Soluci´on num´erica de un modelo de ondas localmente amortiguadas

1 - INTRODUCTION

The linear systems of partial differential equations of the first order with the identical main partsis considered. Аpplying the well-known relation between a normal system of ordinary differentialequations and a linear system of partial differential equations of the first order with the samemain parts, the existence of integral basis of a linear inhomogeneous system of partial differentialequations of the first order adjoining to some solution of the same linear inhomogeneous systemof differential equations with partial derivatives of the first order is proved. A sign at which thenonlinear system of ordinary differential equations has a neighborhood such that any solution withinitial values from it tends to zero is found. Using the equivalence of a linear system of partialdifferential equations of the first order with identical main parts to a linear differential equationwith partial derivatives of the first order, the existence of integral basis of the adjoining to zerolinear homogeneous system of partial differential equations of first order with nonlinear coefficientsis shown.

The dynamics of time-delay systems are governed by delay differential equations, which are infinite dimensional and can pose computational challenges. Several methods have been proposed for studying the stability characteristics of delay differential equations. One such method employs Galerkin approximations to convert delay differential equations into partial differential equations with boundary conditions; the partial differential equations are then converted into systems of ordinary differential equations, whereupon standard ordinary differential equation methods can be applied. The Galerkin approximation method can be applied to a second-order delay differential equation in two ways: either by converting into a second-order partial differential equation and then into a system of second-order ordinary differential equations (the “second-order Galerkin” method) or by first expressing as two first-order delay differential equations and converting into a system of first-order partial differential equations and then into a first-order ordinary differential equation system (the “first-order Galerkin” method). In this paper, we demonstrate that these subtly different formulation procedures lead to different roots of the characteristic polynomial. In particular, the second-order Galerkin method produces spurious roots near the origin, which must then be identified through substitution into the characteristic polynomial of the original delay differential equation. However, spurious roots do not arise if the first-order Galerkin method is used, which can reduce computation time and simplify stability analyses. We describe these two formulation strategies and present numerical examples to highlight their important differences.

Chapter 11 - Applications of Systems of Ordinary Differential Equations

The use of sparse matrix technique in the numerical integration of stiff systems of linear ordinary differential equations

This thesis investigates existence results for boundary value problems for systems of second-order ordinary differential equations with impulses, for the vector ordinary p-Laplacian and also for the finite difference equations arising from the vector ordinary p-Laplacian. The thesis begins with systems of second-order ordinary differential equations x''=f(t,x,x') subject to nonlinear boundary conditions and nonlinear impulses. Many papers on boundary value problems involving this differential equation have been published, but some made strong assumptions on f or presented complicated proofs. Additionally, very few authors have explored the theory of impulsive boundary value problems for such equations, although these problems have a large number of applications in the modelling of physical problems such as mechanical systems with impact. In this thesis, we extend the notion of admissible bounding set O which is a subset of [0,1] x {R}d, of the Hartman-Nagumo conditions and of compatibility to show the problem has a solution xnwith (t,x(t)) enO. To establish the existence, we reformulate the problem as an equivalent nonlinear equation and homotope it to a new nonlinear equation. This equation is equivalent to a simple linear system of ordinary differential equations subject to Dirichlet boundary conditions and impulses. We show that there exists a unique solution to this problem. Then, by the Leray Index Theorem and homotopy invariance, we show that the degree of the nonlinear equation is non-zero, and hence establish existence results for the original problem. Our proof is simpler and requires weaker assumptions on f than those of earlier works closely related to this problem. More importantly, a key technique used in our proof offers a fresh starting point for future research to establish the most natural and general existence results for boundary value problems. Next, we study the Dirichlet problem for the vector ordinary p-Laplacian (║x'║p-2x')] = f(t,x,x').We establish a general result for a broader class of fnthan earlier works about this problem by imposing fewer assumptions on f. We extend the concept of admissible bounding set and introduce the notion of a p-admissible bounding set O to guarantee a priori bounds on potential solutions. Moreover, we introduce the p-Mawhin-Urena-Nagumo condition and the p-Hartman-Nagumo condition to obtain a priori bounds on the derivatives of solutions. We apply our first result to establish the existence results. For p, we turn the vector ordinary p-Laplacian into an equivalent system of second-order ordinary differential equations to prove existence. For pnwe approximate the p-Laplacian and turn the approximation into a system of second-order ordinary differential equations. Then using our first result, we show solutions to the approximation problem exist and converge to solutions to the original problem. Finally, we investigate boundary value problems for systems of finite difference equations arising from discretizing the vector ordinary p-Laplacian. Many papers have presented important results for discrete p-Laplacian problems that do not arise from discretization of the vector ordinary p-Laplacian. However, there appears to have been little research done on the discrete problems arising from the discretization of boundary value problems for the vector ordinary p-Laplacian. This thesis fills this gap in the literature. We establish a more general result than previous works do by imposing weaker assumptions on fnand by removing a strong assumption. Existence and convergence theorems follow, once the assumptions of our second result and the compatibility of boundary conditions are introduced.

In this thesis we investigate the existence of solutions to boundary value problems (BVPs) for nonlinear systems of first-order ordinary differential equations (ODEs) and difference equations. The thesis is divided into two parts. Part one is devoted to the study of difference equations where two types of problems are studied. First, we derive the existence of solutions to first-order systems of difference equations that arise when one applies the trapezoidal rule to approximate solutions of second-order scalar ODEs. Strict discrete lower solutions are used with maximum principle arguments in the discrete problem to obtain a priori bounds on solutions. The a priori bounds on difference quotients for solutions of the discrete problem are obtained from the Bernstein-Nagumo condition. We use homototopy to find solutions of the discrete amroximations. Second, motivated by Ma [60] we consider existence and uniqueness of solutions to difference equations which arise as discrete approximations to three-point BVPs for systems of first-order ODES after employing the Euler method. Several existence and uniqueness results for three-point BVPs are established, using the contraction mapping theorem and the Brouwer fixed point theorem in Rn. In part two, two types of problems for systems of first-order ODES are analyzed. First, we study the existence of a unique solution to a system of first-order differentia1 equations under multi-point BVPs with nonlinear boundary conditions. We prove the existence and uniqueness of solutions for a system of first-order three-point BVP under which the properties of linear three-point BVP are preserved under small nonlinear perturbations of both the differentia1 equation and the boundary conditions. The method of proof of the existence of solutions using the Banach contraction mapping principle is similar to that used by Rodriguez [75], who studied the discrete analogue of this BVP. Second, motivated by Cronin 120,211 we investigate the existence of solutions to three-point BVPs in perturbed systems of first-order ODEs at resonance where the associated homogeneous linear problem has nontrivial solutions. Based on the second part of the investigation of three-point BVPs in perturbed systems of first-order ODEs at resonance, four different results were obtained: (i) The existence of solutions via a version of the Brouwer Fixed Point Theorem which is due to Miranda. (ii) The existence of solutions to the problem through the application of the Implicit Function Theorem. These two results extend the work of Feng and Webb [34], and Gupta [39] for the problems at resonance in Euclidean 2-space. (iii) Existence results for three-point BVPs at resonance for general BVPs through the application of Brouwer degree theory; we show the degree is non zero through the application of Borsuk's Theorem for one problem and through a result of Cronin involving pairs of polynomials in two variables whose terms of highest order have no common factors. (iv) The existence of solutions to the entrainment of frequency problem in Euclidean 2-space. We will investigate conditions under which entrainment of frequency for three-point as well as two-point BVPs by adapting the work of Cronin in periodic case. We will describe such conditions and then show the degree is non zero through a result of Cronin involving pairs of polynomials in two variables whose terms of lowest order have no common factors.

APPROXIMATE SOLUTION OF ELLIPTIC BOUNDARY VALUE PROBLEMS BY SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS

Study of new, not previously observed, processes of formation of axisymmetric regularly shaped craters in permafrost zones requires the creation of models to explain the occurrence of such anomalous phenomena. In this paper, the problem of thermal destruction of the vertical channel (a well) mainly composed of ice and gas-liquid flow is considered. In the developed mathematical model it is assumed that a hot gas is supplied to the channel inlet. When the hot gas moves through the channel, part of its energy is transferred to channel walls, causing thus their thermal destruction. The high-pressure decomposition products of the channel (water and rock) are carried by the flow to the surface. A system of first-order ordinary differential equations is constructed to identify the main parameters of the system “channel-gas flow”: pressure, temperature and flow rate, as well as mass flow and its components (water and rock). Numerical implementation of the upward flow in a vertical borehole consists of two phases. In the first stage, we solve the resulting system of ordinary differential equations using the Runge-Kutta fourth-order method, where the initial values of flow velocity are determined via the method of shooting. With this method, the inlet velocity is taken so that the maximum outlet velocity does not exceed the sound speed at a given local pressure, and the pressure at the end of the channel remains almost the same as the atmospheric one. The parameters of the system of ordinary differential equations are calculated in this case for a fixed radius of the well and its thermal effect. In the second stage, the destruction of channel walls is described. For a given distribution of flow parameters, the time step is performed, and the problem of determining the radii of the borehole and thermal effects is solved. The solution was built using the equation in quasistationary approximation. Simulations give the critical well radius, at which the flow regimes change. The dynamics of parameter changes in the well during its thermal destruction is demonstrated. It has been found that with increasing channel radius the intensity of the channel destruction increases.

We show that systems of second-order ordinary differential equations, , subject to compatible nonlinear boundary conditions and impulses, have a solution x such that lies in an admissible bounding subset of when f satisfies a Hartman-Nagumo growth bound with respect to . We reformulate the problem as a system of nonlinear equations and apply Leray-Schauder degree theory. We compute the degree by homotopying to a new system of nonlinear equations based on the simpler system of ordinary differential equations, , subject to Picard boundary conditions and impulses and using the Leray index theorem. Our proof is simpler than earlier existence proofs involving nonlinear boundary conditions without impulses and requires weak assumptions on f. MSC:34A37, 34A34, 34B15.

We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.

Solving multivariate ordinary differential equations (ODE) systems and partial differential equations (PDE) systems is the key to many complex physics and chemistry problems, such as the combustion in process of reacting flow. However, the traditional numerical methods in solving multivariate ODE and PDE systems are limited by computational cost, and sometimes its impossible to obtain the solution due to the high stiffness of ODE or PDE. Coincident with the development of machine learning has been a growing appreciation of applying neural networks in solving physics models. DeepM&M net was proposed to address complicated problems in fluid mechanics based on another neural network: DeepONet, which is used to predict functional nonlinear operators. Inspired by these two nets, a machine learning way of solving certain ODE and PDE systems is proposed with a similar framework to the DeepM&M net, which takes inputs of the initial conditions and outputs the corresponding solutions. The main ideas of this framework are first to explore the relations among solutions of the system by DeepONets and then to train a deep neural network with the assistance of trained DeepONets. The implicit operators between variables in certain ODE systems are verified to have existed and are well predicted by the DeepONet. The feasibility of the proposed framework is implied by the success in building blocks.

The automatic solution to systems of ordinary differential equations by the tau method

Polynomial Models of Yield Term Structure

The system of ordinary differential equations has many uses in contemporary mathematics and engineering. Finding the numerical solution to a system of ordinary differential equations for any arbitrary interval is very appealing to researchers. The numerical solution of a system of fourth-order ordinary differential equations on any finite interval [a,b] is found in this work using a symmetric Bernstein approximation. This technique is based on the operational matrices of Bernstein polynomials for solving the system of fourth-order ODEs. First, using Chebyshev collocation nodes, a generalised approximation of the system of ordinary differential equations is discretized into a system of linear algebraic equations that can be solved using any standard rule, such as Gaussian elimination. We obtain the numerical solution in the form of a polynomial after obtaining the unknowns. The Hyers–Ulam and Hyers–Ulam–Rassias stability analyses are provided to demonstrate that the proposed technique is stable under certain conditions. The results of numerical experiments using the proposed technique are plotted in figures to demonstrate the accuracy of the specified approach. The results show that the suggested Bernstein approximation method for any interval is quick and effective.