Showing papers on "Integrating factor published in 1989"
Book•
01 Feb 1989
437 citations
Proceedings Article•
01 Dec 1989
93 citations
17 Jul 1989
TL;DR: The reducibility and factorization of linear homogeneous differential equations are of great theoretical and practical importance and their use in an automatic differential equation solver requires a more detailed analysis of the various steps involved.
Abstract: The reducibility and factorization of linear homogeneous differential equations are of great theoretical and practical importance in mathematics. Although it has been known for a long time that factorization is in principle a decision procedure, its use in an automatic differential equation solver requires a more detailed analysis of the various steps involved. Especially important are certain auxiliary equations, the so-called associated equations. An upper bound for the degree of its coefficients is derived. Another important ingredient is the computation of optimal estimates for the size of polynomial and rational solutions of certain differential equations with rational coefficients. Applying these results, the design of the factorization algorithm LODEF and its implementation in the Scratchpad II Computer Algebra System is described.
80 citations
TL;DR: In this paper, Cartan's equivalence method is used to study the differential invariants of a single second order ordinary differential equation relative to the pseudo-group of point transformations, and a simple characterization is given of those second order equations which are linearizable by a point transformation.
74 citations
TL;DR: An existence and uniqueness theorem for solutions of fuzzy differential equations is given and it is shown that there is no such thing as a "f fuzzy solution" to these equations.
60 citations
57 citations
54 citations
44 citations
TL;DR: Interval algorithms as mentioned in this paper compute an interval valued function in which the solution to a system of ordinary differential equations is guaranteed to lie, and use differential inequalities, finite difference approximations with remainders, Taylor series, defect corrections, or contractive iterations.
01 Apr 1989
TL;DR: Oscillation criteria for the second-order nonlinear differential equation x" + a(t)Ix 17sgnx = 0 y' 6 1 were studied in this article.
Abstract: Oscillation criteria for the second-order nonlinear differential equation x" + a(t)Ix 17sgnx = 0 y' 6 1 , are studied where the coefficient a(t) is not assumed to be non-negative New proofs are given to theorems of Butler, and extend earlier results of the author
01 Feb 1989
TL;DR: The proposed arithmetic mean method for solving numerically large sparse sets of linear ordinary differential equations has second-order accuracy in time and is stable and has been tested on the CRAY X-MP/48 utilizing two CPUs.
Abstract: This paper is concerned with the development, analysis and implementation on a computer consisting of two vector processors of the arithmetic mean method for solving numerically large sparse sets of linear ordinary differential equations. This method has second-order accuracy in time and is stable. The special class of differential equations that arise in solving the diffusion problem by the method of lines is considered. In this case, the proposed method has been tested on the CRAY X-MP/48 utilizing two CPUs. The numerical results are largely in keeping with the theory; a speedup factor of nearly two is obtained.
TL;DR: In this paper, a quasilinear second order ordinary differential equation with a small parameter is considered and an appropriate problem is constructed, and an iterative procedure is developed.
Abstract: In this paper we consider a quasilinear second order ordinary differential equation with a small parameter e. Firstly an appropriate problem is constructed. Then an iterative procedure is developed. Finally we give an algorithm whose accuracy is good for arbitrary e>0.
01 Jan 1989
TL;DR: In this article, the existence and uniqueness of boundary value problems for the second-order differential equation in a critical case is proved by using the method of upper and lower solutions, and the boundary value problem with a parameter is investigated.
Abstract: Existence and uniqueness of the solution to some boundary value problems for the second-order differential equation in a critical case is proved by using the method of upper and lower solutions. Further boundary value problems with a parameter are investigated.
TL;DR: In this paper, the Hermite, Laguerre, Legendre, and Chebyshev Equations are studied for which polynomial solutions exist, and these solutions turn out to be generalizations of well-known polynomials.
Abstract: Several classes of ordinary differential equations which have polynomial solutions are studied. In particular, generalizations of the Hermite, Laguerre, Legendre, and Chebyshev equations are given for which such solutions exist. These solutions turn out to be generalizations of well‐known polynomials and enjoy similar properties.
TL;DR: In this article, the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition, and the general computation format is obtained by this method, and its convergence is proved.
Abstract: Many engineering problems can be reduced to the solution of a variable coefficient differential equation. In this paper, the exact analytic method is suggested to solve variable coefficient differential equations under arbitrary boundary condition. By this method, the general computation format is obtained. Its convergence is proved. We can get analytic expressions which converge to exact solution and its higher order derivatives unifornuy. Four numerical examples are given, which indicate that satisfactory results can be obtained by this method.
TL;DR: In this article, the existence of solutions of polynomial growth of ordinary differential equations of type E: dY dX = F(X, Y), where F is of class C 1.
TL;DR: In this paper, the possibility of stable reduction of linear ODEs to other problems was considered. But the results were restricted to systems of linear ordinary differential equations, with regard to their stable reduction to non-linear ODE problems.
Abstract: Certain multipoint problems for systems of linear ordinary differential equations are considered, with regard to the possibility of their stable reduction to other problems.
TL;DR: In this paper, the first integrals of some classes of non-linear equations of the form y + p(t)y + q(t), n ≠ 1 are derived by a direct and simple procedure, where the choice of integrating factors depends on the constraints for certain solutions of the same equation.
Abstract: The first integrals of some classes of non-linear equations of the form y + p(t)y + q(t)y = ƒ(t)yn, n ≠ 1 are derived by a direct and simple procedure. It is shown that the choice of integrating factors depends on the constraints for certain solutions of the same equation. Several known results in the literature are deduced as particular cases. Further, some differential equations involving quadratic non-linearities are discussed. The sufficient conditions for the class of equations y + p(t) y + q(t)y = − y 2 y −1 + ƒ(t)y −1 to be completely integrable and the corresponding integrals are also derived. A few illustrative examples are discussed.
01 Jan 1989
Journal Article•
TL;DR: In this paper, the method of differential inequalities has been applied to study the boundary value problems of nonlinear ordinary differential equation with two parameters, and the asymptotic solutions have been found and the remainders have been estimated.
Abstract: In this paper, the method of differential inequalities has been applied to study the boundary value problems of nonlinear ordinary differential equation with two parameters. The asymptotic solutions have been found and the remainders have been estimated.