Abstract: Abstract Variable-order (VO) fractional differential equations (FDEs) with a time (t), space (x) or other variables dependent order have been successfully applied to investigate time and/or space dependent dynamics. This study aims to provide a survey of the recent relevant literature and findings in primary definitions, models, numerical methods and their applications. This review first offers an overview over the existing definitions proposed from different physical and application backgrounds, and then reviews several widely used numerical schemes in simulation. Moreover, as a powerful mathematical tool, the VO-FDE models have been remarkably acknowledged as an alternative and precise approach in effectively describing real-world phenomena. Hereby, we also make a brief summary on different physical models and typical applications. This review is expected to help the readers for the selection of appropriate definition, model and numerical method to solve specific physical and engineering problems.

Abstract: We consider global efficiency of algorithms for minimizing a sum of a convex function and a composition of a Lipschitz convex function with a smooth map. The basic algorithm we rely on is the prox-linear method, which in each iteration solves a regularized subproblem formed by linearizing the smooth map. When the subproblems are solved exactly, the method has efficiency $$\mathcal {O}(\varepsilon ^{-2})$$ , akin to gradient descent for smooth minimization. We show that when the subproblems can only be solved by first-order methods, a simple combination of smoothing, the prox-linear method, and a fast-gradient scheme yields an algorithm with complexity $$\widetilde{\mathcal {O}}(\varepsilon ^{-3})$$ . We round off the paper with an inertial prox-linear method that automatically accelerates in presence of convexity.

Abstract: In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high-dimensional PDEs. We test the method on different examples from physics, stochastic control and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.

Abstract: This paper describes a new MATLAB software package of iterative regularization methods and test problems for large-scale linear inverse problems. The software package, called IR TOOLS, serves two related purposes: we provide implementations of a range of iterative solvers, including several recently proposed methods that are not available elsewhere, and we provide a set of large-scale test problems in the form of discretizations of 2D linear inverse problems. The solvers include iterative regularization methods where the regularization is due to the semi-convergence of the iterations, Tikhonov-type formulations where the regularization is explicitly formulated in the form of a regularization term, and methods that can impose bound constraints on the computed solutions. All the iterative methods are implemented in a very flexible fashion that allows the problem’s coefficient matrix to be available as a (sparse) matrix, a function handle, or an object. The most basic call to all of the various iterative methods requires only this matrix and the right hand side vector; if the method uses any special stopping criteria, regularization parameters, etc., then default values are set automatically by the code. Moreover, through the use of an optional input structure, the user can also have full control of any of the algorithm parameters. The test problems represent realistic large-scale problems found in image reconstruction and several other applications. Numerical examples illustrate the various algorithms and test problems available in this package.

Abstract: Purpose The purpose of this paper is to examine the combined effects of thermal radiation and magnetic field of molybdenum disulfide nanofluid in a channel with changing walls. Water is considered as a Newtonian fluid and treated as a base fluid and MoS2 as nanoparticles with different shapes (spherical, cylindrical and laminar). The main structures of partial differential equations are taken in the form of continuity, momentum and energy equations. Design/methodology/approach The governing partial differential equations are converted into a set of nonlinear ordinary differential equations by applying a suitable similarity transformation and then solved numerically via a three-stage Lobatto III-A formula. Findings All obtained unknown functions are discussed in detail after plotting the numerical results against different arising physical parameters. The validations of numerical results have been taken into account with other works reported in literature and are found to be in an excellent agreement. The study reveals that the Nusselt number increases by increasing the solid volume fraction for different shapes of nanoparticles, and an increase in the values of wall expansion ratio α increases the velocity profile f′(η) from lower wall to the center of the channel and decreases afterwards. Originality/value In this paper, a numerical method was utilized to investigate the influence of molybdenum disulfide (MoS2) nanoparticles shapes on MHD flow of nanofluid in a channel. The validity of the literature review cited above ensures that the current study has never been reported before and it is quite new; therefore, in case of validity of the results, a three-stage Lobattoo III-A formula is implemented in Matlab 15 by built in routine “bvp4c,” and it is found to be in an excellent agreement with the literature published before.

Abstract: We introduce the framework of continuous--depth graph neural networks (GNNs). Graph neural ordinary differential equations (GDEs) are formalized as the counterpart to GNNs where the input-output relationship is determined by a continuum of GNN layers, blending discrete topological structures and differential equations. The proposed framework is shown to be compatible with various static and autoregressive GNN models. Results prove general effectiveness of GDEs: in static settings they offer computational advantages by incorporating numerical methods in their forward pass; in dynamic settings, on the other hand, they are shown to improve performance by exploiting the geometry of the underlying dynamics.

Abstract: Our aim in this paper is to introduce an extragradient-type method for solving variational inequality with uniformly continuous pseudomonotone operator. The strong convergence of the iterative sequence generated by our method is established in real Hilbert spaces. Our method uses computationally inexpensive Armijo-type linesearch procedure to compute the stepsize under reasonable assumptions. Finally, we give numerical implementations of our results for optimal control problems governed by ordinary differential equations.

Abstract: The main objective of this paper is to investigate an accurate numerical method for solving a biological fractional model via Atangana-Baleanu fractional derivative. We focused our attention on linear and nonlinear Fisher’s equations. We use the spectral collocation method based on the Chebyshev approximations. This method reduced the nonlinear equations to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them by using the finite difference method. This is the first time that this method is used to solve nonlinear equations in Atangana-Baleanu sense. We present the effectiveness and accuracy of the proposed method by computing the absolute error and the residual error functions. The results show that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non-local singular kernels.

Abstract: The main goal of this work is to find the solutions of linear and nonlinear fractional differential equations with the Mittag-Leffler nonsingular kernel. An accurate numerical method to search this problem has been constructed. The theoretical results are proved by utilizing two experiments.

Abstract: The present work is dedicated to an accurate modeling of violent Fluid-Structure-Interaction (FSI) problems using a coupled Lagrangian particle method combining a multi-phase δ-SPH scheme and a Total-Lagrangian-Particle (TLP) method. Advanced numerical techniques, e.g. Adaptive-Particle-Refinement (APR), have been included in the particle method for improving the local accuracy and the overall numerical efficiency. On one hand, this paper aims to demonstrate the capability of the proposed numerical method in modeling FSI flows with large density-ratios, strong fluid impacts, complex interfacial evolutions and considerable wall-boundary movements and deformations; On the other hand, the numerical results presented in this paper show the importance of considering the existence of air-phase in some complex FSI problems. The entrapped air-bubble, after the free-surface rolling and closing, plays an important role in the overall flow evolution and hence the hydrodynamic load on the structure. Although a density ratio as large as 1000 has been adopted, clear and sharp multi-phase interfaces, which undergo violent breakups and reconnections, are present in the numerical results, and more importantly, stable and smooth pressure fields are obtained. This contributes to an accurate prediction of the structural response, as validated by both the experimental data and other numerical results.

Abstract: In this paper, based on the projection strategy, we propose a derivative-free iterative method for large-scale nonlinear monotone equations with convex constraints, which can generate a sufficient descent direction at each iteration. Due to its lower storage and derivative-free information, the proposed method can be used to solve large-scale non-smooth problems. The global convergence of the proposed method is proved under the Lipschitz continuity assumption. Moreover, if the local error bound condition holds, the proposed method is shown to be linearly convergent. Preliminary numerical comparison shows that the proposed method is efficient and promising.

Abstract: Starting from the observation that artificial neural networks are uniquely suited to solving optimization problems, and most physics problems can be cast as an optimization task, we introduce a novel way of finding a numerical solution to wide classes of differential equations. We find our approach to be very flexible and stable without relying on trial solutions, and applicable to ordinary, partial and coupled differential equations. We apply our method to the calculation of tunneling profiles for cosmological phase transitions, which is a problem of relevance for baryogenesis and stochastic gravitational wave spectra. Comparing our solutions with publicly available codes which use numerical methods optimized for the calculation of tunneling profiles, we find our approach to provide at least as accurate results as these dedicated differential equation solvers, and for some parameter choices, even more accurate and reliable solutions. In particular, we compare the neural network approach with two publicly available profile solvers, CosmoTransitions and BubbleProfiler, and give explicit examples where the neural network approach finds the correct solution while dedicated solvers do not.We point out that this approach of using artificial neural networks to solve equations is viable for any problem that can be cast into the form Fðx Þ ¼ 0, and is thus applicable to various other problems in perturbative and nonperturbative quantum field theory.

Abstract: The present work considers a nonlinear system of singularly perturbed delay differential equation whose each component of the solution has multiple layers. Here, we provide an a posteriori based convergence analysis for the adaptation of these layer phenomena. We derive a parameter uniform a posteriori error estimate which will lead to a layer adaptive mesh by moving a fixed number of mesh points. It is theoretically shown that the layer adaptive solution on the a posteriori generated mesh will uniformly converge to the exact solution with optimal order accuracy where the optimality is measured with respect to the continuous problem discretization. The comparison results with the existing methods based on a priori meshes show that the proposed method on the a posteriori mesh is highly effective.

Abstract: Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential equations known as aggregation-diffusion equations. Over the past 15 years, aggregation-diffusion equations have become widespread in biological applications and have also attracted significant mathematical interest, due to their competing forces at different length scales. These competing forces lead to rich dynamics, including symmetrization, stabilization, and metastability, as well as sharp dichotomies separating well-posedness from finite time blow-up. In the present work, we review known analytical results for aggregation-diffusion equations and consider singular limits of these equations, including the slow diffusion limit, which leads to the constrained aggregation equation, and localized aggregation and vanishing diffusion limits, which lead to metastability behavior. We also review the range of numerical methods available for simulating solutions, with special attention devoted to recent advances in deterministic particle methods. We close by applying such a method—the blob method for diffusion—to showcase key properties of the dynamics of aggregation-diffusion equations and related singular limits.

Abstract: In this new work, the free motion of a coupled oscillator is investigated. First, a fully description of the system under study is formulated by considering its classical Lagrangian, and as a result, the classical Euler-Lagrange equations of motion are constructed. After this point, we extend the classical Lagrangian in fractional sense, and thus, the fractional Euler-Lagrange equations of motion are derived. In this new formulation, we consider a recently introduced fractional operator with Mittag-Leffler nonsingular kernel. We also present an efficient numerical method for solving the latter equations in a proper manner. Due to this new powerful technique, we are able to obtain remarkable physical thinks; indeed, we indicate that the complex behavior of many physical systems is realistically demonstrated via the fractional calculus modelling. Finally, we report our numerical findings to verify the theoretical analysis.

Abstract: One of the important classes of coupled systems of algebraic, differential and fractional differential equations (CSADFDEs) is fractional differential algebraic equations (FDAEs). The main difference of such systems with other class of CSADFDEs is that their singularity remains constant in an interval. However, complete classifying and analyzing of these systems relay mainly to the concept of the index which we introduce in this paper. For a system of linear differential algebraic equations (DAEs) with constant coefficients, we observe that the solvability depends on the regularity of the corresponding pencils. However, we show that in general, similar properties of DAEs do not hold for FDAEs. In this paper, we introduce some practical applications of systems of FDAEs in physics such as a simple pendulum in a Newtonian fluid and electrical circuit containing a new practical element namely fractors. We obtain the index of introduced systems and discuss the solvability of these systems. We numerically solve the FDAEs of a pendulum in a fluid with three different fractional derivatives (Liouville–Caputo’s definition, Caputo–Fabrizio’s definition and with a definition with Mittag–Leffler kernel) and compare the effect of different fractional derivatives in this modeling. Finally, we solved some existing examples in research and showed the effectiveness and efficiency of the proposed numerical method.

Abstract: In this work, we propose an algorithm for solving system of nonlinear equations. The idea is a combination of the descent Dai-Liao method by Babaie-Kafaki and Gambari (Optim. Meth. Soft. 29(3), 583–591, 2014) and the hyperplane projection method. Using the monotonicity and Lipschitz continuity assumptions, we prove that the proposed method is globally convergent. Examples of numerical experiment show that the method is promising and efficient compared to the method proposed by Sun et al. (Journal of Inequalities and Applications 236, 1–8, 2017).

Abstract: The studies of the dynamic behaviors of nonlinear models arising in ocean engineering play a significant role in our daily activities. In this study, we investigate the coupled Boussinesq equation which arises in the shallow water waves for two-layered fluid flow. The modified exp $$(-\varphi (\zeta ))$$ -expansion function method is utilized in reaching the solutions to this equation such as the topological kink-type soliton and singular soliton solutions. The interesting 2D and 3D graphics of the obtained analytical solutions in this study are presented. Via one of the reported analytical solutions, the finite forward difference method is used in obtaining the approximate numerical and exact solutions to this equation. The Fourier–Von Neumann analysis is used in checking the stability of the used numerical method with the studied model. The $$L_{2}$$ and $$L_{\infty }$$ error norms are computed. We finally present a comprehensive conclusion to this study.

Abstract: In this article, a parameter uniform numerical method is developed for a two-parameter singularly perturbed parabolic partial differential equation with discontinuous convection coefficient and source term. The presence of perturbation parameter and the discontinuity in the convection coefficient and source term lead to the boundary and interior layers in the solution. On the spatial domain, an adaptive mesh is introduced before discretizing the continuous problem. The present method observes a uniform convergence in maximum norm which is almost first-order in space and time irrespective of the relation between convection and diffusion parameters. Numerical experiment is carried out to validate the present scheme.

Abstract: This express brief proposes two new iterative approaches for solving the power flow problem in direct current networks as efficient alternatives to the classical Gauss–Seidel and Newton–Raphson methods. The first approach works with the set of nonlinear equations by rearranging them into a conventional fixed point form, generating a successive approximation methodology. The second approach is based on Taylors series expansion method by using a set of decoupling equations to linearize the problem around the desired operating point; these linearized equations are recursively solved until reach the solution of the power flow problem with minimum error. These two approaches are comparable to the classical Gauss–Seidel method and the classical Newton–Raphson method, respectively. Simulation results show that the proposed approaches have a better performance in terms of solution precision and computational requirements. All the simulations were conducted via MATLAB software by using its programming interface.

Abstract: In this article, some second-order time discrete schemes covering parameter 𝜃 combined with Galerkin finite element (FE) method are proposed and analyzed for looking for the numerical solution of nonlinear cable equation with time fractional derivative. At time tk−𝜃, some second-order 𝜃 schemes combined with weighted and shifted Grunwald difference (WSGD) approximation of fractional derivative are considered to approximate the time direction, and the Galerkin FE method is used to discretize the space direction. The stability of second-order 𝜃 schemes is derived and the second-order time convergence rate in L2-norm is proved. Finally, some numerical calculations are implemented to indicate the feasibility and effectiveness for our schemes.

Abstract: We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, in the spirit of classical numerical analysis. We demonstrate that conventional machine learning models and algorithms, such as the random feature model, the two-layer neural network model and the residual neural network model, can all be recovered (in a scaled form) as particular discretizations of different continuous formulations. We also present examples of new models, such as the flow-based random feature model, and new algorithms, such as the smoothed particle method and spectral method, that arise naturally from this continuous formulation. We discuss how the issues of generalization error and implicit regularization can be studied under this framework.

Abstract: The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in a triangulated computational domain and can overlap the time-independent background mesh in an arbitrary way. The numerical method is based on finite difference discretizations of time derivatives and a standard geometrically unfitted finite element method with an additional stabilization term in the spatial domain. The performance and analysis of the method rely on the fundamental extension result in Sobolev spaces for functions defined on bounded domains. This paper includes a complete stability and error analysis, which accounts for discretization errors resulting from finite difference and finite element approximations as well as for geometric errors coming from a possible approximate recovery of the physical domain. Several numerical examples illustrate the theory and demonstrate the practical efficiency of the method.