Solution Of ODE Research Articles

Overview of Artificial Neural Network-Based Solution for Ordinary and Partial Differential Equations by Feed Forward Method Using Python

In the modern era, models such as Neural Networks are increasingly used to deal with ordinary and partial differential equations (ODEs-PDEs) because of their related complexity. The main aim of the paper is to focus on Artificial Neural Networks as a method to solve these equations since it is possible to approximate complex nonlinear relationships with great accuracy and adaptively reduce computational costs, thus enhancing the accuracy of solutions within highly dimensional and nonlinear problems. The traditional methods are not applicable to partial differential equations due to the problem of nonlinearity and high dimensionality. However, the Feed Forward method implemented in Python is quite a powerful alternative that can easily approximate complicated functions and very well have a hold on nonlinear equations. This research paper contains differential equations as nonlinear and linear equations and Feed Forward neural network models for both solutions of ODEs and PDEs. Python enables the integration of AI-based algorithms for the enhancement of more accurate and efficient results. This work compares the existing solutions based on ANNs and further highlights the possible advantages of using ANNs in ordinary as well as complex differential equations problems. The results indicate that ANNs have immense potential for further developing computational algorithms applicable to solving ODEs and PDEs, particularly in real-time applications that demand high precision and adaptability.

A class of explicit second derivative general linear methods for non-stiff ODEs

In this paper, we construct explicit second derivative general linear methods (SGLMs) with quadratic stability and a large region of absolute stability for the numerical solution of non-stiff ODEs. The methods are constructed in two different cases: SGLMs with p = q = r = s and SGLMs with p = q and r = s = 2 in which p, q, r and s are respectively the order, stage order, the number of external stages and the number of internal stages. Examples of the methods up to order five are given. The efficiency of the constructed methods is illustrated by applying them to some well-known non-stiff problems and comparing the obtained results with those of general linear methods of the same order and stage order.

Innovative binary nanofluid approach with copper (Cu - EO) and Magnetite ([formula omitted] - EO) for enhanced thermal performance

In this study, we present an innovative and detailed analysis of the motion of a binary nanofluid, specifically copper (Cu - EO) and Magnetite (Fe3O4 - EO), in a porous medium under solar radiation. Our research is driven by the potential to significantly enhance the thermal performance of a Parabolic Trough Solar Collector (PTSC) using a binary nanofluid. The boundary conditions, partial differential restrictions, and the governing PDEs are transformed into the ODEs system with appropriate similarity transformations. The finite element method (FEM) and semi-analytical technique memad Akbari-Ganji's Method (AGM) are used to solve the approximate solutions of ODEs. The findings of evaluating Copper (Cu - EO) and Magnetite (Fe3O4 - EO) were the two different nanofluids based on engine oil that were presented. Furthermore, the model's overall entropy variations are enhanced using Reynolds. Moreover, our study contributes to understanding fluidity and viscosity alterations in such systems. We use the Brinkman number to observe these alterations. We reveal that the thermal efficiency of (Cu - EO) is lower than (Fe3O4 - EO) by a range of 0.005–0.6. This indicates a substantial enhancement in the thermal performance of the PTSC.

Picard-type theorem and the solutions of certain ODE and PDE

Picard-type theorem and the solutions of certain ODE and PDE

A chemistry load balancing model for OpenFOAM

Efficient simulation tools are crucial for studying complex systems such as reacting flows where computational costs of computing chemical reaction rates can vastly exceed the costs for the integration of the convective and diffusive transport terms. Load imbalance in parallel computing poses a significant challenge for massively parallel reacting flow simulations. In response, a novel load balancing library has been developed to enhance OpenFOAM's solver performance in parallel environments. This library seamlessly integrates with OpenFOAM, offering ease of use and applicability to any OpenFOAM reacting solver incorporating finite-rate chemistry. In addition, it supports the standard and the dynamic adaptive chemistry model (TDAC) of OpenFOAM.The newly developed load-balanced standard and TDAC models address significant load imbalances by exchanging information between processes via MPI calls and tracking ODE solution times on a cell level. The TDAC model introduces dual tables on each core and enables immediate addition of computed solutions, enhancing computational efficiency. Validation on various test cases, including simulations on the HLRS Hawk supercomputer with up to 8000 cores, confirms identical results compared to the original unbalanced models, with notable speed-up factors of up to 6 for the standard and 5 for the TDAC model. Despite non-linear scaling at lower cell count per processor, load-balanced models consistently outperform unbalanced counterparts, making them the preferred choice for reacting flow simulations in OpenFOAM.

Left-Invariance for Smooth Vector Fields and Applications

Let X={X0,…,Xm}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X = \\{X_0,\\ldots ,X_m\\}$$\\end{document} be a family of smooth vector fields on an open set Ω⊆RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subseteq \\mathbb {R}^N$$\\end{document}. Motivated by applications to the PDE theory of Hörmander operators, for a suitable class of open sets Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document}, we find necessary and sufficient conditions on X for the existence of a Lie group (Ω,∗)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\Omega ,*)$$\\end{document} such that the operator L=∑i=1mXi2+X0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L=\\sum _{i = 1}^mX_i^2+X_0$$\\end{document} is left-invariant with respect to the operation ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$*$$\\end{document}. Our approach is constructive, as the group law is constructed by means of the solution of a suitable ODE naturally associated to vector fields in X. We provide an application to a partial differential operator appearing in the Finance.

The capability of a deep learning based ODE solution for low temperature plasma chemistry

The capability of a deep learning based ODE solution for low temperature plasma chemistry

On some $q$-Bessel type continuous wavelet transform

In this paper we continue as in \cite{Rezguietal} to exploit the modified variants of Bessel function in the framework of $q$-theory to construct wavelet operators. A generalized $q$-Bessel type function has been introduced leading to an associated mother wavelet which in turns has induced a continuous wavelet transform. Finally, Plancherel/Parceval type relations have been proved. Such variants of wavelets permit in some sense to approximate solutions of ODEs and PDEs by transforming them to recurrent sequences.

New Generalized Jacobi Polynomial Galerkin Operational Matrices of Derivatives: An Algorithm for Solving Boundary Value Problems

In this study, we present a novel approach for the numerical solution of high-order ODEs and MTVOFDEs with BCs. Our method leverages a class of GSJPs that possess the crucial property of satisfying the given BCs. By establishing OMs for both the ODs and VOFDs of the GSJPs, we integrate them into the SCM, enabling efficient and accurate numerical computations. An error analysis and convergence study are conducted to validate the efficacy of the proposed algorithm. We demonstrate the applicability and accuracy of our method through eight numerical examples. Comparative analyses with prior research highlight the improved accuracy and efficiency achieved by our approach. The recommended approach exhibits excellent agreement between approximate and precise results in tables and graphs, demonstrating its high accuracy. This research contributes to the advancement of numerical methods for ODEs and MTVOFDEs with BCs, providing a reliable and efficient tool for solving complex BVPs with exceptional accuracy.

A new closed-form expression for the solution of ODEs in a ring of distributions and its connection with the matrix algebra

A new expression for solving homogeneous linear ODEs based on a generalization of the Volterra composition was recently introduced. In this work, we extend such an expression, showing that it corresponds to inverting an infinite matrix. This is done by studying a particular subring and connecting it with a subalgebra of infinite matrices.

A generalization of the rational rough Heston approximation

Previously, in Gatheral and Radoičić (Rational approximation of the rough Heston solution. Int. J. Theor. Appl. Finance, 2019, 22(3), 1950010), we derived a rational approximation of the solution of the rough Heston fractional ODE in the special case λ = 0 , which corresponds to a pure power-law kernel. In this paper we extend this solution to the general case of the Mittag-Leffler kernel with λ ≥ 0 . We provide numerical evidence of the convergence of the solution.

Numerical Integration of Highly Oscillatory Functions with and without Stationary Points

This paper proposes an original approach to calculating integrals of rapidly oscillating functions, based on Levin’s algorithm, which reduces the search for an anti-derivative function to solve an ODE with a complex coefficient. The direct solution of the differential equation is based on the method of integrating factors. The reduction in the original integration problem to a two-stage method for solving ODEs made it possible to overcome the instability that arises in the standard (in the form of solving a system of linear algebraic equations) approach to the solution. And due to the active use of Chebyshev interpolation when using the collocation method on Gauss–Lobatto grids, it is possible to achieve high speed and stability when taking into account a large number of collocation points. The presented spectral method of integrating factors is both flexible and reliable and allows for avoiding the ambiguities that arise when applying the classical method of collocation for the ODE solution (Levin) in the physical space. The new method can serve as a basis for solving ordinary differential equations of the first and second orders when creating high-efficiency software, which is demonstrated by solving several model problems.

Abaqus implementation of a large family of finite viscoelasticity models

In this paper, we introduce an Abaqus UMAT subroutine for a family of constitutive models for the viscoelastic response of isotropic elastomers of any compressibility – including fully incompressible elastomers – undergoing finite deformations. The models can be chosen to account for a wide range of non-Gaussian elasticities, as well as for a wide range of nonlinear viscosities. From a mathematical point of view, the structure of the models is such that the viscous dissipation is characterized by an internal variable Cv, subject to the physically-based constraint detCv=1, that is solution of a nonlinear first-order ODE in time. This ODE is solved by means of an explicit Runge–Kutta scheme of high order capable of preserving the constraint detCv=1 identically. The accuracy and convergence of the code is demonstrated numerically by comparison with an exact solution for several of the Abaqus built-in hybrid finite elements, including the simplicial elements C3D4H and C3D10H and the hexahedral elements C3D8H and C3D20H. The last part of this paper is devoted to showcasing the capabilities of the code by deploying it to compute the homogenized response of a bicontinuous rubber blend.

The mixed convection thermally radiated hybrid nanofluid flow through an inclined permeable shrinking plate with slip condition and inclined magnetic effect

In this paper, we consider the slip boundary condition to analyze the radiative inclined magnetohydrodynamics mixed convection hybrid nanofluid (Al2O3–Cu/H2O) flow across an inclined shrinking permeable plate. The following model’s governing PDEs are transformed into nonlinear ODEs with the help of similarity transformations. To achieve the numerical solution of ODEs, the boundary-value problem of 4th order accuracy (bvp4c) is applied. With appropriate values for the copper volume fraction, magnetic parameter, slip parameter, and radiation parameter, the numerical results are described and graphically represented in velocity and temperature profiles. Due to the shrinking plate, the dual solutions are to be observed. For higher values of copper volume fraction, slip parameter, and magnetic parameter the first solution in the velocity profile increases and the second solution in the velocity profile decreases by increasing the value of copper volume fraction and magnetic parameter. By increasing the magnetic parameter, and slip parameter first solution of the temperature profile decreases while both the solution increases by increasing the copper volume fraction and radiation parameter.

An enhanced Sample-Partitioning Adaptive Reduced Chemistry method with a-priori error estimation

Reactor-based approaches for handling the Turbulence–Chemistry-Interactions closure have the advantage of embedding finite-rate chemistry in the combustion model of RANS and LES simulations, which might be crucial for the solution accuracy when complex combustion regimes are investigated. However, the numerical solution of the chemical ODEs is burdened with stiffness and increased dimensionality, especially when large detailed mechanisms are required. To this end, the Sample-Partitioning Adaptive Reduced Chemistry (SPARC) methodology couples adaptive chemistry and machine learning to speed-up the chemistry integration in reactive flows simulations. It consists in building a library of skeletal mechanisms, associated to clusters of similar thermo-chemical states identified in a training dataset, and then, at run-time, each computational cell is assigned to a specific cluster, whose skeletal mechanism is retrieved and employed for the time integration. Such workflow builds on four interacting blocks, i.e., training dataset generation, clustering, mechanism simplification, and classification, and its success tightly depends on the quality of each block, which generally results from a combination of theoretical, methodological, and computational choices. In this paper, we explore the effects of the mechanism simplification strategy on the SPARC performance and we develop an ad-hoc procedure that automatically identifies the cluster-wise optimal reduction parameters, delivering a higher global reduction and therefore a larger computational speed-up compared to a standard approach, along with an explicit a-priori control on accuracy. We implement and test this novel procedure on a RANS simulation of the Adelaide Jet-in-hot-coflow (AJHC) burner, and we attain a ∼2x CPU time improvement with respect to the simulation obtained with a 36-species detailed mechanism.Novelty and significance This work makes a contribution towards the acceleration of chemistry integration in reactive flows simulations. More specifically, the Sample-Partitioning Adaptive Reduced Chemistry method, which couples adaptive chemistry and machine learning, is enhanced with automatic target species definition and a-priori error estimation. The novelties lie in the utilization of the computational singular perturbation (CSP) reduction algorithm, which provides means for automatically identifying an adaptive set of target species, and in the definition of a novel strategy for assessing the performance of the reduced mechanisms in the pre-processing phase, with the goal of estimating a measure of accuracy of the upcoming CFD simulation.

A modified Runge–Kutta method for increasing stability properties

In control theory, we are interested mainly in investigating asymptotic stability of systems. In this paper explicit Runge–Kutta methods are investigated for numerical solutions of nonlinear ODEs with known strict Lyapunov functions. In order to increase stability properties, a modified version of explicit Runge–Kutta methods is presented. The modified Runge–Kutta method is explicit. It is different from the projection methods in literature which are implicit. Comparing with the standard Runge–Kutta method, the modified Runge–Kutta one has better stability properties. Numerical examples are provided to illustrate the effectiveness of the modified Runge–Kutta method.

Inclined magnetic force impact on cross nanoliquid flowing with widening shallow and heat generating by using artificial neural network (ANN)

Artificial neural networks (ANNs) have a wide range of applications in science and technology. Artificial neural networks (ANNs) widely utilized in image and speech recognition, neural language processing, drug discovery, genomics and bioinformatics, Robotics and control systems, material science and energy and power system. Due to the above applications, the main focus of this article is to scrutinize the impact of inclined magnetic field on a cross nanofluid flow with slip velocity and convective boundary conditions over a variable porosity. Gold (Au) nanoparticles are suspended in base liquid blood. Thermal stratification and heat generation with joule heating impact is taken in the energy equation in order to see the heat transfer. To transform the non-linear partial differential equations (PDE's) into non-linear ordinary differential equations (ODE's) utilized the von-Karman similarity transformation parameters. For the solutions of the non-linear ODE's, the 4th- order Runge-Kutta numerical method is employed. A dataset for the employed neural network back propagated Levenberg-Marquard scheme (NN-BLMS) is generated for various estimations of the embedded parameters like Weissenberg number, magnetic parameter, angle of inclination, porosity and permeability parameters, thermal stratification parameter by utilizing the 4th-order Runge-Kutta numerical scheme. The testing, validation and training methods of NN-BLMS are utilized to scrutinize the approximate solution of cross nanofluid with slip velocity and convective boundary conditions. The performance of the proposed NN-BLMS to successfully solved the cross nanofluid model is endorsed via mean squared error, error histogram and regression analysis. Streamlines and heatlines are plotted for different parameters including in velocity and temperature equations.

Euler scheme for approximation of solution of nonlinear ODEs under inexact information

We investigate error of the Euler scheme in the case when the right-hand side function of the underlying ODE satisfies nonstandard assumptions such as local one-sided Lipschitz condition and local Hölder continuity. Moreover, we assume two cases in regards to information availability: exact and noisy with respect to the right-hand side function. Optimality analysis of the Euler scheme is also provided. Finally, we present the results of some numerical experiments.

Dynamics of breathers in the Gardner hierarchy: universality of the variational characterization

We present a new variational characterization of breather solutions of any equation of the focusing Gardner hierarchy. This hierarchy is characterized by a nonnegative index n, and 2n + 1 represents the order of the corresponding PDE member. In this paper, we first show the existence of such breathers, and that they are solutions of the (2n+1)th-order Gardner equation. Then we prove a variational universality property, in the sense that all these breather solutions satisfy the same fourth order stationary elliptic ODE, regardless the order of the hierarchy member. This fact also characterizes them as critical points of the same Lyapunov functional, that we also construct here. As by product of our approach, we find breather solutions of the hierarchy of (2n+1)th-order mKdV equations, as well as a respective characterization of them as solutions of a fourth order stationary elliptic ODE. We also extend part of these results to the periodic setting, presenting new breather solutions for the 5th and 7th mKdV members of the hierarchy. Finally, we prove ill-posedness results for the whole Gardner hierarchy, by using appropiately their breather solutions.

General Solutions of First-Order Algebraic ODEs in Simple Constant Extensions

General Solutions of First-Order Algebraic ODEs in Simple Constant Extensions