Necessary and Sufficient Conditions for the Numerical Approximation of a Partial Differential Equation Depending on a Small Parameter

2016 ◽  
pp. 161-168 ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Small Parameter ◽  
Numerical Approximation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Partial Differential ◽  
Necessary And Sufficient

scholarly journals Inextensible Flows of Spacelike Curves According to Equiform Frame in 4-dimensional Minkowski Space R41

2021 ◽  
Vol 19 ◽  
pp. 683-698
Author(s):  
W. M. Mahmoud ◽  
Alaa Hassan Noreldeen
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Minkowski Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Spacelike Surface ◽  
Partial Differential ◽  
Dimensional Minkowski Space ◽  
Inextensible Flows ◽  
Necessary And Sufficient

In this paper, we study inextensible flows of spacelike curves lying fully on a spacelike surface Ω according to equiform frame in 4-dimensional Minkowski space ℝ1 4 . We give necessary and sufficient conditions for this inextensible flows which are expressed as a partial differential equation involving the equiform curvature functions in 4-dimensional Minkowski space ℝ1 4 . Finally we give an application of inextensible flows of spacelike curves in ℝ1 4 .

Symmetry-based algorithms to relate partial differential equations: I. Local symmetries

1990 ◽  
Vol 1 (3) ◽  
pp. 189-216 ◽  
Author(s):  
G. W. Bluman ◽  
S. Kumei
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Linear Partial Differential Equation ◽  
Variable Coefficients ◽  
Necessary And Sufficient Conditions ◽  
Partial Differential ◽  
Flow Equations ◽  
Local Symmetries ◽  
Necessary And Sufficient

Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV equations.

scholarly journals Hypoelliptic differential operators with generalized constant coefficients

1998 ◽  
Vol 41 (1) ◽  
pp. 47-60 ◽  
Author(s):  
M. Nedeljkov ◽  
S. Pilipović
Keyword(s):  
Small Parameter ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Necessary And Sufficient Conditions ◽  
Order Estimate ◽  
Colombeau Generalized Functions ◽  
Constant Coefficients ◽  
Necessary And Sufficient

The space of Colombeau generalized functions is used as a frame for the study of hypoellipticity of a family of differential operators whose coefficients depend on a small parameter ε.There are given necessary and sufficient conditions for the hypoellipticity of a family of differential operators with constant coefficients which depend on ε and behave like powers of ε as ε→0. The solutions of such family of equations should also satisfy the power order estimate with respect to ε.

scholarly journals Distributional properties of solutions of dVt = Vt-dUt + dLt with Lévy noise

2011 ◽  
Vol 43 (3) ◽  
pp. 688-711 ◽  
Author(s):  
Anita Diana Behme
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Autocorrelation Function ◽  
Sufficient Conditions ◽  
Stationary Solutions ◽  
Necessary And Sufficient Conditions ◽  
Tail Behavior ◽  
Absolutely Continuous ◽  
Distributional Properties ◽  
Necessary And Sufficient

For a given bivariate Lévy process (Ut, Lt)t≥0, distributional properties of the stationary solutions of the stochastic differential equation dVt = Vt-dUt + dLt are analysed. In particular, the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described. In the case where U has jumps of size −1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given.

Extremal Positive Solutions of Semilinear Schrödinger Equations

1983 ◽  
Vol 26 (2) ◽  
pp. 171-178 ◽  
Author(s):  
C. A. Swanson
Keyword(s):  
Differential Equation ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Exterior Domains ◽  
Hölder Continuous ◽  
Semilinear Differential Equation ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions are proved for the existence of maximal and minimal positive solutions of the semilinear differential equation Δu = -ƒ(x, u) in exterior domains of Euclidean n-space. The hypotheses are that ƒ(x, u) is nonnegative and Hölder continuous in both variables, and bounded above and below by ugi(| x |, u), i = 1, 2, respectively, where each gi(r, u) is monotone in u for each r > 0.

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