Initial-value Problems In Ordinary Differential Equations Research Articles

Parsimonious physics-informed random projection neural networks for initial value problems of ODEs and index-1 DAEs.

We present a numerical method based on random projections with Gaussian kernels and physics-informed neural networks for the numerical solution of initial value problems (IVPs) of nonlinear stiff ordinary differential equations (ODEs) and index-1 differential algebraic equations (DAEs), which may also arise from spatial discretization of partial differential equations (PDEs). The internal weights are fixed to ones while the unknown weights between the hidden and output layer are computed with Newton's iterations using the Moore-Penrose pseudo-inverse for low to medium scale and sparse QR decomposition with L 2 regularization for medium- to large-scale systems. Building on previous works on random projections, we also prove its approximation accuracy. To deal with stiffness and sharp gradients, we propose an adaptive step-size scheme and address a continuation method for providing good initial guesses for Newton iterations. The "optimal" bounds of the uniform distribution from which the values of the shape parameters of the Gaussian kernels are sampled and the number of basis functions are "parsimoniously" chosen based on bias-variance trade-off decomposition. To assess the performance of the scheme in terms of both numerical approximation accuracy and computational cost, we used eight benchmark problems (three index-1 DAEs problems, and five stiff ODEs problems including the Hindmarsh-Rose neuronal model of chaotic dynamics and the Allen-Cahn phase-field PDE). The efficiency of the scheme was compared against two stiff ODEs/DAEs solvers, namely, ode15s and ode23t solvers of the MATLAB ODE suite as well as against deep learning as implemented in the DeepXDE library for scientific machine learning and physics-informed learning for the solution of the Lotka-Volterra ODEs included in the demos of the library. A software/toolbox in Matlab (that we call RanDiffNet) with demos is also provided.

Classical Theory of Linear Multistep Methods for Volterra Functional Differential Equations

Based on the linear multistep methods for ordinary differential equations (ODEs) and the canonical interpolation theory that was presented by Shoufu Li who is exactly the second author of this paper, we propose the linear multistep methods for general Volterra functional differential equations (VFDEs) and build the classical stability, consistency, and convergence theories of the methods. The methods and theories presented in this paper are applicable to nonneutral, nonstiff, and nonlinear initial value problems in ODEs, Volterra delay differential equations (VDDEs), Volterra integro-differential equations (VIDEs), Volterra delay integro-differential equations (VDIDEs), etc. At last, some numerical experiments verify the correctness of our theories.

Insight into 2-step continuous block method for solving mixture model and SIR model

Understanding of the solutions of first-order ordinary differential equations (ODEs), mixture model and SIR model in order to develop deep insight and exploration are major problems before the experts, biologists, scientists, and mathematicians. In all these problems, the governing equations are either single first-order or coupled ODEs kind of initial value problem (IVP). In this paper, a polynomial function q(x) that passes through the points (xn, yn), (xn+1, yn+1), . . . , (xn+2, yn+2) was adopted as the basis function that leads to third derivatives continuous 2-step block method suitable to solve first order initial value problems of ODEs. Upon using the newly proposed scheme to solve linear ODEs (i.e., mixture theory) and nonlinear ODE (i.e., SIR model), it is worth concluding that the algorithm is not only efficient but minimises error.

On some boundary value methods for stiff IVPs in ODEs

A boundary value method (BVM) leads to a discrete boundary value problem with some conditions imposed at the initial points and the remaining ones at the final points. The interest in this class of methods is the fact that BVMs overcome the limitations of the well-known Dahlquist order and stability barrier for an A-stable linear multistep method (LMM). Yet, A-stability is fundamental to the integration of stiff ordinary differential equations (ODEs). This presents multiple families of linear multistep formulas (LMFs) with future points based on extended backward differentiation formula (BDF) which can be used as a main method in a BVM implementation for the solution of initial value problems (IVPs) in ODEs.

A family of highly stable second derivative block methods for stiff IVPs in ODEs

This paper considers a class of highly stable block methods for numerically solving initial value problems (IVPs) in ordinary differential equations (ODEs). The boundary locus of the proposed parallel one-block, r-output point algorithms shows that the new schemes are A-stable for output points r = 2(2)8 and A(α)-stable for output points r = 10(2)20, where r is the number of processors in a particular block method in the family. Numerical results of the block methods are compared with the second derivative linear multistep method in [8].

Some General Linear Methods for the Numerical Solution of Non-Stiff IVPs in ODEs

In this paper, we consider the construction of explicit General Linear Methods (GLM) for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The discrete coefficients of the methods are obtained from the continuous coefficients of the continuous schemes which are derived using collocation and interpolation approach. One of the new method's stability matrix M( z) has a single nonzero eigenvalue similar to that of the third order Runge Kutta method (RKM), while other eigenvalues being zero, such general linear method is referred to as GLM with Runge Kutta stability. Numerical results are provided to illustrate the application of the new schemes. The new schemes are found to have competitive accuracy with the existing Almost Runge Kutta Methods (ARKM) discussed in Butcher [9].

On the construction of high order A(α)-stable hybrid linear multistep methods for stiff IVPs in ODEs

In this paper, we present a class of A(α)-stable hybrid linear multistep methods for numerical solving stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The method considered uses a second derivative like the Enright’s second derivative linear multistep methods for stiff IVPs in ODEs.

A class of continuous hybrid linear multistep methods for stiff IVPs in ODEs

In this paper, we present a class of continuous hybrid linear multistep methods (CHLMM) for stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The constuction of these methods are based on the approach of collocation and interpolation. The interval of absolute stability of the method is investigated, using the root locus method. Numerical results of the methods solving stiff IVPs in ODEs are compared with that from the state-of-the-art Ode15s Matlab ODEs code. Mathematics Subject Classification 2000: 65L05, 65L06.

Precise time integration for linear two-point boundary value problems

The precise time integration (PTI) method is proposed to solve the linear two-point boundary value problem (TPBVP). By employing the method of dimensional expanding, the non-homogeneous ordinary differential equations (ODEs) can be transformed into homogeneous ones and then the original PTI algorithms can be applied directly to the TPBVP. The PTI consists of two methods: the method of matrix exponential and the method of Riccati equations, both of which utilize the merit of 2 N algorithm and guarantee high precise numerical results. The method of matrix exponential follows the similar scheme proposed for initial value problem of ODEs, and uses the matrix exponential to link boundary conditions between the two points. The method of Riccati equations employs the Riccati equations to express the relationships of two point boundary conditions. And then in terms of the relationships of boundary conditions at two points, the full conditions at initial point can be obtained and then the TPBVP can be transformed into initial value problem and then solved by direct time marching scheme. With some modifications, the above algorithms can be directly extended to the infinite-interval problem and the variant coefficient ODEs problem. In the program implementation, the object-oriented (OO) design of PTI is proposed to demonstrate the applicability and easy maintenance of OO techniques in numerical computation. Finally, four selected numerical examples are given to show the high precision characteristics of PTI.

Coefficients for studying one-step rational schemes for IVPs in ODEs: III. Extrapolation methods

This paper considers extrapolation of rational methods for the numerical solution of initial value problems (IVPs) in ordinary differential equations (ODEs). The extrapolation code, DIFEX2 of Fatunla [1], which is a modification of the automatic extrapolation code, DIFEXI of Deuflhard [2], is further modified to accommodate the basic integrators. The extrapolation method which we refer to as DIFEX2+ for reference purposes is compared with results from DIFEXI and DIFEX2, and GBS extrapolation method.

Detecting and locating a singular point in the numerical solution of IVPs for ODEs

Many authors have worked on approaches for solving Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) whose solutions contain one or more singular points within the interval of integration. Their approaches, however, assumed that the user knows in advance that the problem is singular. Hence they introduced new formulas to cope with this difficulty. In this paper, a new approach to detect and locate a singularity is suggested. This approach, which does not require the changing of the underlying formula, is comprised of two stages. The first is a preliminary singularity detection stage. The second stage is the confirmation stage which gathers more information about the existence and location of the singular point. We justify the first state and introduce three different techniques for confirming the existence of a singularity. The numerical results show that our approach is effective.