First-Order System Least Squares for Second-Order Partial Differential Equations: Part I

1994 ◽  
Vol 31 (6) ◽  
pp. 1785-1799 ◽  
Author(s):  
Z. Cai ◽  
R. Lazarov ◽  
T. A. Manteuffel ◽  
S. F. McCormick
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Least Squares ◽  
Second Order ◽  
Order System ◽  
Partial Differential ◽  
First Order

scholarly journals XV. On a general method in dynamics; by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function

1834 ◽  
Vol 124 ◽  
pp. 247-308 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Characteristic Function ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Mathematical Science ◽  
Partial Differential ◽  
First Order ◽  
Present Essay

The theoretical development of the laws of motion of bodies is a problem of such interest and importance, that it has engaged the attention of all the most eminent mathematicians, since the invention of dynamics as a mathematical science by Galileo, and especially since the wonderful extension which was given to that science by Newton. Among the successors of those illustrious men, Lagrange has perhaps done more than any other analyst, to give extent and harmony to such deductive researches, by showing that the most varied consequences respecting the motions of systems of bodies may be derived from one radical formula; the beauty of the method so suiting the dignity of the results, as to make of his great work a kind of scientific poem. But the science of force, or of power acting by law in space and time, has undergone already another revolution, and has become already more dynamic, by having almost dismissed the conceptions of solidity and cohesion, and those other material ties, or geometrically imaginable conditions, which Lagrange so happily reasoned on, and by tending more and more to resolve all connexions and actions of bodies into attractions and repulsions of points: and while the science is advancing thus in one direction by the improvement of physical views, it may advance in another direction also by the invention of mathematical methods. And the method proposed in the present essay, for the deductive study of the motions of attracting or repelling systems, will perhaps be received with indulgence, as an attempt to assist in carrying forward so high an inquiry. In the methods commonly employed, the determination of the motion of a free point in space, under the influence of accelerating forces, depends on the integration of three equations in ordinary differentials of the second order; and the determination of the motions of a system of free points, attracting or repelling one another, depends on the integration of a system of such equations, in number threefold the number of the attracting or repelling points, unless we previously diminish by unity this latter number, by considering only relative motions. Thus, in the solar system, when we consider only the mutual attractions of the sun and of the ten known planets, the determination of the motions of the latter about the former is reduced, by the usual methods, to the integration of a system of thirty ordinary differential equations of the second order, between the coordinates and the time; or, by a transformation of Lagrange, to the integration of a system of sixty ordinary differential equations of the first order, between the time and the elliptic elements: by which integrations, the thirty varying coordinates, or the sixty varying elements, are to be found as functions of the time. In the method of the present essay, this problem is reduced to the search and differentiation of a single function, which satisfies two partial differential equations of the first order and of the second degree: and every other dynamical problem, respecting the motions of any system, however numerous, of attracting or repelling points, (even if we suppose those points restricted by any conditions of connexion consistent with the law of living force,) is reduced, in like manner, to the study of one central function, of which the form marks out and characterizes the properties of the moving system, and is to be determined by a pair of partial differential equations of the first order, combined with some simple considerations. The difficulty is therefore at least transferred from the integration of many equations of one class to the integration of two of another: and even if it should be thought that no practical facility is gained, yet an intellectual pleasure may result from the reduction of the most complex and, probably., of all researches respecting the forces and motions of body, to the study of one characteristic function, the unfolding of one central relation.

scholarly journals IV. On simultaneous partial differential equations

1901 ◽  
Vol 195 (262-273) ◽  
pp. 151-191 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Single Equation ◽  
Second Order ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Two Independent Variables

In this paper, without touching on the question of the existence of integrals of systems of simultaneous partial differential equations, I have given a method by which the problem of finding their complete primitives may he attacked. The cases discussed are two: that of a pair of equations of the first order in two dependent and two independent variables, and that of a single equation of the second order, with one dependent and two independent variables.

Spurious roots of delay differential equations using Galerkin approximations

2020 ◽  
Vol 26 (15-16) ◽  
pp. 1178-1184
Author(s):  
Shanti S Kandala ◽  
Surya Samukham ◽  
Thomas K Uchida ◽  
C. P. Vyasarayani
Keyword(s):  
Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Galerkin Method ◽  
Delay Differential Equations ◽  
Second Order ◽  
Partial Differential ◽  
First Order ◽  
Galerkin Approximations ◽  
Delay Differential

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.

scholarly journals I. Memoir on the integration of partial differential equations second order in three independent variables when an intermediary integral does not exist in general

1898 ◽  
Vol 191 ◽  
pp. 1-86 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Second Order ◽  
Original Equation ◽  
Partial Differential ◽  
First Order ◽  
Independent Variables ◽  
Dependent Variables ◽  
Independent Variable ◽  
Two Independent Variables

The general feature of most methods for the integration of partial differential equations in two independent variables is, in some form or other, the construction of a set of subsidiary equations in only a single independent variable; and this applies to all orders. In particular, for the first order in any number of variables (not merely in two), the subsidiary system is a set of ordinary equations in a single independent variable, containing as many equations as dependent variables to be determined by that subsidiary system. For equations of the second order which possess an intermediary integral, the best methods (that is, the most effective as giving tests of existence) are those of Boole, modified and developed by Imschenetsky, and that of Goursat, initially based upon the theory of characteristics, but subsequently brought into the form of Jacobian systems of simultaneous partial equations of the first order. These methods are exceptions to the foregoing general statement. But for equations of the second order or of higher orders, which involve two independent variables and in no case possess an intermediary integral, the most general methods are that of Ampere and that of Darboux, with such modifications and reconstruction as have been introduced by other writers; and though in these developments partial differential equations of the first order are introduced, still initially the subsidiary system is in effect a system with one independent variable expressed and the other, suppressed during the integration, playing a parametric part. In oilier words, the subsidiary system practically has one independent variable fewer than the original equation. In another paper I have given a method for dealing with partial differential equations of the second order in three variables when they possess an intermediary integral; and references will there be found to other writers upon the subject. My aim in the present paper has been to obtain a method for partial differential equations of the second order in three variables when, in general, they possess no intermediary integral. The natural generalisation of the idea in Darboux’s method has been adopted, viz., the construction of subsidiary equations in which the number of expressed independent variables is less by unity than the number in the original equation; consequently the number is two. The subsidiary equations thus are a set of simultaneous partial differential equations in two independent variables and a number of dependent variables.

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