computational analysis of time fractional diffusion wave equation via novel B-splines-based technique

<abstract><p>In this paper, the authors propose and utilize a novel B-splines-based technique to solve time fractional diffusion wave equations numerically. The present approach employs the arbitrary-order fractional derivative in conjunction with the $ \theta $-weighted scheme. The integration of the Atangana-Baleanu fractional derivative into time fractional diffusion wave equations represents a novel advancement within B-spline methodologies. Furthermore, the stability and convergence of the presented schemes are given. The suggested approach not only exhibits unconditional stability, it also attains second-order convergence in both temporal and spatial dimensions. We conclude by applying our theoretical results to certain numerical examples to demonstrate their efficiency and accuracy.</p></abstract>

A hybrid collocation method for the approximation of 2D time fractional diffusion-wave equation

<p>The multi-term time-fractional order diffusion-wave equation (MT-TFDWE) is an important mathematical model for processes exhibiting anomalous diffusion and wave propagation with memory effects. This article develops a robust numerical technique based on the Chebyshev collocation method (CCM) coupled with the Laplace transform (LT) to solve the time-fractional diffusion-wave equation. The CCM is utilized to discretize the spatial domain, which ensures remarkable accuracy and excellent efficiency in capturing the variations of spatial solutions. The LT is used to handle the time-fractional derivative, which converts the problem into an algebraic equation in a simple form. However, while using the LT, the main difficulty arises in calculating its inverse. In many situations, the analytical inversion of LT becomes a cumbersome job. Therefore, the numerical techniques are then used to obtain the time domain solution from the frequency domain solution. Various numerical inverse Laplace transform methods (NILTMs) have been developed by the researchers. In this work, we use the contour integration method (CIM), which is capable of handling complex inversion tasks efficiently. This hybrid technique provides a powerful tool for the numerical solution of the time-fractional diffusion-wave equation. The accuracy and efficiency of the proposed technique are validated through four test problems.</p>

Time two-grid technique combined with temporal second order difference method for semilinear fractional diffusion-wave equations

Time two-grid technique combined with temporal second order difference method for semilinear fractional diffusion-wave equations

Well-posedness and blow-up results for a time-space fractional diffusion-wave equation

<abstract><p>In this paper, we demonstrate the local well-posedness and blow up of solutions for a class of time- and space-fractional diffusion wave equation in a fractional power space associated with the Laplace operator. First, we give the definition of the solution operator which is a noteworthy extension of the solution operator of the corresponding time-fractional diffusion wave equation. We have analyzed the properties of the solution operator in the fractional power space and Lebesgue space. Next, based on some estimates of the solution operator and source term, we prove the well-posedness of mild solutions by using the contraction mapping principle. We have also investigated the blow up of solutions by using the test function method. The last result describes the properties of mild solutions when $ \alpha\rightarrow1^- $. The main feature of the proof is the reasonable use of continuous embedding between fractional space and Lebesgue space.</p></abstract>