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This article presents an application of approximate product integration (API) to find the numerical solution of fractional order Volterra integro‐differential equation based on Caputo non‐integer derivative of order ν$$ \nu $$, where 0<ν<1$$ 0<\nu <1 $$. Also, the idea is extended to a class of fractional order Volterra integro‐differential equation with a weakly singular kernel. For this purpose, two numerical schemes are established by utilizing the concept of the API method, and L1 and L1‐2 formulae. We applied L1 and L1‐2 discretization to approximate the Caputo non‐integer derivative. At the same time, Taylor's series expansion of an unknown function is taken into consideration when approximating the Volterra part in the considered mathematical model using the API method. Combination of API method with L1 and L1‐2 formula provided the order of convergence (2−ν)$$ \left(2-\nu \right) $$ and (3−ν)$$ \left(3-\nu \right) $$ for Scheme‐I and Scheme‐II, respectively. The derived techniques reduced the proposed model to a set of algebraic equations that can be resolved using well‐known numerical algorithms. Furthermore, the unconditional stability, convergence, and numerical stability of the formulated schemes have been rigorously investigated. Finally, we conducted some numerical experiments to validate our theoretical findings and guarantee the accuracy and efficiency of the recommended schemes. The comparison between the numerical outcomes obtained by proposed schemes and existing numerical techniques has also been provided through tables and graphs. [ABSTRACT FROM AUTHOR]

Discrete inverse problems correspond to solving a system of equations in a stable way with respect to noise in the data. A typical approach to select a meaningful solution is to introduce a regularizer. While for most applications the regularizer is convex, in many cases it is neither smooth nor strongly convex. In this paper, we propose and study two new iterative regularization methods, based on a primal-dual algorithm, to regularize inverse problems efficiently. Our analysis, in the noise free case, provides convergence rates for the Lagrangian and the feasibility gap. In the noisy case, it provides stability bounds and early stopping rules with theoretical guarantees. The main novelty of our work is the exploitation of some a priori knowledge about the solution set: we show that the linear equations determined by the data can be used more than once along the iterations. We discuss various approaches to reuse linear equations that are at the same time consistent with our assumptions and flexible in the implementation. Finally, we illustrate our theoretical findings with numerical simulations for robust sparse recovery and image reconstruction. We confirm the efficiency of the proposed regularization approaches, comparing the results with state-of-the-art methods. [ABSTRACT FROM AUTHOR]

This paper employs a numerical approach to investigate the impact of liquidity shocks on European options in modeling markets. To accurately capture the behavior of European options under liquidity shocks, a coupled system of differential equations is employed, consisting of a degenerate parabolic equation and a diffusion-free equation. The primary focus is on developing and analyzing implicit-explicit methods for numerically simulating European option pricing, specifically considering the presence of liquidity shocks while ensuring the positivity of the solution. The paper also includes convergence analysis and establishes the discrete comparison principle for the developed methods. Numerical experiments are conducted using both uniform and nonuniform meshes to validate the theoretical findings, demonstrating the efficiency and accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

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]

The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes to discretize the spatially nonlocal operator, and apply the explicit difference scheme to approximate the temporal derivative to achieve a fully discrete infinity system. After that, we construct the Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs), to reduce the infinite discrete system into a finite discrete system. To do so, we first adopt the idea in Du et al. (2018, Commun. Comput. Phys. , 24 , 1049–1072) and Du et al. (2018, SIAM J. Sci. Comp. , 40 , A1430–A1445) to derive the Dirichlet-to-Dirichlet (DtD)-type mappings for one- and two-dimensional cases, respectively. We then use the discrete nonlocal Green's first identity to achieve the discrete DtN-type mappings from the DtD-type mappings. The resulting DtN-type mappings make it possible to perform the stability and convergence analysis of the reduced problem. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

A novel space-time discretization for the (linear) scalar-valued dissipative wave equation is presented. It is a structured approach, namely, the discretization space is obtained tensorizing the Virtual Element (VE) discretization in space with the Discontinuous Galerkin (DG) method in time. As such, it combines the advantages of both the VE and the DG methods. The proposed scheme is implicit and it is proved to be unconditionally stable and accurate in space and time. [ABSTRACT FROM AUTHOR]

This paper studies two efficient numerical methods for the generalized tempered integrodifferential equation with respect to another function. The proposed methods approximate the unknown solution through two phases. First, the backward Euler (BE) method and first-order interpolation quadrature rule are adopted to approximate the temporal derivative and generalized tempered integral term to construct a semi-discrete BE scheme. Second, the backward differentiation formula (BDF) and second-order interpolation quadrature rule are adopted to establish a semi-discrete second-order BDF (BDF2) scheme. Additionally, the stability and convergence of two semi-discrete methods are deduced in detail. To further demonstrate the effectiveness of proposed techniques, fully discrete BE and BDF2 finite difference schemes are formulated. Subsequently, the theoretical results of two fully discrete difference schemes are presented. Finally, the numerical results demonstrate the accuracy and competitiveness of the theoretical analysis. [ABSTRACT FROM AUTHOR]

In last decades, there have been drastic environmental transformations and mutations happening all around the world. Due to the continuous mass transfer process, for example, CO2{{\rm{CO}}}_{2} mass transfer, which in this case, takes the form of greenhouse gas emissions, unusual and extreme kinds of phenomena have been occurring here and there, disturbing our ecosystems and causing damage and chaos on their paths. Reducing or stopping these gas emissions has become one of the major topics in our planet. We investigate the solvability of a mathematical model describing the mass transport process in nature and where additional perturbations parameters have been considered. Besides addressing the stability of the model, its convergence analysis is also given with the use of Crank–Nicholson numerical method, in order to assess its efficiency and perform some numerical simulations. The results obtained show that the model’s dynamic is characterized by many grouping (accumulation) zones, where mass (of CO2{{\rm{CO}}}_{2}, for instance) accumulates in an increasing way. This result is important in controlling how CO2{{\rm{CO}}}_{2} can be stored in this growingly perturbed environment that surrounds us.

In this paper, we present a three-step iterative approach for solving the fractional Brusselator model. The fractional derivatives are described in the Caputo, Caputo-Fabrizio, and Atangana-Baleanu senses. The fractional differential equation is converted into an analogous integral equation and then obtained using the three-step iterative approach. The method consists of first implementing the trapezoidal rule and then performing the iteration approach on the resulting equation. In the fractional Brusselator model, we examine the stability of equilibrium points. For each fractional order operator, simulations were performed and analyzed in order to verify the method's high feasibility and efficiency, as well as to confirm the theoretical aspect. The obtained solution simulations suggest that the methodology utilised is robust, time-efficient, and appropriate for dealing with the system of fractional equations of small fractional order from a numerical standpoint. [ABSTRACT FROM AUTHOR]

The construction of efficient numerical schemes with uniform convergence order for time-fractional diffusion equations (TFDEs) is an important research problem. We are committed to study an efficient uniform accuracy scheme for TFDEs. Firstly, we use the piecewise quadratic interpolation to construct an efficient uniform accuracy scheme for the fractional derivative of time. And the local truncation error of the efficient scheme is also given. Secondly, the full discrete numerical scheme for TFDEs is given by combing the spatial center second order scheme and the above efficient time scheme. Thirdly, the efficient scheme's stability and error estimates are strictly theoretical analysis to obtain that the unconditionally stable scheme is $ 3-\beta $ convergence order with uniform accuracy in time. Finally, some numerical examples are applied to show that the proposed scheme is an efficient unconditionally stable scheme.

We formulate a hybridizable discontinuous Galerkin method for parabolic equations with non-linear tensor-valued coefficients and jump conditions (Henry's law). The analysis of the proposed scheme indicates the optimal convergence order for mildly non-linear problems. The same order is also obtained in our numerical studies for simplified settings. A series of numerical experiments investigate the effect of choosing different order approximation spaces for various unknowns. [ABSTRACT FROM AUTHOR]

The construction of efficient numerical schemes with uniform convergence order for time-fractional diffusion equations (TFDEs) is an important research problem. We are committed to study an efficient uniform accuracy scheme for TFDEs. Firstly, we use the piecewise quadratic interpolation to construct an efficient uniform accuracy scheme for the fractional derivative of time. And the local truncation error of the efficient scheme is also given. Secondly, the full discrete numerical scheme for TFDEs is given by combing the spatial center second order scheme and the above efficient time scheme. Thirdly, the efficient scheme's stability and error estimates are strictly theoretical analysis to obtain that the unconditionally stable scheme is 3 − β convergence order with uniform accuracy in time. Finally, some numerical examples are applied to show that the proposed scheme is an efficient unconditionally stable scheme. The construction of efficient numerical schemes with uniform convergence order for time-fractional diffusion equations (TFDEs) is an important research problem. We are committed to study an efficient uniform accuracy scheme for TFDEs. Firstly, we use the piecewise quadratic interpolation to construct an efficient uniform accuracy scheme for the fractional derivative of time. And the local truncation error of the efficient scheme is also given. Secondly, the full discrete numerical scheme for TFDEs is given by combing the spatial center second order scheme and the above efficient time scheme. Thirdly, the efficient scheme's stability and error estimates are strictly theoretical analysis to obtain that the unconditionally stable scheme is convergence order with uniform accuracy in time. Finally, some numerical examples are applied to show that the proposed scheme is an efficient unconditionally stable scheme. [ABSTRACT FROM AUTHOR]

This paper presents a high order time discretization method by combining the time splitting scheme with semi-implicit spectral deferred correction method for the space-fractional Swift-Hohenberg equation (SFSH). Based on the operator splitting method, the original problem is split into linear and nonlinear subproblems, respectively. The Fourier spectral method is adopted for the linear part, and a first-order accurate finite difference scheme in time together with the Fourier spectral method in space is used for the nonlinear part. The stability and convergence of the obtained numerical scheme are analyzed theoretically. Moreover, the spectral deferred correction (SDC) method is then employed to improve the temporal accuracy. Various two and three dimensional numerical experiments are performed to validate the theoretical results and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This study intends to investigate the impacts of scheme type, time step, and error threshold on the stability of numerical simulation in the groundwater modeling. Hence, a two-dimensional finite element (FE) was implemented to simulate groundwater flow in a synthetic test case and a real-world study (Birjand aquifer). To verify the proposed model in both cases, the obtained results were compared with analytical solutions and observed values. The stability of numerical results was analyzed through different schemes and time-step sizes. Besides, the effect of the error threshold was examined by considering different threshold values. The results confirmed that the FE model has a good capacity to simulate groundwater fluctuations even for the real problem with more complexities. Examination of implicit outputs indicated that groundwater simulations based on this scheme have good accuracy, stability, and proper convergence in all time intervals. However, in the explicit and Crank–Nicolson schemes the time interval should be less than or equal to 0.001 and 0.1 day, respectively. Also, results reveal that for making stability in all schemes the value of the error threshold should not be more than 0.0001 m. Moreover, it derived that the boundary conditions of the aquifer influence the stability of numerical outputs. Finally, it was comprehended that as time interval and error threshold increases, the oscillation rate propagated. [ABSTRACT FROM AUTHOR]

In this work, we present a high-order Discontinuous Galerkin time integration scheme for second-order (in time) differential systems that typically arise from the space discretization of the elastodynamics equation. By rewriting the original equation as a system of first-order differential equations we introduce the method and show that the resulting discrete formulation is well-posed, stable, and retains a super-optimal rate of convergence with respect to the discretization parameters, namely the time step and the polynomial approximation degree. A set of two- and three-dimensional numerical experiments confirm the theoretical bounds. Finally, the method is applied to real geophysical simulations. [ABSTRACT FROM AUTHOR]

The piecewise quadratic polynomial collocation is used to approximate the nonlocal model, which generally leads to a nonsymmetric indefinite system (Chen et al. (2021) [5]). In this case, the discrete maximum principle is not satisfied, which might be trickier for the stability analysis of the high-order numerical schemes (D'Elia et al. (2020) [10] ; Leng et al. (2021) [26]). Here, we present a modified (shifted-symmetric) piecewise quadratic polynomial collocation for solving the linear nonlocal diffusion model, which leads to a symmetric positive definite system and satisfies the discrete maximum principle. Using Faulhaber's formula and Riemann zeta function, the perturbation error for symmetric positive definite system and nonsymmetric indefinite system are given. Then rigorous convergence analysis for the nonlocal models are provided under the general horizon parameter δ = O (h β) , with β ≥ 0. More concretely, the global error is O (h min ⁡ { 2 , 1 + β } ) if δ is not set as a grid point, while it recovers O (h max ⁡ { 2 , 4 − 2 β } ) when δ is set as a grid point. We also prove that the shifted-symmetric scheme is asymptotically compatible, which has the global error O (h min ⁡ { 2 , 2 β } ) as δ , h → 0. The numerical experiments (including two-dimensional case) are performed to verify the convergence. [ABSTRACT FROM AUTHOR]

The numerical approximation of incompressible fluid–structure interaction problems with Lagrange multiplier is generally based on strongly coupled schemes. This delivers unconditional stability, but at the expense of solving a computationally demanding coupled system at each time step. For the case of the coupling with immersed thin-walled solids, we introduce a class of semi-implicit coupling schemes that avoids strongly coupling without compromising stability and accuracy. A priori energy and error estimates are derived. The theoretical results are illustrated through numerical experiments in an academic benchmark. [ABSTRACT FROM AUTHOR]

This paper is concerned with the convergence analysis of the PH heterogeneous asynchronous time integrators algorithm, proposed by Prakash and Hjelmstad [Int. J. Numer. Methods Eng. 61 (2004) 2183–2204], and devoted to transient dynamic problems for structural analysis. According to PH method, the time discretization is performed using the well-known Newmark schemes, where the time step ratio, i.e., the ratio of the macro time step to the micro time step, of two subdomains separated by an interface is a positive integer. The analysis is restricted to linear problems with two subdomains for simplification. We show that L∞-uniform convergence of the approximated solutions is achieved taking into account damping terms and suitable regularities on load terms. We shall also give some error estimates of the method. [ABSTRACT FROM AUTHOR]

In this work, we design and analyze a discrete model to approximate the solutions of a parabolic partial differential equation in multiple dimensions. The mathematical model considers a nonlinear reaction term and a space-dependent diffusion coefficient. The system has a Gibbs' free energy, we establish rigorously that it is non-negative under suitable conditions, and that it is dissipated with respect to time. The discrete model proposed in this work has also a discrete form of the Gibbs' free energy. Using a fixed-point theorem, we prove the existence of solutions for the numerical model under suitable assumptions on the regularity of the component functions. We prove that the scheme preserves the positivity and the dissipation of the discrete Gibbs' free energy. We establish theoretically that the discrete model is a second-order consistent scheme. We prove the stability of the method along with its quadratic convergence. Some simulations illustrating the capability of the scheme to preserve the dissipation of Gibb's energy are presented. [ABSTRACT FROM AUTHOR]

In this paper, we study a high-order unconditionally stable numerical method to approximate the class of multi-term time-fractional diffusion equations. This type of problem appears in the modelling of transport of certain quantities such as heat, mass, energy, solutes in ground water and soils. The multi-term time-fractional derivative is approximated by using the Crank–Nicolson method for the Caputo's time derivative. The space derivative is approximated by using the collocation method based on quintic B-spline basis functions. We have established the stability and convergence analysis of the proposed numerical scheme thoroughly, and it is shown that the order of convergence in space variable is almost four and in the time variable is O (Δ t 2 − max { γ , γ i }). To prove the accuracy and efficiency of the developed method, we consider four numerical examples and perform the numerical simulation. The developed algorithm works well andvalidate the theoretical results. The developed method is fourth-order convergent in the space variable, which is almost two orders of magnitude higher than the other spline collocation methods. [ABSTRACT FROM AUTHOR]

The viscous Cahn–Hilliard equation (VCH) is a parameter-dependent model encapsulated Cahn–Hilliard (CH) model and constrained Allen–Cahn (CAC) model. In this paper, we propose a fast explicit time-splitting spectral scheme for the VCH with nonlocal diffusion operator. The numerical scheme is constructed based on the second-order symmetric time-splitting method, which results in a splitting of the original problem into nonlocal linear part and nonlinear part, respectively. The nonlocal linear subproblem can be solved exactly by applying the Fourier spectral discretization in space and thus has no stability restriction on the time-step size. The nonlinear subproblem is solved via a second-order strong stability preserving Runge–Kutta method. The new scheme is time symmetric, fully explicit and conserves the discrete mass exactly. Theoretical analysis shows that the stability of the scheme depends on both the viscosity coefficient and the range of nonlocal interactions. In addition, a rigorously convergence analysis of the scheme is established, and the convergence rate is proved to be second-order in time and spectral-order in space, respectively. Four numerical experiments are performed to demonstrate the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical solutions of the multi-dimensional tempered fractional integrodifferential equation. First, the second-order backward differentiation formula (BDF2) and the second-order convolution quadrature rule are utilized for the temporal discretization, and the finite difference method (FDM) and alternating direction implicit (ADI) technique are used for the spatial discretization, in which the ADI algorithm is mainly employed to reduce the computational cost of high-dimensional non-local problems. Then, the fully discrete BDF2 ADI difference scheme for multi-dimensional tempered non-local problems can be obtained. After that, the stability and convergence of the BDF2 ADI difference scheme in multi-dimensional cases are proved by means of the energy method. In the numerical experiments, some examples validate the theoretical analysis. [ABSTRACT FROM AUTHOR]

For the first time, a new dissipation-preserving scheme is proposed and analyzed to solve a Caputo–Riesz time-space-fractional multidimensional nonlinear wave equation with generalized potential. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hyper cube. We introduce an energy-type functional and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. A discrete form of the continuous energy is proposed and the discrete operator is shown to satisfy a conservation law, in agreement with its continuous counterpart. We employ a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carry out a number of numerical simulations to verify the validity of our theoretical results. [ABSTRACT FROM AUTHOR]

In this work, we introduce and theoretically analyze various computational techniques to approximate the solutions of solve a fractional extension of a double condensate system. More precisely, the continuous model extends the well-known Gross–Pitaevskii equation to the fractional scenario, and considering two interacting condensates. The mathematical system considers two complex-valued regimes with coupling, and a mass and energy functions are associated to this model. Both are constant in time. Here, various discretizations are analyzed to solve this system. Some of them are able to preserve the mass and the energy, some are not. We discuss the existence of solutions, the consistency of the models, the stability and the convergence. Finally, from the computational point of view, some algorithms are simpler to code than others. In fact, those for which the mass and the energy are conserved are more difficult to implement. We discuss here pros and cons. [ABSTRACT FROM AUTHOR]

In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the accuracy of the proposed method, test examples with different scale and weight functions are considered, and the obtained numerical results are compared with the existing methods in the literature. It is observed that the proposed approach works very well with the generalized fractional derivatives (GFDs), as the presence of scale and weight functions in a generalized fractional derivative (GFD) cause difficulty for its discretization and further analysis. [ABSTRACT FROM AUTHOR]

This paper presents a formally second‐order backward differentiation formula (BDF2) Sinc‐collocation method for solving the Volterra integro‐differential equation with a weakly singular kernel. In the time direction, the time derivative is discretized via the BDF2 and the second‐order convolution quadrature rule is used to approximate the Riemann–Liouville fractional integral term. Then a fully discrete scheme is established via the Sinc approximation based on the double exponential transformation in space. The convergence and stability analysis are derived by the energy method. Numerical examples are provided to illustrate the effectiveness of proposed method and it can be found that our scheme is super‐exponentially convergent in space and order 1 + α convergent in time with 0 < α < 1, respectively. Meanwhile, the numerical results based on the single exponential transformation are compared with the proposed method to illustrate the high accuracy of our method. [ABSTRACT FROM AUTHOR]

The theoretical convergence, exploratory, and stability analysis of any heuristic algorithm is an important aspect to make it more efficient and reliable to the research community. The artificial electric field algorithm (AEFA) (Yadav et al. ,Swarm Evol Comput 48:93–108, 54) is a new optimization algorithm in the class of heuristics optimization algorithms. It is inspired by the Coulombs law of electrostatic force. In this article, a study of the convergence and stability analysis of the particle trajectory of the AEFA algorithm is established. A theoretical study of first and second-order stability of the AEFA algorithm is established which is represented by a stochastic recurrence relation. The convergence of expectation and variance of particle positions is proved and discussed in detail. Further, the boundary conditions for the convergence of expectation and variance of particle positions are established along with their first and second-order stability. These boundary conditions suggest better parameter values for the AEFA algorithm. The coefficient boundaries for particle positions associated with different kinds of oscillation behavior e.g. mono-oscillation, harmonic, and zigzagging are discussed in both the time and frequency domain. Also, the theoretical findings are validated by solving 23 benchmark optimization problems and some real-world optimization problems. [ABSTRACT FROM AUTHOR]

The paper aims to develop the stable numerical schemes for generalized time-fractional diffusion equations (GTFDEs) with smooth and non-smooth solutions on the non-uniform grid. In time, the generalized Caputo derivative is discretized by a difference scheme of order (2 − α) on a non-uniform grid where 0 < α < 1. Choosing the non-uniform meshes in the case of the smooth and non-smooth solution is also essential, so we graded the mesh in both cases separately. Stability and convergence for smooth as well as non-smooth solutions are obtained in L 2 -norm and L ∞ -norm respectively. Several numerical results are presented to show how the grading of meshes is essential. Also, numerical results validate the efficiency and effectiveness of proposed schemes and show how a non-uniform grid produces better results. [ABSTRACT FROM AUTHOR]

For the first time, a new dissipation-preserving scheme is proposed and analyzed to solve a Caputo–Riesz time-space-fractional multidimensional nonlinear wave equation with generalized potential. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hyper cube. We introduce an energy-type functional and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. A discrete form of the continuous energy is proposed and the discrete operator is shown to satisfy a conservation law, in agreement with its continuous counterpart. We employ a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carry out a number of numerical simulations to verify the validity of our theoretical results.

The present work introduces a numerical scheme that preserves the dissipation of energy of the Higgs boson equation in the de Sitter space-time. More precisely, the model considered in this work is a mathematical generalization of Higgs' model which includes a general time-dependent diffusion coefficient and a generalized potential. The mathematical system is dissipative, and we propose an implicit discrete method which approximates consistently the radially symmetric solutions of the continuous system. At the same time, a discrete energy functional is presented, and we prove that, as its continuous counterpart, the numerical technique dissipates the energy of the discrete system. The properties of consistency, stability and convergence of the numerical model are proved rigorously. To confirm the theoretical results, we approximate some radially symmetric solutions of the classical Higgs boson equation in the de Sitter space-time. In particular, the numerical results confirm the stability and the formation of bubble-like solutions. [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]

In this paper, an approximate method combining the finite difference and collocation methods is studied to solve the generalized fractional diffusion equation (GFDE). The convergence and stability analysis of the presented method are also established in detail. To ensure the effectiveness and the accuracy of the proposed method, test examples with different scale and weight functions are considered, and the obtained numerical results are compared with the existing methods in the literature. It is observed that the proposed approach works very well with the generalized fractional derivatives (GFDs), as the presence of scale and weight functions in a generalized fractional derivative (GFD) cause difficulty for its discretization and further analysis.

In this paper, a weighted and shifted Grünwald-Letnikov difference (WSGD) Legendre spectral method is proposed to solve the two-dimensional nonlinear time fractional mobile/immobile advection-dispersion equation. We introduce the correction method to deal with the singularity in time, and the stability and convergence analysis are proven. In the numerical implementation, a fast method is applied based on a globally uniform approximation of the trapezoidal rule for the integral on the real line to decrease the memory requirement and computational cost. The memory requirement and computational cost are O(Q) and O(QK), respectively, where K is the number of the final time step and Q is the number of quadrature points used in the trapezoidal rule. Some numerical experiments are given to confirm our theoretical analysis and the effectiveness of the presented methods. [ABSTRACT FROM AUTHOR]

We propose and analyze an accelerated iterative dual diagonal descent algorithm for the solution of linear inverse problems with strongly convex regularization and general data-fit functions. We develop an inertial approach of which we analyze both convergence and stability properties. Using tools from inexact proximal calculus, we prove early stopping results with optimal convergence rates for additive data terms and further consider more general cases, such as the Kullback--Leibler divergence, for which different type of proximal point approximations hold. [ABSTRACT FROM AUTHOR]

In this article, we develop a Crank–Nicolson alternating direction implicit finite volume method for time‐dependent Riesz space‐fractional diffusion equation in two space dimensions. Norm‐based stability and convergence analysis are given to show that the developed method is unconditionally stable and of second‐order accuracy both in space and time. Furthermore, we develop a lossless matrix‐free fast conjugate gradient method for the implementation of the numerical scheme, which only has O(N) memory requirement and O(NlogN) computational complexity per iteration with N being the total number of spatial unknowns. Several numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed scheme for large‐scale modeling and simulations. [ABSTRACT FROM AUTHOR]

This paper is dedicated to numerically solving the Sobolev equations that have several applications in physics and mechanical engineering. First, the temporal derivative is discretized by the Crank-Nicolson finite difference technique to obtain a semi-discrete scheme in the temporal direction. Afterward, the stability and convergence analysis of the time semi-discrete scheme are proven by applying the energy method. It also implies that the convergence order in the temporal direction is O (d t 2). Second, a fully discrete formula has been acquired by discretizing the spatial derivatives via Legendre spectral element method (LSEM). This method applies the Lagrange polynomial based on the Gauss-Legendre-Lobatto (GLL) points. Moreover, an error estimation is given for the obtained fully discrete scheme. Eventually, the two-dimensional Sobolev equations are solved by using the proposed procedure. The accuracy and efficiency of the mentioned procedure are demonstrated by several numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the Crank–Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such discretization are that the proposed numerical method is unconditionally stable and its global truncation error is of O τ 2 + h 4 - γ with 0 < γ < 1 , where τ and h are the discretization sizes in the temporal and spatial dimensions respectively. Also we develop the conjugate gradient squared method to solving the resulting discretized nonsymmetric and indefinite systems arising from time-dependent nonlocal problems including two-dimensional cases. By using additive and multiplicative Cauchy kernels in nonlocal problems, structured coefficient matrix-vector multiplication can be performed efficiently in the conjugate gradient squared iteration. Numerical examples are given to illustrate our theoretical results and demonstrate that the computational cost of the proposed method is of O (M log M) operations where M is the number of collocation points. [ABSTRACT FROM AUTHOR]

In the paper, a space fractional Benjamin-Bona-Mahony (BBM) equation is proposed. For the direct problem, we develop the Chebyshev-Legendre spectral scheme for the proposed equation. The given approach is based on Legendre Galerkin formulation while the Chebyshev-Gauss-Lobatto (CGL) nodes are used in the computation. By using the presented method, the computational complexity is reduced and both accuracy and efficiency are achieved compared with the Legendre spectral method. Stability and convergence analysis of the numerical method are proven. For the inverse problem, the Bayesian method is developed to estimate some relevant parameters based on the spectral format of the direct problem. The convergence analysis on the Kullback-Leibler distance between the true posterior distribution and the approximation is derived. Some numerical experiments are included to demonstrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this work, we introduce and theoretically analyze a relatively simple numerical algorithm to solve a double-fractional condensate model. The mathematical system is a generalization of the famous Gross–Pitaevskii equation, which is a model consisting of two nonlinear complex-valued diffusive differential equations. The continuous model studied in this manuscript is a multidimensional system that includes Riesz-type spatial fractional derivatives. We prove here the relevant features of the numerical algorithm, and illustrative simulations will be shown to verify the quadratic order of convergence in both the space and time variables.

This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally, we implemented the scheme computationally to confirm the validity of the mass and energy conservation properties, obtaining satisfactory results.

A collocation method based on B-spline is developed for solving Convection-Reaction-Diffusion equation subjected to Dirichlet's boundary conditions. The method comprises an explicit finite difference associated with extended cubic B-spline collocation method. We analyze the convergence and stability of the presented method. The proposed method is applied to various test examples and results are reported in Tables. The obtained numerical results have been compared with the results of the existing methods. Numerical results verify the efficiency and applicability of the scheme. [ABSTRACT FROM AUTHOR]

This paper presents a numerical solution of time-fractional nonlinear advection–diffusion equations (TFADEs) based on the local discontinuous Galerkin method. The trapezoidal quadrature scheme (TQS) for the fractional order part of TFADEs is investigated. In TQS, the fractional derivative is replaced by the Volterra integral equation which is computed by the trapezoidal quadrature formula. Then the local discontinuous Galerkin method has been applied for space-discretization in this scheme. Additionally, the stability and convergence analysis of the proposed method has been discussed. Finally some test problems have been investigated to confirm the validity and convergence of the proposed method. [ABSTRACT FROM AUTHOR]

We consider the time‐dependent magnetic induction model where the sought magnetic field interacts with a prescribed velocity field. This coupling results in an additional force term and time dependence in Maxwell's equation. We propose two different magnetic diffusivity stabilized continuous nodal‐based finite element methods for this problem. The first formulation simply adds artificial magnetic diffusivity to the partial differential equation, whereas the second one uses a local projected magnetic diffusivity as stabilization. We describe those methods and analyze them semi‐discretized in space to get bounds on stabilization parameters where we distinguish equal‐order elements and Taylor‐Hood elements. Different numerical experiments are performed to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

The main aim of the current paper is to solve the multi-dimensional generalized modified anomalous sub-diffusion equation by using a new spectral element method. At first, the time variable has been discretized by a finite difference scheme with second-order accuracy. The stability and convergence of the time-discrete scheme have been investigated. We show that the time-discrete scheme is unconditionally stable and the convergence order is O (τ 2) in the temporal direction. Secondly, the Galerkin spectral element method has been combined with alternating direction implicit idea to discrete the space variable. The unconditional stability and convergence of the full-discrete scheme have been proved. By developing the proposed scheme, we need to calculate one-dimensional integrals for two-dimensional problems and two-dimensional integrals for three-dimensional problems. Thus, the used CPU time for the presented numerical procedure is lower than the two- and three-dimensional Galerkin spectral element methods. Also, the proposed method is suitable for computational domains obtained from the tensor product. Finally, two examples are analyzed to check the theoretical results. [ABSTRACT FROM AUTHOR]

In the current manuscript, we consider a generalized fractional reaction–diffusion equation. The considered model is based on the time fractional derivative. The developed scheme is based on two procedures. At first, we obtain a semi-discrete scheme for the temporal direction. The time fractional derivative is discretized by using a difference scheme with second-order accuracy. Thus, the final time-discrete formula has second-order accuracy in the temporal direction. Then, we obtain a fully discrete scheme by applying the mixed finite element method (MFEM). For the constructed numerical technique, we prove the unconditional stability and also obtain an error bound for the full-discrete scheme by using the energy method. We employ some test problems to show the accuracy of the proposed technique. [ABSTRACT FROM AUTHOR]