scholarly journals A Singular Initial-Value Problem for Second-Order Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Afgan Aslanov
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Local Existence ◽  
Second Order ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Existence Of A Solution ◽  
Local Existence And Uniqueness ◽  
Singular Initial Value Problem

We are interested in the existence of solutions to initial-value problems for second-order nonlinear singular differential equations. We show that the existence of a solution can be explained in terms of a more simple initial-value problem. Local existence and uniqueness of solutions are proven under conditions which are considerably weaker than previously known conditions.

scholarly journals Singular Initial Value Problem for Certain Classes of Systems of Ordinary Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Josef Diblík ◽  
Josef Rebenda ◽  
Zdeněk Šmarda
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
Singular Initial Value Problem ◽  
The Right

The paper is devoted to the study of the solvability of a singular initial value problem for systems of ordinary differential equations. The main results give sufficient conditions for the existence of solutions in the right-hand neighbourhood of a singular point. In addition, the dimension of the set of initial data generating such solutions is estimated. An asymptotic behavior of solutions is determined as well and relevant asymptotic formulas are derived. The method of functions defined implicitly and the topological method (Ważewski's method) are used in the proofs. The results generalize some previous ones on singular initial value problems for differential equations.

MATHEMATICAL ANALYSIS OF A REACTIVE-DIFFUSIVE MODEL OF THE DISPERSAL OF A CHEMICAL TRACER WITH NONLINEAR CONVECTION

1995 ◽  
Vol 05 (01) ◽  
pp. 29-46 ◽  
Author(s):  
STEVE COHN ◽  
J. DAVID LOGAN
Keyword(s):  
Initial Value Problem ◽  
Local Existence ◽  
Heteroclinic Orbit ◽  
Traveling Wave Solutions ◽  
Sobolev Embedding ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Contraction Mapping Theorem ◽  
Weakly Nonlinear ◽  
Local Existence And Uniqueness

We formulate and analyze a nonlinear reaction-convection-diffusion system that models the dispersal of solutes, or chemical tracers, through a one-dimensional porous medium. A similar set of model equations also arises in a weakly nonlinear limit of the combustion equations. In particular, we address two fundamental questions with respect to the model system: first, the existence of wavefront type traveling wave solutions, and second, the local existence and uniqueness of solutions to the pure initial value problem. The solution to the wavefront problem is obtained by showing the existence of a heteroclinic orbit in a two-dimensional phase space. The existence argument for the initial value problem is based on the contraction mapping theorem and Sobolev embedding. In the final section we prove non-negativity of the solution.

scholarly journals Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

2018 ◽  
Vol 5 (1) ◽  
pp. 102-112 ◽  
Author(s):  
Shekhar Singh Negi ◽  
Syed Abbas ◽  
Muslim Malik
Keyword(s):  
Time Scales ◽  
Initial Value Problem ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Type Inequality ◽  
Second Order ◽  
Oscillation Criterion ◽  
Initial Value ◽  
Nonlinear Dynamic Equation ◽  
Singular Initial Value Problem

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

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