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This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0, 1). By providing an a priori estimate of the exact solution, we have established the continuous dependence on the initial data and uniqueness of the solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2-type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings. [ABSTRACT FROM AUTHOR]

We establish a unified framework to study the conforming and nonconforming virtual element methods (VEMs) for a class of time dependent fourth-order reaction–subdiffusion equations with the Caputo derivative. To resolve the intrinsic initial singularity we adopt the nonuniform Alikhanov formula in the temporal direction. In the spatial direction three types of VEMs, including conforming virtual element, |$C^0$| nonconforming virtual element and fully nonconforming Morley-type virtual element, are constructed and analysed. In order to obtain the desired convergence results, the classical Ritz projection operator for the conforming virtual element space and two types of new Ritz projection operators for the nonconforming virtual element spaces are defined, respectively, and the projection errors are proved to be optimal. In the unified framework we derive a prior error estimate with optimal convergence order for the constructed fully discrete schemes. To reduce the computational cost and storage requirements, the sum-of-exponentials (SOE) technique combined with conforming and nonconforming VEMs (SOE-VEMs) are built. Finally, we present some numerical experiments to confirm the theoretical correctness and the effectiveness of the discrete schemes. The results in this work are fundamental and can be extended into more relevant models. [ABSTRACT FROM AUTHOR]

This work is to analyze a Legendre collocation approximation for third-kind Volterra integro-differential equations. The rigorous error analysis in the L ∞ and L ω 0 , 0 2 -norms is provided for the proposed method. In fact when converting the original equation to an equivalent second kind one, the integral operator of the obtained equation contains two singularities and may become non-compact under certain conditions. In addition, in order to avoid the low-order accuracy caused by the singularity of the solution at the initial point, we adopted the idea of smooth transformation at the beginning to convert the original equation into a new equation with a more regular solution. Finally, the validity and applicability of the method are verified by several numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we first construct a linearized Galerkin‐Legendre spectral method for the one‐dimensional nonlinear fractional Ginzburg‐Landau equation, where a three‐level linearized Crank‐Nicolson scheme is used for time discretization. The unique solvability and boundedness properties of the fully discrete scheme are analyzed. It is shown that the method is unconditionally convergent in the maximum norm with second‐order accuracy in time and spectral accuracy in space. Then, two‐dimensional problems are considered and a split‐step alternating direction implicit Galerkin‐Legendre spectral method is introduced without theoretical analysis. Finally, some numerical examples are presented to illustrate the effectiveness of the two proposed schemes. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the numerical solutions of the coupled fractional Klein-Gordon-Schrödinger equation. The numerical schemes are constructed by combining the Crank-Nicolson/leap-frog difference methods for the temporal discretization and the Galerkin finite element methods for the spatial discretization. We give a detailed analysis of the conservation properties in the senses of discrete mass and energy. Then the numerical solutions are shown to be unconditionally bounded in L2 −norm, H α 2 − semi-norm and L ∞ − norm, respectively. Based on the well-known Brouwer fixed-point theorem and the mathematical induction, the unique solvability of the discrete solutions is proved. Moreover, the schemes are proved to be unconditionally convergent with the optimal order O τ 2 + h r + 1 , where τ is the temporal step, h is the spatial grid size, and r is the order of the selected finite element space. Furthermore, by using the proposed decoupling and iterative algorithms, several numerical examples are included to support theoretical results and show the effectiveness of the schemes. Finally, the fast Krylov subspace solver with suitable circulant preconditioner is designed to effectively solve the Toeplitz-like linear systems. In each iterative step, this method can effectively reduce the memory requirement of above each finite element scheme from O M 2 to O(M), and the computational complexity from O M 3 to O (M log M) , where M is the number of grid nodes. Numerical tests are carried out to show that this fast algorithm is more practical than the traditional backslash and LU factorization/Cholesky decomposition methods, in terms of memory requirement and computational cost. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability properties of Runge-Kutta methods for Volterra integro-differential equations. Both the basic test equation and a convolution test equation are considered. Some fixed order recurrence relations and the corresponding stability conditions are derived for general methods. The concept of V 0 -stability is introduced for the convolution test equation and some V 0 -stable one-stage methods are found. The A 0 -stability and V 0 -stability of the fully implicit discretized collocation methods with one or two stages are investigated in details. Finally, some numerical experiments are given to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate multistep Runge–Kutta methods for Volterra integro-differential equations. First, we derive order conditions for methods of order p and stage order q = p − 1 for p ⩽ 4. Then, stability properties of these methods with respect to the basic and convolution test equations are analyzed. V 0 -stable methods of order 2 and 3 are obtained. Finally, some numerical experiments are given to verify the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a two-parameter Milstein method for stochastic Volterra integral equations is introduced. First, the method is proved to be strongly convergent with order one in L p norm (p ≥ 1). Then, we investigate the mean square stability of the exact and numerical solutions of a stochastic convolution test equation. Stability conditions are derived. Based on these conditions, analytical and numerical stability regions are plotted and compared with each other. The results show that additional implicitness offers benefits for numerical stability. Finally, some numerical experiments are carried out to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a new notion for solving algebraic nonlinear equations, which is derived from the well known waveform relaxation iterative method, and is well suited to couple different numerical method for nonlinear equations, such as classical Newton’s method, quasi-Newton method, Conjugate-Gradient method, etc. We show in this paper that the arithmetic obtained by coupling the classical Newton’s method has essential capability for parallel computation and converges globally. Numerical results validate the theoretical analysis very well. [Copyright &y& Elsevier]

This paper aims to provide a rigorous analysis of exponential convergence of the Chebyshev collocation method for third kind linear Volterra integral equations. Different from Volterra integral equations of the second kind, the integral operator in third kind equations is noncompact under certain conditions, which brings special challenges to numerical analysis. The key idea of the proposed method is to adopt a smoothing transformation for the Chebyshev collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the solution of the resulting equation will possess better regularity and then the numerical method can achieve the spectral accuracy. Moreover, in order to show the applicability and efficiency of the method, several examples with non-smooth solutions are illustrated. [ABSTRACT FROM AUTHOR]

This paper investigates a nonlinear time-fractional mixed sub-diffusion and diffusion-wave equation with delay. The problem is particularly challenging due to its nonlinear nature, the presence of a time delay, and the incorporation of both fractional diffusion and fractional wave terms, introducing computational complexities for numerical analysis. To address this, we transform the model into a new generalized form involving Riemann–Liouville fractional integrals and a Caputo derivative of order α ∈ (0 , 1). Subsequently, a temporal second-order linearized difference scheme is presented to approximate the model, and its unconditional stability is rigorously proven based on discrete Gronwall's inequality. To reduce computational and storage costs, we extend the discussion to a fast variant of the proposed difference scheme. The sum of exponentials approach is employed to approximate the kernel function in fractional-order operators, leading to fast difference analogs for the Riemann–Liouville fractional integral and the Caputo derivative. A fast variant of the developed direct method is introduced based on these analogs. Numerical results are provided to validate our theoretical findings and assess the accuracy and efficiency of the difference schemes. • Investigation of the nonlinear time-fractional delayed MSDDWE. • A second-order temporal L2-type difference scheme. • Demonstration of the unconditional stability of the numerical scheme. • A fast second-order scheme for the nonlinear time-fractional delayed MSDDWE. • Numerical examples to validate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper investigates a class of the time-fractional diffusion-wave equation (TFDWE), which incorporates a fractional derivative in the Caputo sense of order α + 1 where 0 < α < 1. Initially, the original problem is transformed into a new model that incorporates the fractional Riemann–Liouville integral. Subsequently, a more generalized model is considered and investigated, which includes the generalized fractional integral with memory kernel. The paper establishes the existence and uniqueness of a numerical solution for the problem. We introduce a discretization technique for approximating the generalized fractional Riemann–Liouville integral and study its fundamental properties. Based on this technique, we propose a second-order numerical scheme for approximating the generalized model. The proof of the stability of the proposed difference scheme when considered in terms of the L 2 norm is established. Furthermore, a difference scheme is devised for solving a nonlinear generalized model with time delay. The convergence and stability analysis of this particular case are established using the discrete Gronwall inequality. Numerical examples are provided to evaluate the effectiveness and reliability of the proposed methods and to support our theoretical analysis. [ABSTRACT FROM AUTHOR]