Solution of 1<sup>st</sup> Order Stiff Ordinary Differential Equations Using Feed Forward Neural Network and Bayesian Regularization Algorithm

An adaptive discontinuous Galerkin method for very stiff systems of ordinary differential equations

A new Trapezoidal-type scheme is proposed for the direct numerical integration of time-dependent partial differential equations. The evolving system of ordinary differential equations after discretization is usually stiff, so it is desirable for the use of a numerical method in solving it to have good properties concerning stability. The method proposed in this article is L-stable and at least of algebraic order two. It is increasingly common to encounter partial differential equations (PDEs) posed on manifolds, and standard numerical methods are not available for such novel situations. Here, an L-Stable implicit Trapezoidal-like numerical integrator was developed for solving partial differential equations on manifolds. This approach allows the immediate use of familiar finite difference methods for the discretization and numerical solution. Presented here are the motivation and details of the method, illustration of its numerical convergence and stability properties for a general case. Numerical experiments illustrate the performance of the new method on different stiff systems of ODEs after discretizing in the space variable of some PDE problems. Results show accuracy of maximum error for various space and time steps. Key words: Trapezoidal-type method, partial differential equations, L-stability, implicit surfaces, manifolds, stiff systems.

Implicit methods applied to the numerical solution of systems of ordinary differential equations (ODEs) with an identically singular matrix multiplying the derivative of the sought-for vector-function are considered. The effects produced by losing L-stability of a classical implicit Euler scheme when solving such stiff systems are discussed.

10 - Applications of Systems of Ordinary Differential Equations

We study here a matrix-free method for solving stiff systems of ordinary differential equations (ODE’s). In the numerical time integration of stiff ODE initial value problems by BDF methods, the resulting nonlinear algebraic system is usually solved by a modified Newton method and an appropriate linear system algorithm. In place of that, we substitute Newton’s method (unmodified) coupled with an iterative linear system method. The latter is a projection method called the Incomplete Orthogonalization Method (IOM), developed mainly by Y. Saad. A form of IOM, with scaling included to enhance robustness, is studied in the setting of Inexact Newton Methods. The implementation requires no Jacobian matrix storage whatever. Tests on several stiff problems, of sizes up to 16,000, show the method to be quite effective and much more economical, in both computational cost and storage, than standard solution methods, at least when the problem has a certain amount of clustering in its spectrum.

For each associative, commutative, and unitary algebra over the field of real or complex numbers and an integrable nonlinear ordinary differential equation we can to construct integrable systems of ordinary differential equations and integrable systems of partial differential equations. In this paper we consider in some sense the inverse problem. Determine the conditions under which a given system of ordinary differential equations can be represented as a differential equation in some associative, commutative and unitary algebra. It is also shown that associativity is not a necessary condition.

Remarks on the choice of a stiff ordinary differential equation solver

On the comparison of numerical methods for the integration of kinetic equations in atmospheric chemistry and transport models

The performance of the Block Generalised Multistep Adams and Backward Differentiation Formulae (BGMBDF) as compared with Generalised Multistep Adams and Backward Differentiation Formulae (GMBDF) is presented. These methods are used to solve initial value problems (IVPs) for stiff and nonstiff systems of ordinary differential equations (ODEs). The results obtained show that the BGMBDF reduces the total number of steps in computation. Keywords: Block; backward differentiation formulae; steps. Closure matroid.

The non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form q ˙ i = ∂ H ∂ p i , p ˙ i = − ∂ H ∂ q i + Γ i ( t , q i , p i ) ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.

Separation of variables is a well‐known technique for solving differential equations. However, it is seldom used in practical applications since it is impossible to carry out a separation of variables in most cases. In this paper, we propose the amplitude–shape approximation (ASA) which may be considered as an extension of the separation of variables method for ordinary differential equations. The main idea of the ASA is to write the solution as a product of an amplitude function and a shape function, both depending on time, and may be viewed as an incomplete separation of variables. In fact, it will be seen that such a separation exists naturally when the method of lines is used to solve certain classes of coupled partial differential equations. We derive new conditions which may be used to solve the shape equations directly and present a numerical algorithm for solving the resulting system of ordinary differential equations for the amplitude functions. Alternatively, we propose a numerical method, similar to the well‐established exponential time differencing method, for solving the shape equations. We consider stability conditions for the specific case corresponding to the explicit Euler method. We also consider a generalization of the method for solving systems of coupled partial differential equations. Finally, we consider the simple reaction diffusion equation and a numerical example from chemical kinetics to demonstrate the effectiveness of the method. The ASA results in far superior numerical results when the relative errors are compared to the separation of variables method. Furthermore, the method leads to a reduction in CPU time as compared to using the Rosenbrock semi‐implicit method for solving a stiff system of ordinary differential equations resulting from a method of lines solution of a coupled pair of partial differential equations. The present amplitude–shape method is a simplified version of previous ones due to the use of a linear approximation to the time dependence of the shape function. Copyright © 2007 John Wiley & Sons, Ltd.

Extended Fourier Neural Operators to learn stiff chemical kinetics under unseen conditions

Tau leaping of stiff stochastic chemical systems via local central limit approximation

This research proposes a novel technique for fake news classification using natural language processing (NLP) methods. The proposed technique, TwIdw (Term weight–inverse document weight), is used for feature extraction and is based on TfIdf, with the term frequencies replaced by the depth of the words in documents. The effectiveness of the TwIdw technique is compared to another feature extraction method—basic TfIdf. Classification models were created using the random forest and feedforward neural networks, and within those, three different datasets were used. The feedforward neural network method with the KaiDMML dataset showed an increase in accuracy of up to 3.9%. The random forest method with TwIdw was not as successful as the neural network method and only showed an increase in accuracy with the KaiDMML dataset (1%). The feedforward neural network, on the other hand, showed an increase in accuracy with the TwIdw technique for all datasets. Precision and recall measures also confirmed good results, particularly for the neural network method. The TwIdw technique has the potential to be used in various NLP applications, including fake news classification and other NLP classification problems.

Eigensolutions of boundary value problems using inverse iteration