Showing papers on "Integrating factor published in 1974"
Book•
01 Jan 1974
TL;DR: In this paper, the authors define the notion of groups of transformations and prove that a one-parameter group essentially contains only one infinitesimal transformation and is determined by it.
Abstract: 1. Ordinary Differential Equations.- 1.0. Ordinary Differential Equations.- 1.1. Example: Global Similarity Transformation, Invariance and Reduction to Quadrature.- 1.2. Simple Examples of Groups of Transformations Abstract Definition.- 1.3. One-Parameter Group in the Plane.- 1.4. Proof That a One-Parameter Group Essentially Contains Only One Infinitesimal Transformation and Is Determined by It.- 1.5. Transformations Symbol of the Infinitesimal Transformation U.- 1.6. Invariant Functions and Curves.- 1.7. Important Classes of Transformations.- 1.8. Applications to Differential Equations Invariant Families of Curves.- 1.9. First-Order Differential Equations Which Admit a Group Integrating Factor Commutator.- 1.10. Geometric Interpretation of the Integrating Factor.- 1.11. Determination of First-Order Equations Which Admit a Given Group.- 1.12. One-Parameter Group in Three Variables More Variables.- 1.13. Extended Transformation in the Plane.- 1.14. A Second Criterion That a First-Order Differential Equation Admits a Group.- 1.15. Construction of All Differential Equations of First-Order Which Admit a Given Group.- 1.16. Criterion That a Second-Order Differential Equation Admits a Group.- 1.17. Construction of All Differential Equations of Second-Order Which Admit a Given Group.- 1.18. Examples of Application of the Method.- 2. Partial Differential Equations.- 2.0. Partial Differential Equations.- 2.1. Formulation of Invariance for the Special Case of One dependent and Two Independent Variables.- 2.2. Formulation of Invariance in General.- 2.3. Fundamental Solution of the Heat Equation Dimensional Analysis.- 2.4. Fundamental Solutions of Heat Equation Global Affinity.- 2.5. The Relationship Between the Use of Dimensional Analysis and Stretching Groups to Reduce the Number of Variables of a Partial Differential Equation.- 2.6. Use of Group Invariance to Obtain New Solutions from Given Solutions.- 2.7. The General Similarity Solution of the Heat Equation.- 2.8. Applications of the General Similarity Solution of the Heat Equation,.- 2.9. -Axially-Symmetric Wave Equation.- 2.10. Similarity Solutions of the One-Dimensional Fokker-Planck Equation.- 2.11. The Green's Function for an Instantaneous Line Particle Source Diffusing in a Gravitational Field and Under the Influence of a Linear Shear Wind - An Example of a P.D.E. in Three Variables Invariant Under a Two-Parameter Group.- 2.12. Infinite Parameter Groups - Derivation of the Poisson Kernel.- 2.13. Far Field of Transonic Flow.- 2.14. Nonlinear and Other Examples.- 2.15. Construction of Partial Differential Equations Invariant Under a Given Multi-parameter Group.- Appendix. Solution of Quasilinear First-Order Partial Differential Equations.- Bibliography. Part 1.- Bibliography. Part 2.
1,037 citations
178 citations
TL;DR: In this paper, a generalization of facts familiar for ordinary differential equations and retarded FDEs is presented, where the derivative may appear with a time lag, and a corresponding exponential dichotomy of the solutions of the homogeneous equation (2.4), the solutions in one subspace being O(e ǫlt as t --j + co (0~~ < a), while in a complementary subspace the solutions exist for all t < 0 and are O(eaat) as t -+ cc (aa > CX).
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TL;DR: In this paper, the authors consider the circumstances under which the two concepts are or are not equivalent, and show that strongly limit-point is not equivalent to the notion of self-adjoint extension.
Abstract: : The concept of limit-point for a formally self-adjoint ordinary differential operator of the second order is associated with the existence of a unique self-adjoint extension. A concept of strongly limit-point has also been defined, and this paper considers the circumstances under which the two concepts are or are not equivalent. (Author)
TL;DR: In this article, the local behavior near constant solutions of local integral manifolds have been determined in a systematic way, and a theorem on stability under constantly acting disturbances has been proved using these methods.
TL;DR: In this article, the existence theorems for boundary value problems with disconjugate differential operators of order n were established. But these results were used to establish comparison results for linear differential equations with order n.
Abstract: In this paper we present some existence theorems for boundary value problems for the equation $Ly = f(x,y)$, where L is a disconjugate differential operator of order n. These results are used to establish comparison theorems for linear differential equations of order n.
TL;DR: In this paper, it was shown that if (1) has an oscillatory solution, then there are two linearly independent oscillatory solutions of (1), whose zeros separate and such that any solution of ( 1) is a linear combination of them.
Journal Article•
TL;DR: In this article, Liapunov's direct method is applied to continuous differentiable functions, where the initial conditions depend continuously on initial conditions, and the solution of a solution depends on the initial condition.
Abstract: (1) x' = f(t, x) (' = djdt) where f : [0, oo) χ R->R is continuous and has the property that Solutions of (1) depend continuously on initial conditions. Often we will ask that f (i, 0) == 0 so that x(t) = 0 is a solution of (1) on [0, oo). In the classical theory of Liapunov's direct method, for a continuously differentiable function F: [0, oo) χ [0, oo), where S is a neighborhood of the origin in /?, one definesF' = grad F · f H—^—, so that if x(t) is a solution d of (1) then-^F(f,z(0)= V'(t,x(t)). If V is positive definite and V <£ 0, then the
01 Jan 1974
TL;DR: A step-size monitor is presented for use in numerically solving ordinary differential equations by extrapolation methods using the information present in the extrapolation lozenge to determine the “optimal” step- size and order.
Abstract: A step-size monitor is presented for use in numerically solving ordinary differential equations by extrapolation methods. The monitor uses the information present in the extrapolation lozenge to determine the “optimal” step-size and order. This allows the monitor to adjust both the order and step-size to the local behavior of the solution in a reasonably “optimal” fashion. The monitor is particularly useful when obtaining low-precision solutions which require radical step-size changes. The results of this monitor are compared quite favorably with previous proposals.
TL;DR: In this paper, the authors consider stabilization procedures, i.e. modifications of the original differential equations such that the solutions of the modified are Liapunov stable and thus the new system is better suited for numerical integration.
Abstract: For the solutions of a system of ordinary differential equations restricted Liapunov stability and the notion of Liapunov stable sets are discussed, in particular for separable and almost separable Hamiltonian systems. We consider stabilization procedures, i.e. modifications of the original differential equations such that the solutions of the modified are Liapunov stable and thus the new system is better suited for numerical integration. In this paper some theoretical aspects rather than numerical methods are pointed out.
TL;DR: In this paper, the stability properties of subsets of Rn were examined using a family of Liapunov functions and the invariance properties of the sets were analyzed using a set of invariant sets.
Abstract: The stability properties of subsets of Rn are examined using a family of Liapunov functions and the invariance properties of the sets.
TL;DR: In this paper, a system of ordinary differential equations with a small parameter in the neighborhood of a fixed solution was studied and a normal form for such a system was found, and it was shown that the formal integral manifold is not always analytic.
Abstract: In this paper we study a system of ordinary differential equations with a small parameter in the neighborhood of a fixed solution. We find a normal form for such a system. Then for the case of a small parameter and a single resonance we show that the formal integral manifold, found by V. I. Arnol'd (see Referativnyi Zhurnal Matematika, 8B678), is not always analytic. We discuss the conditions under which it is analytic.