Ordinary Differential Equation Solver Research Articles

Three-dimension holographic numerical simulation of gravity anomalies

In gravity-anomaly-based prospecting, it is difficult to achieve large-scale and high-precision inversion imaging of complex geological models. To address this issue, this paper proposes a three-dimensional holographic numerical simulation method for gravity anomalies. This method transforms the 3D partial differential equation of gravity potential into many independent 1D differential equations with different wavenumbers by performing a 2D Fourier transform along the horizontal direction. It decomposes a large-scale 3D numerical modeling problem into many 1D numerical modeling problems, which greatly reduces the computation and memory requirements of modeling, and each 1D differential equation is independent, so it has high parallelism. The vertical direction is reserved as the spatial domain, which has strict upper and lower boundary conditions; The horizontal 2D Fourier transform uses a holographic Fourier transform (Holo-FT) with a full interval of integration, consistent with the physical boundary in the horizontal direction, thus the physical information of the gravity potential described by the three-dimensional partial differential equation is fully and accurately simulated. Using the high parallelism of the algorithm, the CPU is used for parallel solving ordinary differential equations, while the GPU is used for parallel computing of Holo-FT, which implements the CPU-GPU parallel acceleration scheme and further improves the efficiency of this algorithm.

Solving Ordinary Differential Equations with Boundary Conditions Numerically

This article elucidates the method and implementation of finite differences, aiming to enhance clarity for students and provide detailed insights into its application, as many authors assume a pre-existing understanding of numerical analysis. It includes two case studies employing Dirichlet and Cauchy boundary conditions, demonstrating the precision and effectiveness of this numerical tool in obtaining results.

A Sparse Differential Algebraic Equation (DAE) and Stiff Ordinary Differential Equation (ODE) Solver in Maple

This paper implements efficient numerical methods in Maple to solve index-1 nonlinear Differential Algebraic Equations (DAEs) and stiff Ordinary Differential Equations (ODEs) systems. Single-step methods (like Trapezoid (TR), Implicit-mid point (IMP), Euler-backward (EB), Radau IIA (Rad) methods, TRBDF2, TRX2) and backward-difference formula of order 2 are implemented with adaptive time-stepping methods in Maple to solve index-1 nonlinear DAEs. Maple’s robust and efficient ability to search within a list/set is exploited to identify the sparsity pattern and automatically calculate the analytic Jacobian. The algorithm and implementation are robust and efficient for index-1 DAE problems and scale well for finite difference/finite element discretization of two-dimensional models with system size up to 10,000 nonlinear DAEs and solve the same in a few seconds. The computational efficiency of the proposed algorithm (provided as an open-access code) compares favorably with the commercial solver gPROMs, one of the most commonly used sparse DAE solvers in the industry.

Accurate error estimation for model reduction of nonlinear dynamical systems via data-enhanced error closure

Accurate error estimation is crucial in model order reduction, to obtain small reduced-order models as well as to certify their accuracy when deployed in downstream applications such as digital twins. In existing a posteriori error estimation approaches, knowledge about the time integration scheme is mandatory, e.g., the residual-based error estimators proposed for the reduced basis method. This poses a challenge when automatic ordinary differential equation solver libraries are used to perform the time integration. To address this, we present a data-enhanced approach for a posteriori error estimation. Our new formulation enables residual-based error estimators to be independent of any time integration method. To achieve this, we introduce a corrected reduced-order model which takes into account a data-driven closure term for improved accuracy. The closure term, subject to mild assumptions, is related to the local truncation error of the corresponding time integration scheme. We propose efficient computational schemes for approximating the closure term, at the cost of a modest amount of training data. Furthermore, the new error estimator is incorporated within a greedy process to obtain parametric reduced-order models. Numerical results on three different systems show the accuracy of the proposed error estimation approach and its ability to produce ROMs that generalize well.

On problematic practice of using normalization in self-modeling/multivariate curve resolution (S/MCR)

This short communication paper is briefly dealing with greater or lesser misused normalization in self-modeling/multivariate curve resolution (S/MCR) practice. The importance of the correct use of the ODE (ordinary differential equation) solvers and apt kinetic illustrations are elucidated. The new terms, external and internal normalizations are defined and interpreted. The problem of reducibility of a matrix is touched. Improper generalization/development of normalization-based methods are cited as examples. The position of the extreme values of the signal contribution function is clarified. An Executable Notebook with Matlab Live Editor was created for algorithmic explanations and depictions.

Hyperbolic modeling of gradient damage and one-dimensional finite volume simulations

This study provides a new formulation of gradient damage model which allows an efficient explicit numerical solution of dynamics problems. The proposed methodology is based on an ”extended Lagrangian approach” developed by one of the authors for the nondissipative and dispersive shallow water equation. By using this strategy, the global minimization problem commonly derived for gradient damage models is recast as a purely local hyperbolic one with source terms that can be easily solved using finite volumes. The numerical solution of the governing system is then based on a fractional-step method consisting of a classical Godunov-type scheme and an implicit Ordinary Differential Equation solver for the local source terms. Numerical results are presented on one-dimensional multi-fragmentation and spalling tests for illustration purpose.

A fast and efficient computation method for reflective diffraction simulations

We present a new computation method for simulating reflection high-energy electron diffraction and the total-reflection high-energy positron diffraction experiments. The two experiments are used commonly for the structural analysis of material surface. The present paper improves the conventional numerical method, the multi-slice method, for faster computation, since the present method avoids the matrix-eigenvalue solver for the computation of matrix exponentials and can adopt higher-order ordinary differential equation solvers. Moreover, we propose a high-performance implementation based on multi-thread parallelization and cache-reusable subroutines. In our tests, this new method performs up to 2,000 times faster than the conventional method without compromising accuracy.

Applications of particle swarm optimization for numerical simulation of Fisher’s equation using RBF

This research article focused on optimising the parameter related to radial basis function (RBF) by solving the Fisher’s equation numerically. To obtain the numerical solutions, Particle Swarm Optimisation (PSO) algorithm is used along with radial basis functions (RBFs). PSO has been used by a number of researchers to obtain the optimised results in various scientific applications. In this work, using RBF-pseudospectral method, partial differential equations (PDEs) are transformed into a system of ordinary differential equations (ODEs) then are solved in MATLAB using the ODE solver. Two numerical examples are presented to demonstrate the effectiveness and accuracy of the proposed method by computingL∞, L2, and Lrms error norms with their order of convergence. Also, it can be demonstrated from the tables that the obtained results are in conformity with the results available in the literature.

Mathematical solutions in internal dose assessment: A comparison of Python-based differential equation solvers in biokinetic modeling

In biokinetic modeling systems employed for radiation protection, biological retention and excretion have been modeled as a series of discretized compartments representing the organs and tissues of the human body. Fractional retention and excretion in these organ and tissue systems have been mathematically governed by a series of coupled first-order ordinary differential equations (ODEs). The coupled ODE systems comprising the biokinetic models are usually stiff due to the severe difference between rapid and slow transfers between compartments. In this study, the capabilities of solving a complex coupled system of ODEs for biokinetic modeling were evaluated by comparing different Python programming language solvers and solving methods with the motivation of establishing a framework that enables multi-level analysis. The stability of the solvers was analyzed to select the best performers for solving the biokinetic problems. A Python-based linear algebraic method was also explored to examine how the numerical methods deviated from an analytical or semi-analytical method. Results demonstrated that customized implicit methods resulted in an enhanced stable solution for the inhaled 60Co (Type M) and 131I (Type F) exposure scenarios for the inhalation pathway of the International Commission on Radiological Protection (ICRP) Publication 130 Human Respiratory Tract Model (HRTM). The customized implementation of the Python-based implicit solvers resulted in approximately consistent solutions with the Python-based matrix exponential method (expm). The differences generally observed between the implicit solvers and expm are attributable to numerical precision and the order of numerical approximation of the numerical solvers. This study provides the first analysis of a list of Python ODE solvers and methods by comparing their usage for solving biokinetic models using the ICRP Publication 130 HRTM and provides a framework for the selection of the most appropriate ODE solvers and methods in Python language to implement for modeling the distribution of internal radioactivity.

A NEW APPROACH FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS

A new method for solving ordinary differential equations is suggested. As it is shown the differential equation can be solved not only by means of integration its differentiation as well. The developed method is applied for consideration of corresponding equations for the exponential and trigonometric functions. In the framework of the developed method the simple approach for derivation of Taylor series of a function is demonstrated. Based on the geometric definition the connections between trigonometric functions and their derivatives are obtained as well. The well-known Euler formula is also produced, which establishes a connection between trigonometric and exponential functions.

Structural reliability analysis of aircraft wing rib fatigue cracking using surrogate dynamic Bayesian network

AbstractDynamic Bayesian networks (DBNs) are crucial for evaluating time‐sensitive structural risks in aircraft, yet their inherent complexity can create substantial computational demands. This research introduces a surrogate model, specifically a two‐layer artificial neural network (ANN), to replace the most computationally intensive node in the DBN, typically associated with an ordinary differential equation solver. The surrogate model was chosen after comparing 24 regression machine learning models and optimizing hyperparameters, leading to a significant reduction in computational time and an improvement over traditional methods such as Monte Carlo simulations. The surrogate model demonstrates practical significance with its conservative risk estimations and successful application to a real‐world structural fatigue issue in an F‐4E aircraft intermediate rib. The unique aspect of this research lies in the strategic application of ANN to mitigate computational challenges, thereby enhancing the performance of DBN in fatigue risk analysis.

Exact soliton solutions and soliton diffusion of two kinds of stochastic KdV equations with variable coefficients

In this paper, we consider the exact solutions and soliton diffusion phenomenon of two kinds of stochastic KdV equations with variable coefficients. Firstly, according to symmetric reduction, stochastic KdV equations with variable coefficients are transformed into a coupled system of a deterministic KdV-type equation with variable coefficients and a solvable stochastic ordinary differential equation. Then, by the generalized wave transformation and the Clarkson-Kruskal direct method, we obtain the exact solutions of the deterministic KdV-type equation with variable coefficients. By coupling with the exact solutions of the stochastic ordinary differential equation, the exact solutions of stochastic KdV equations with variable coefficients are obtained. Compared to Wick-type stochastic KdV equations, our research work does not require additional inverse transformations, but solves stochastic partial differential equations more concisely, systematically, and directly. Secondly, two examples are given to verify the correctness of the theoretical analysis, and the soliton diffusion phenomenon of the system is discussed. Finally, by the Zabusky-Kruskal finite difference scheme, numerical simulations are provided to demonstrate the effectiveness of the analytic methods. The results indicate that the soliton diffusion phenomenon is subject to noise influence. In particular, the wave speed accelerates the soliton diffusion over time in the multiplicative noise background, and the wave speed slows down the soliton diffusion over time in the additive noise background.

Evolutionary Neural Networks for Option Pricing: Multi-Assets Option and Exotic Option

This paper presents a novel framework based on the evolutionary neural network to solve the generalized Black-Scholes equation arising in the financial market efficiently and accurately. We first employ evolutionary neural networks to parameterize the Partial Differential Equations (PDEs) involved in option pricing. This approach allows us to simplify the Black-Scholes PDEs and convert them into the corresponding Ordinary Differential Equations (ODEs). Thus we can use standard ODE solvers such as Euler’s method to solve the simplified ODE problem. The proposed framework is flexible and can handle various boundary conditions and terminal conditions, allowing customization based on specific market requirements and scenarios. Moreover, our method offers a reliable and deterministic solution methodology for the pricing framework, as it does not rely on stochastic training. Unlike other approaches that incorporate stochastic elements in their training process, our method eliminates the need for such training and provides consistent results. This deterministic nature enhances the reliability and stability of our approach, making it well-suited for real-world applica- tions in option pricing and financial markets. The experiments on multiple settings are carried out to illustrate the applicability and accuracy of the proposed framework.

On Implementation of Numerical Methods for Solving Ordinary Differential Equations in Computer Algebra Systems

On Implementation of Numerical Methods for Solving Ordinary Differential Equations in Computer Algebra Systems

On-chip all-optical second-order ordinary differential equation solver based on a single microdisk resonator.

This paper proposes an all-optical second-order ordinary differential equation (SODE) solver based on a single microdisk resonator. We validate the feasibility of our structure for constant and complex coefficient SODE solutions for Gaussian and super-Gaussian pulses. The results demonstrate a good agreement between the solutions obtained with the designed structure and those obtained through mathematical calculations for both constant and complex coefficient SODEs. We also discuss the influence of input optical signal pulse width on solution result deviations. Furthermore, we validate the capability of the designed structure to achieve tunable solutions for complex-coefficient SODEs with a tuning power of less than 10 mW. The device footprint is approximately 20×30 μm2, and it is 3-4 times smaller than the current smallest solving unit. The maximum Q-factor reaches 9.8×104. The proposed device avoids the traditional approach of cascading two resonators for SODE solving. Moreover, achieving mode alignment within the same resonator reduces the process challenges associated with aligning multiple devices in a cascade. Furthermore, it offers wider applicability for solving SODEs, namely, the ability to solve both constant and complex coefficient SODEs with complete derivative terms.

Wideband parametric baseband macromodeling of linear and passive photonic circuits via complex vector fitting

A novel wideband parametric baseband macromodeling technique for passive photonic devices and circuits is presented. It allows to efficiently estimate the baseband scattering representations of a linear, passive photonic system as a function of a set of design variables, such as geometrical layout or substrate features. The proposed technique relies on the interpolation of macromodels computed via a complex vector fitting (CVF) algorithm, by adopting a methodology based on amplitude and frequency scaling that preserves, by construction, the physical properties of the system, such as causality, stability and passivity. For a specified combination of the design parameters, a rational CVF model is derived that can be simulated by a wide range of ordinary differential equation (ODE) solvers or circuit simulators. Additionally, time-domain simulations using the computed model can be performed at arbitrary optical carrier frequencies, thus allowing for the simulation of multi-wavelength systems. Two application examples are presented to demonstrate the flexibility and advantages of the proposed method.

Data-Driven Tabulation for Chemistry Integration Using Recurrent Neural Networks.

Due to the wide range of time scales involved in the ordinary differential equations (ODEs) describing chemical reaction kinetics, multidimensional numerical simulation of chemical reactive flows using detailed combustion mechanisms is computationally expensive. To confront this issue, this article presents an economic data-driven tabulation algorithm for fast combustion chemistry integration. It uses the recurrent neural networks (RNNs) to construct the tabulation from a series of current and past states to the next state, which takes full advantage of RNN in handling long-term dependencies of time series data. The training data are first generated from direct numerical integrations to form an initial state space, which is divided into several subregions by the K-means algorithm. The centroid of each cluster is also determined at the same time. Next, an Elman RNN is constructed in each of these subregions to approximate the expensive direct integration, in which the integration routine obtained from the centroid is regarded as the basis for a storing and retrieving solution to ODEs. Finally, the alpha-shape metrics with principal component analysis (PCA) are used to generate a set of reduced-order geometric constraints that characterize the applicable range of these RNN approximations. For online implementation, geometric constraints are frequently verified to determine which RNN network to be used to approximate the integration routine. The advantage of the proposed algorithm is to use a set of RNNs to replace the expensive direct integration, which allows to reduce both the memory consumption and computational cost. Numerical simulations of a H₂/CO-air combustion process are performed to demonstrate the effectiveness of the proposed algorithm compared to the existing ODE solver.

On Implementation of Numerical Methods for Solving Ordinary Differential Equations in Computer Algebra Systems

This paper presents an original package for investigating numerical solutions of ordinary differential equations, which is built in the Sage computer algebra system. This project is focused on a closer integration of numerical and symbolic methods while primarily aiming to create a convenient tool for working with numerical solutions in Sage. The package defines two new classes: initial problems and approximate solutions. The first class defines tools for symbolic computations related to initial problems, while the second class defines tools for interpolating values of symbolic expressions on an approximate solution and estimating the error with the use of the Richardson method. An implementation of the Runge–Kutta method is briefly described, with its main feature being the possibility of working with arbitrary Butcher tableaux and arbitrary numeric fields.

IdopNetwork: A network tool to dissect spatial community ecology

Abstract Network models have been used as a tool to characterize internal workings of complex systems. The amount of topological and functional information extracted from a network depend on the method of network inference and the type of network data. An interdisciplinary computational model has been proposed to reconstruct informative, dynamic, omnidirectional and personalized networks (idopNetwork) from any data domains including static data. We implement idopNetwork as an R‐based cartographic tool to characterize spatially varying interspecies interaction networks using the abundance data of multiple species from different geographical locations. This tool provides a unified framework for integrating power curve fitting based on allometrical scaling law, functional clustering, LASSO‐based variable selection, quasi‐dynamic ordinary differential equation solving, species abundance decomposition and network visualization. It coalesces all species from different spaces into location‐specific networks. We demonstrate the utility of this tool by analysing different organs that are spatially interconnected via microbiomes within the host using two datasets from the gut microbiota and plant microbiota. Given that biodiversity and organization vary biogeographically at different scales, idopNetwork will find its widespread application to modelling and estimating interspecific interactions with differing functions across space.

An Extended Generalized Average Modeling Framework for Power Converters

The Generalized Averaged Modeling (GAM) technique is traditionally employed to capture the dynamic performance of power electronic converters. This paper proposes an improved version of it, named the Extended-GAM (EGAM) technique, which supports the multiplication of two Double Fourier Series (DFS) signals in the time domain. Multiplication of DFS signals in the time domain translates to the 2D-convolution of coefficients of the DFS terms of their equivalent Discrete Fourier Image (DFI) representations. Thus, the proposed EGAM technique, capable of capturing many harmonics present in the output of a power converter, effectively captures the dynamic behavior of power converters excited by two distinct frequencies. The proposed technique is then converted into an algorithm suitable for numerical platforms, which typically use Ordinary Differential Equation (ODE) solvers. The proposed algorithm is validated based on the observations of the effects of harmonic truncation. The efficacy of the proposed technique is assessed through a case study, wherein a single-phase inverter employs LC filters on both the dc-link and the ac-side. Finally, it is shown that the results obtained with the proposed method show an excellent congruence between simulation and hardware experimental models. Additionally, the proposed algorithm is packaged into a MATLAB toolbox and shared for future implementations.