Time-fractional Diffusion-wave Equation Research Articles

A novel explicit fast numerical scheme for the Cauchy problem for integro-differential equations with a difference kernel and its application

The present study focuses on designing a second-order novel explicit fast numerical scheme for the Cauchy problem incorporating memory associated with an evolutionary equation, where the integral term's kernel is a discrete difference operator. The Cauchy problem under consideration is related to a real finite-dimensional Hilbert space and includes a self-adjoint operator that is both positive and definite. We introduce a transformative technique for converting the Cauchy problem incorporating memory, into a local evolutionary system of equations by approximating the difference kernel using the sum of exponentials (SoE) approach. A second-order explicit scheme is then constructed to solve the local system. We thoroughly investigate the stability of this explicit scheme, and present the necessary conditions for the stability of the scheme. Moreover, we extended our investigation to encompass time-fractional diffusion-wave equations (TFDWEs) involving a fractional Caputo derivative with an order ranging between (1,2). Initially, we transform the main TFDWE model into a new model that incorporates the fractional Riemann-Liouville integral. Subsequently, we expand the applicability of our idea to develop an explicit fast numerical algorithm for approximating the model. The stability properties of this fast scheme for solving TFDWEs are assessed. Numerical simulations including a two-dimensional Cauchy problem as well as one-dimensional and two-dimensional TFDWE models are provided to validate the accuracy and experimental order of convergence of the schemes.

Mainardi smoothing homotopy method for solving nonlinear optimal control problems

This paper proposes a new type of smoothing homotopy method for solving general nonlinear optimal control problems via the indirect method, leveraging the Mainardi kernel as the smoothing kernel. The Mainardi kernel is firstly derived from the fundamental solution of the time-fractional diffusion-wave equation, representing a generalized form of the Gaussian kernel. By altering the fractional derivative order, the kernel can seamlessly switch between non-Gaussian and Gaussian forms. Then, parts of the two-point boundary value problems are convolved with the smoothing kernel, and the resulting surrogates are incorporated into the necessary conditions, replacing the terminal state and costate variables. The homotopy process is used for the smoothing parameter: increasing this parameter enhances the smoothing effect, simplifying the homotopy problems, while a parameter value of zero implies no smoothing, reverting to the original problems. Additionally, explicit expressions for the Mainardi kernel at specific derivative orders are derived, avoiding the inefficiencies associated with fractional derivatives. Simulation examples demonstrate that the proposed method offers significant advantages in flexibility and convergence compared to the traditional Gaussian smoothing homotopy method.

Analysis of a meshless generalized finite difference method for the time-fractional diffusion-wave equation

In this paper, a generalized finite difference method (GFDM) is proposed and analyzed for meshless numerical solution of the time-fractional diffusion-wave equation. Two (3−α)-order accurate temporal discretization schemes are presented by using the L1 formula and the original H2N2 or fast H2N2 formulas to discretize the time-fractional derivative of order α∈(1,2). The stability of the temporal discretization schemes is analyzed. Then, the time-fractional diffusion-wave initial-boundary value problem is transformed into a series of time-independent integer-order boundary value problems, and discrete linear algebraic systems are built by the application of the GFDM. Accuracy analysis of the GFDM with both original H2N2 and fast H2N2 formulas is presented in theory, and numerical experimental results are provided to verify the theoretical results and the effectiveness of the proposed meshless 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.

DECAY RATES AND INITIAL VALUES FOR TIME-FRACTIONAL DIFFUSION-WAVE EQUATIONS

We consider a solution u(·,t) to an initial boundary value problem for time-fractional diffusion-wave equation with the order α ∈ (0,2)\ {1} where t is a time variable. We first prove that a suitable norm of u(·,t) is bounded by the rate t −α for 0 α 1 and t 1−α for 1 α 2 for all large t > 0. Second, we characterize initial values in the cases where the decay rates are faster than the above critical exponents. Differently from the classical diffusion equation α = 1, the decay rate can keep some local characterization of initial values. The proof is based on the eigenfunction expansions of solutions and the asymptotic expansions of the Mittag-Leffler functions for large time.

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.

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.

A fast compact finite difference scheme for the fourth-order diffusion-wave equation

In this paper, the H 2 N 2 method and compact finite difference scheme are proposed for the fourth-order time-fractional diffusion-wave equations. In order to improve the efficiency of calculation, a fast scheme is constructed with utilizing the sum-of-exponentials to approximate the kernel t 1 − γ . Based on the discrete energy method, the Cholesky decomposition method and the reduced-order method, we prove the stability and convergence. When K 1 < 3 2 , the convergence order is O ( τ 3 − γ + h 4 + ϵ ) , where K 1 is diffusion coefficient, γ is the order of fractional derivative, τ is the parameters for the time meshes, h is the parameters for the space meshes and ε is tolerance error. Numerical results further verify the theoretical analysis. It is find that the CPU time is extremely little in our scheme.

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

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

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

&lt;abstract&gt;&lt;p&gt;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.&lt;/p&gt;&lt;/abstract&gt;

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

&lt;p&gt;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.&lt;/p&gt;

The quasi-boundary regularization method for identifying the source term of time-fractional diffusion wave equation

The quasi-boundary regularization method for identifying the source term of time-fractional diffusion wave equation

Numerical Investigation of the Fractional Diffusion Wave Equation with the Mittag–Leffler Function

A spline is a sufficiently smooth piecewise curve. B-spline functions are powerful tools for obtaining computational outcomes. They have also been utilized in computer graphics and computer-aided design due to their flexibility, smoothness and accuracy. In this paper, a numerical procedure dependent on the cubic B-spline (CuBS) for the time fractional diffusion wave equation (TFDWE) is proposed. The standard finite difference (FD) approach is utilized to discretize the Atangana–Baleanu fractional derivative (ABFD), while the derivatives in space are approximated through the CuBS with a θ-weighted technique. The stability of the propounded algorithm is analyzed and proved to be unconditionally stable. The convergence analysis is also studied, and it is of the order O(h2+(Δt)2). Numerical solutions attained by the CuBS scheme support the theoretical solutions. The B-spline technique gives us better results as compared to other numerical techniques.

Symmetric fractional order reduction method with L1 scheme on graded mesh for time fractional nonlocal diffusion-wave equation of Kirchhoff type

Symmetric fractional order reduction method with L1 scheme on graded mesh for time fractional nonlocal diffusion-wave equation of Kirchhoff type

Jacobi polynomials for the numerical solution of multi-dimensional stochastic multi-order time fractional diffusion-wave equations

In this paper, the one- and two-dimensional stochastic multi-order fractional diffusion-wave equations are introduced and a collocation procedure based on the shifted Jacobi polynomials is established to find their numerical solutions. Through this way, some operational matrices regarding classical and stochastic integrals as well as fractional and classical differentiations of these polynomials, are obtained. By representing the problem solution using an expansion of these polynomials (in which the coefficients of the expansion are unknown) and substituting it into the first problem, as well as by employing the obtained operational matrices, a system containing algebraic equations is obtained. Eventually, the coefficients of expansion and subsequently the solution of the original problem are found by solving this system. The correctness of the procedure is studied by solving four examples.

Single-term and multi-term nonuniform time-stepping approximation methods for two-dimensional time-fractional diffusion-wave equation

The aim of this work is to propose two efficient schemes to handle the accuracy near the singularity at t=0 in solving two-dimensional time-fractional diffusion-wave equation (TFDWE). The considered time fractional derivative is in the Caputo sense of order α(1<α<2). For the approximation of the time-fractional Caputo derivative (TFCD), we use Nonuniform L1 method (single-step) and Nonuniform Crank-Nicolson L1−2 method (multi-step). The L1 method has order of convergence (OC) min⁡(3−α,γα), where γ is the mesh grading parameter used in construction of the nonuniform mesh, and L1−2 method has second OC. We consider nonuniform time mesh to compensate the lack of smoothness caused by the presence of singularity in TFCD at t=0. After that, we adopt these two methods to approximate TFCD and apply the central difference operator for the space direction derivative approximations to get the system of equations for considered model. Then, we use the Alternating Direction Implicit (ADI) approach to develop two kinds of fully discrete schemes under the regularity conditions to solve the TFDWE. Further, we prove the stability analysis of these two schemes. Two numerical examples are given for one-dimensional (1D) and two-dimensional (2D) TFDWE with smooth and non-smooth exact solutions to indicate the accuracy of ADI schemes. The illustrated examples show that both schemes have second-order accuracy in space direction, and in temporal direction the schemes achieve min⁡(3−α,γα) and second order convergence, respectively for all 1<α<2. The corresponding absolute error is plotted to see the advantage of nonuniform time meshes at the initial singularity t=0.

Simultaneous inversion for a fractional order and a time source term in a time-fractional diffusion-wave equation

Abstract In this article, we consider an inverse problem for determining simultaneously a fractional order and a time-dependent source term in a multi-dimensional time-fractional diffusion-wave equation by a nonlocal condition. Based on a uniformly bounded estimate of the Mittag-Leffler function given in this paper, we prove the uniqueness of the inverse problem and the Lipschitz continuity properties for the direct problem. Then we employ the Levenberg–Marquardt method to recover simultaneously the fractional order and the time source term, and establish a finite-dimensional approximation algorithm to find a regularized numerical solution. Moreover, a fast tensor method for solving the direct problem in the three-dimensional case is provided. Some numerical results in one and multidimensional spaces are presented for showing the robustness of the proposed algorithm.

Numerical approximation of the Cauchy non-homogeneous time-fractional diffusion-wave equation with Caputo derivative using shifted Chebyshev polynomials

In this paper, we introduce a numerical technique for solving the Cauchy non-homogeneous time-fractional diffusion-wave equation with the Caputo derivative operator. The key idea behind our approach is to employ the roots of shifted Chebyshev polynomials to collocate the problem in the time dimension. By introducing certain functions to homogenize the resulting system of ordinary differential equations, we obtain an approximate solution to the problem. To assess the reliability of our method, we conduct a thorough comparison with existing numerical techniques available in the literature. Through extensive experimentation, we demonstrate the effectiveness and accuracy of our proposed technique. Notably, the results highlight the efficiency and precision of our method in approximating solutions for the considered problem. Furthermore, we investigate the convergence of our method and provide theoretical insights into its performance. Our findings support the validity and robustness of the proposed approach. Overall, this paper contributes a valuable numerical tool for addressing the Cauchy non-homogeneous time-fractional diffusion-wave equation, offering enhanced efficiency and accuracy compared to existing methods.

The quasi-reversibility regularization method for backward problems of the time-fractional diffusion-wave equation

This paper is devoted to two backward problems in a time-fractional diffusion-wave equation, with the aid of extra measurement data at a final time. Since these two problems are ill-posed, a quasi-reversibility method is obtained by employing eigenfunction expansion, and the existence, uniqueness and regularity of the regularized solutions are also proven. We can get order-optimal error estimates from the a-priori parameter choice rule. Finally, several numerical examples of both one- and two-dimension show that the proposed regularization method is effective and stable.