Approximation Of Differential Equations Research Articles

Numerical solutions of stochastic functional differential equations with impulsive perturbations and Markovian switching

This paper is focused on analysis of Euler–Maruyama-type approximations for stochastic functional differential equations with impulsive perturbations and Markovian switching. The motivation stems from a wide range of applications. The paper establishes mean square convergence of the approximations under a local Lipschitz condition and a linear growth condition. In addition, this work ascertains the rates of convergence of the numerical solutions under the global Lipschitz condition. Finally, the performance of the algorithm is demonstrated through a numerical example.

Investigation of Heat and Mass Transfer in the Process of Fuel Preparation from Biomass Particles with High Moisture

Currently, in a number of countries an urgent task for development of fuel and energy complexes is to increase the share of generation by involving solid fuels in circulation. Among such projects, those that allow the disposal of waste from various industries are particularly significant. Expired food products in this context are represented as a renewable local energy resource. However, such products require serious activities to prepare them for incineration or other type of high-temperature processing in order to obtain energy. The purpose of the present work is to improve methods of preparing fuel from recycled carrot fruits (unsuitable for use in the food sector). During the fuel preparation of carrots, the drying stage is limiting for the rational organization of its processing in boilers. In addition, the drying stage is extremely energy-consuming, so reliable prediction of its kinetics largely determines the efficiency of the entire technological process. In the course of the study, the following tasks were solved: a numerical method was developed for describing the processes of internal and external heat and mass transfer problems using an explicit difference approximation of differential equations of heat and mass transfer; parametric identification of the proposed one-dimensional mathematical model was performed using empirical dependencies known from literature sources; empirical verification of the proposed mathematical model was carried out by comparing the calculated forecasts obtained with the results of their own field experiments. The fact that the proposed mathematical model and the results of the full-scale experiment are independent, while the calculated forecasts and experimental data are in good agreement, makes us possible to consider the proposed calculation method as a reliable scientific basis for a computer method for calculating of heat and mass transfer processes when organizing the preparation of fuel from carrot fruits.

Approximations of Time-Dependent Nonlinear Partial Differential Equations using Galerkin Optimal Auxiliary Function Method

The purpose of this research work is to employ the Optimal Auxiliary Function Method (OAFM) for obtaining numerical approximations of time-dependent nonlinear partial differential equations (PDEs) that arise in many disciplines of science and engineering. The initial and first approximations of parabolic nonlinear PDEs associated with initial conditions have been generated by utilizing this method. Then the Galerkin method is applied to estimate the coefficients that remain unknown. Finally, the values of the coefficients generated by the Galerkin method have been inserted into the first approximation. In each example, all numerical computations and corresponding absolute errors are provided in schematic representations. The rate of convergence attained by the proposed method is depicted in tabular form.
 GANIT J. Bangladesh Math. Soc. 43.1 (2023) 001- 016

A Galerkin finite element method for the space Hadamard fractional partial differential equation

In this paper, we study the Galerkin finite element approximation for the space Hadamard fractional partial differential equation. We first introduce a modified Fourier transform to analyse the Hadamard fractional calculus, construct the fractional derivative spaces and fractional Sobolev space. Furthermore, we investigate the existence and uniqueness of the weak solution in the fractional Sobolev space. Then using a newly defined log-Lagrangian polynomial as shape function, we discuss the convergence analysis of the semi-discrete scheme. Together with the Crank–Nicolson scheme in time, we present a fully discrete scheme, analyse the stability and convergence. Finally a numerical example is displayed which support the theoretical analysis.

Rate of Convergence in the Smoluchowski-Kramers Approximation for Mean-field Stochastic Differential Equations

In this paper we study a second-order mean-field stochastic differential systems describing the movement of a particle under the influence of a time-dependent force, a friction, a mean-field interaction and a space and time-dependent stochastic noise. Using techniques from Malliavin calculus, we establish explicit rates of convergence in the zero-mass limit (Smoluchowski-Kramers approximation) in the Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document}-distances and in the total variation distance for the position process, the velocity process and a re-scaled velocity process to their corresponding limiting processes.

A novel method to approximate fractional differential equations based on the theory of functional connections

A novel method to approximate fractional differential equations based on the theory of functional connections

ВЫБОР ОПТИМАЛЬНЫХ ПАРАМЕТРОВ ПРОСТРАНСТВЕННО-ВРЕМЕННОЙ СЕТКИ ПРИ МАТЕМАТИЧЕСКОМ МОДЕЛИРОВАНИИ НАГРЕВА ЗАГОТОВОК В ПРОМЫШЛЕННЫХ ПЕЧАХ

The efficiency of two numerical methods for solving multidimensional problems of the theory of thermal conductivity ‒ the fractional steps method and the Liebman method - is evaluated. The efficiency of these methods is compared for the operating conditions of industrial furnaces by the example of calculating the symmetrical heating of a two-dimensional cylinder and a three-dimensional plate made of materials with different thermophysical properties (corundum, ceramic brick and carbon steel). A two-dimensional axisymmetric temperature field for a cylinder and a three-dimensional temperature field for a workpiece in the form of a parallelepiped were found by the grid method under boundary conditions of the II and III genera. The difference approximation of differential equations and boundary conditions is performed by the control volume method according to an implicit finite difference scheme. The developed algorithms for solving multidimensional problems of internal heat transfer by fractional steps and the Liebman method are implemented in the form of computer programs in the Object Pascal programming environment. When comparing the effectiveness of solving multidimensional problems with these methods, the criterion of the effectiveness of difference schemes (CERS) proposed by V.V. Bukhmirov and T.E. Sozinova was used as an optimization criterion. Nomograms are constructed to select the optimal parameters of the space-time grid and the best numerical method for solving multidimensional problems for a specific process of heating (cooling) a solid body

Strong convergence of a fractional exponential integrator scheme for finite element discretization of time-fractional SPDE driven by fractional and standard Brownian motions

The aim of this work is to provide the first strong convergence result of a numerical approximation of a general time-fractional second order stochastic partial differential equation involving a Caputo derivative in time of order α∈(12,1) and driven simultaneously by a multiplicative standard Brownian motion and additive fBm with Hurst parameter H∈(12,1), more realistic to model the random effects on transport of particles in medium with thermal memory. We prove the existence and uniqueness results, and perform the spatial discretization using the standard finite element and the temporal discretization based on a generalized exponential time differencing method (GETD). We provide the temporal and spatial convergence proofs for our fully discrete scheme and the result shows that the convergence orders depend on the regularity of the initial data, the power of the fractional derivative, and the Hurst parameter H. Numerical results are provided to illustrate our theoretical results.

ESSAYS ON STRONG AND WEAK APPROXIMATIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS

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Error estimates and physics informed augmentation of neural networks for thermally coupled incompressible Navier Stokes equations.

Physics Informed Neural Networks (PINNs) are shown to be a promising method for the approximation of partial differential equations (PDEs). PINNs approximate the PDE solution by minimizing physics-based loss functions over a given domain. Despite substantial progress in the application of PINNs to a range of problem classes, investigation of error estimation and convergence properties of PINNs, which is important for establishing the rationale behind their good empirical performance, has been lacking. This paper presents convergence analysis and error estimates of PINNs for a multi-physics problem of thermally coupled incompressible Navier-Stokes equations. Through a model problem of Beltrami flow it is shown that a small training error implies a small generalization error. Posteriori convergence rates of total error with respect to the training residual and collocation points are presented. This is of practical significance in determining appropriate number of training parameters and training residual thresholds to get good PINNs prediction of thermally coupled steady state laminar flows. These convergence rates are then generalized to different spatial geometries as well as to different flow parameters that lie in the laminar regime. A pressure stabilization term in the form of pressure Poisson equation is added to the PDE residuals for PINNs. This physics informed augmentation is shown to improve accuracy of the pressure field by an order of magnitude as compared to the case without augmentation. Results from PINNs are compared to the ones obtained from stabilized finite element method and good properties of PINNs are highlighted.

On the Approximation of Fractional-Order Differential Equations Using Laplace Transform and Weeks Method

Differential equations of fractional order arising in engineering and other sciences describe nature sufficiently in terms of symmetry properties. In this article, a numerical method based on Laplace transform and numerical inverse Laplace transform for the numerical modeling of differential equations of fractional order is developed. The analytic inversion can be very difficult for complex forms of the transform function. Therefore, numerical methods are used for the inversion of the Laplace transform. In general, the numerical inverse Laplace transform is an ill-posed problem. This difficulty has led to various numerical methods for the inversion of the Laplace transform. In this work, the Weeks method is utilized for the numerical inversion of the Laplace transform. In our proposed numerical method, first, the fractional-order differential equation is converted to an algebraic equation using Laplace transform. Then, the transformed equation is solved in Laplace space using algebraic techniques. Finally, the Weeks method is utilized for the inversion of the Laplace transform. Weeks method is one of the most efficient numerical methods for the computation of the inverse Laplace transform. We have considered five test problems for validation of the proposed numerical method. Based on the comparison between analytical results and the Weeks method results, the reliability and effectiveness of the Weeks method for fractional-order differential equations was confirmed.

High resolution numerical algorithm of order (2, 4, 4) for 2D quasi-linear hyperbolic equations based on half-step discretization on an unequal grid

This study suggests a novel three-level implicit half-step scheme of order two and four in time and space direction respectively, utilizing an unequal grid for the approximation of 2D quasi-linear hyperbolic partial differential equations on an unequal grid. The stability analysis of the model Telegraphic equation for unequal grid has been discussed and it has been demonstrated that the presented method is unconditionally stable for the Telegraphic equation. The alternating direction implicit (ADI) techniques are discussed for linear difference equations on a dissimilar domain. The developed method investigated several well-known physical problems on an unequal mesh to demonstrate its accuracy and efficiency.

Weak Approximations of Stochastic Partial Differential Equations with Fractional Noise

Weak Approximations of Stochastic Partial Differential Equations with Fractional Noise

A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations

Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named computational problems. There are a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of them prove convergence without convergence rates and some of these mathematical results even rigorously establish convergence rates but there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the prescribed approximation accuracy ε > 0 \varepsilon > 0 and the PDE dimension d ∈ N d \in \mathbb {N} . We thereby prove, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs.

Sharp lower error bounds for strong approximation of SDEs with discontinuous drift coefficient by coupling of noise

In the past decade, an intensive study of strong approximation of stochastic differential equations (SDEs) with a drift coefficient that has discontinuities in space has begun. In the majority of these results it is assumed that the drift coefficient satisfies piecewise regularity conditions and that the diffusion coefficient is globally Lipschitz continuous and nondegenerate at the discontinuities of the drift coefficient. Under this type of assumptions the best Lp-error rate obtained so far for approximation of scalar SDEs at the final time is 3/4 in terms of the number of evaluations of the driving Brownian motion. In the present article, we prove for the first time in the literature sharp lower error bounds for such SDEs. We show that for a huge class of additive noise driven SDEs of this type the Lp-error rate 3/4 can not be improved. For the proof of this result we employ a novel technique by studying equations with coupled noise: we reduce the analysis of the Lp-error of an arbitrary approximation based on evaluation of the driving Brownian motion at finitely many times to the analysis of the Lp-distance of two solutions of the same equation that are driven by Brownian motions that are coupled at the given time-points and independent, conditioned on their values at these points. To obtain lower bounds for the latter quantity, we prove a new quantitative version of positive association for bivariate normal random variables (Y,Z) by providing explicit lower bounds for the covariance Cov(f(Y),g(Z)) in case of piecewise Lipschitz continuous functions f and g. In addition it turns out that our proof technique also leads to lower error bounds for estimating occupation time functionals ∫01f(Wt)dt of a Brownian motion W, which substantially extends known results for the case of f being an indicator function.

A new algorithm for computing path integrals and weak approximation of SDEs inspired by large deviations and Malliavin calculus

The paper gives a novel path integral formula inspired by large deviation theory and Malliavin calculus. The proposed finite-dimensional approximation of integrals on path space will be a new higher-order weak approximation of multidimensional stochastic differential equations where the dominant part of the local expansion is governed by Varadhan's geodesic distance and the correction terms are given as Malliavin weights. An optimal truncation of asymptotic expansion is used to reduce computational effort. Kusuoka's estimate is applied to justify the finite-dimensional approximation of path integrals. An efficient simulation method is provided with the algorithm. Numerical results are shown to verify the effectiveness.

An h-Adaptive Poly-Sinc-Based Local Discontinuous Galerkin Method for Elliptic Partial Differential Equations

For the purpose of solving elliptic partial differential equations, we suggest a new approach using an h-adaptive local discontinuous Galerkin approximation based on Sinc points. The adaptive approach, which uses Poly-Sinc interpolation to achieve a predetermined level of approximation accuracy, is a local discontinuous Galerkin method. We developed an a priori error estimate and demonstrated the exponential convergence of the Poly-Sinc-based discontinuous Galerkin technique, as well as the adaptive piecewise Poly-Sinc method, for function approximation and ordinary differential equations. In this paper, we demonstrate the exponential convergence in the number of iterations of the a priori error estimate derived for the local discontinuous Galerkin technique under the condition that a reliable estimate of the precise solution of the partial differential equation at the Sinc points exists. For the purpose of refining the computational domain, we employ a statistical strategy. The numerical results for elliptic PDEs with Dirichlet and mixed Neumann-Dirichlet boundary conditions are demonstrated to validate the adaptive greedy Poly-Sinc approach.

A deep Gaussian process model for seismicity background rates

SUMMARYThe spatio-temporal properties of seismicity give us incisive insight into the stress state evolution and fault structures of the crust. Empirical models based on self-exciting point processes continue to provide an important tool for analysing seismicity, given the epistemic uncertainty associated with physical models. In particular, the epidemic-type aftershock sequence (ETAS) model acts as a reference model for studying seismicity catalogues. The traditional ETAS model uses simple parametric definitions for the background rate of triggering-independent seismicity. This reduces the effectiveness of the basic ETAS model in modelling the temporally complex seismicity patterns seen in seismic swarms that are dominated by aseismic tectonic processes such as fluid injection rather than aftershock triggering. In order to robustly capture time-varying seismicity rates, we introduce a deep Gaussian process (GP) formulation for the background rate as an extension to ETAS. GPs are a robust non-parametric model for function spaces with covariance structure. By conditioning the length-scale structure of a GP with another GP, we have a deep-GP: a probabilistic, hierarchical model that automatically tunes its structure to match data constraints. We show how the deep-GP-ETAS model can be efficiently sampled by making use of a Metropolis-within-Gibbs scheme, taking advantage of the branching process formulation of ETAS and a stochastic partial differential equation (SPDE) approximation for Matérn GPs. We illustrate our method using synthetic examples, and show that the deep-GP-ETAS model successfully captures multiscale temporal behaviour in the background forcing rate of seismicity. We then apply the results to two real-data catalogues: the Ridgecrest, CA 2019 July 5 Mw 7.1 event catalogue, showing that deep-GP-ETAS can successfully characterize a classical aftershock sequence; and the 2016–2019 Cahuilla, CA earthquake swarm, which shows two distinct phases of aseismic forcing concordant with a fluid injection-driven initial sequence, arrest of the fluid along a physical barrier and release following the largest Mw 4.4 event of the sequence.

Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a large-scale and ill-conditioned linear system must be solved, that, in case of Maxwell equations, is particularly challenging, because of the presence of both H(div) and H(curl) discretization spaces. We propose here a parallel preconditioner, that exploits the Schur complement block factorization of the linear system matrix and consists of a Jacobi preconditioner for the H(div) block and an auxiliary space preconditioner for the H(curl) block. Several parallel numerical tests have been performed to study the robustness of the solver with respect to mesh refinement, shape of polyhedral elements, time step size and the VEM stabilization parameter.

Wealth dynamics in a market with information asymmetries.

Financial markets are usually investigated through time series such as prices and volumes that describe the behavior of the system as a whole. Such observables emerge from microeconomic interactions between market participants. Agent-based models have been utilized to shed light on this process. The model's ability to produce statistics frequently found in empirical data is evidence of some correspondence with real markets. Here, an agent-based market model is proposed. Different trader profiles with short- and long-term motivations are considered, and limitations on the agents' skills to manipulate information are inserted into the model. According to their profile and limitations, agents are rational. A differential equationapproximation is employed to find the value to which the price converges and the timescale of this process. The relationship between agents' attributes and the evolution of their wealth is explored in different scenarios. Agent's average wealth was not significantly affected by information processing accuracy, but the standard deviation was. The increased risk is the main consequence of low accuracy. The model yielded price series with multifractal behavior and heavy-tailed return distributions, which are nontrivial statistics frequently observed in empirical series.