Energy bounds for some nonstandard problems in partial differential equations

Özer, Özge
This thesis is a survey of the studies of Ames,Payne and Schaefer about the partial differential equations with nonstandard auxiliary conditions; this is where the values of the solution are prescribed as a combination of initial time t=0 and at a later time t=T. The first chaper is introductory and contains some historical background of the problem,basic definitions and theorems.In Chapter 2 energy bounds and pointwise bounds for the solutions of the nonstandard hyperbolic problems have been investigated and by means of energy bound the uniqueness of solutions is examined. Similar discussions for the nonstandard parabolic problems have been presented in Chapter 3. Lastly in Chapter 4 a new continuous dependence result has been derived for the nonstandard problem.


Asymptotic integration of impulsive differential equations
Doğru Akgöl, Sibel; Ağacık, Zafer; Özbekler, Abdullah; Department of Mathematics (2017)
The main objective of this thesis is to investigate asymptotic properties of the solutions of differential equations under impulse effect, and in this way to fulfill the gap in the literature about asymptotic integration of impulsive differential equations. In this process our main instruments are fixed point theorems; lemmas on compactness; principal and nonprincipal solutions of impulsive differential equations and Cauchy function for impulsive differential equations. The thesis consists of five chapters....
Periodic solutions and stability of differential equations with piecewise constant argument of generalized type
Büyükadalı, Cemil; Akhmet, Marat; Department of Mathematics (2009)
In this thesis, we study periodic solutions and stability of differential equations with piecewise constant argument of generalized type. These equations can be divided into three main classes: differential equations with retarded, alternately advanced-retarded, and state-dependent piecewise constant argument of generalized type. First, using the method of small parameter due to Poincaré, the existence and stability of periodic solutions of quasilinear differential equations with retarded piecewise constant...
Periodic solutions and stability of linear impulsive delay differential equations
ALZabut, Jehad; Ağacık, Zafer; Department of Mathematics (2004)
In this thesis, we investigate impulsive differential systems with delays of the form And more generally of the form The dissertation consists of five chapters. The first chapter serves as introduction, contains preliminary considerations and assertions that will be encountered in the sequel. In chapter 2, we construct the adjoint systems and obtain the variation of parameters formulas of the solutions in terms of fundamental matrices. The asymptotic behavior of solutions of systems satisfying the Perron co...
Sturm comparison theory for impulsive differential equations
Özbekler, Abdullah; Ağacık, Zafer; Department of Mathematics (2005)
In this thesis, we investigate Sturmian comparison theory and oscillation for second order impulsive differential equations with fixed moments of impulse actions. It is shown that impulse actions may greatly alter the oscillation behavior of solutions. In chapter two, besides Sturmian type comparison results, we give Leightonian type comparison theorems and obtain Wirtinger type inequalities for linear, half-linear and non-selfadjoint equations. We present analogous results for forced super linear and super...
Boundary value problems for higher order linear impulsive differential equations
Uğur, Ömür; Akhmet, Marat; Taşeli, Hasan; Department of Mathematics (2003)
The theory of impulsive di®erential equations has become an important area of research in recent years. Linear equations, meanwhile, are fundamental in most branches of applied mathematics, science, and technology. The theory of higher order linear impulsive equations, however, has not been studied as much as the cor- responding theory of ordinary di®erential equations. In this work, higher order linear impulsive equations at xed moments of impulses together with certain boundary conditions are investigated...
Citation Formats
Ö. Özer, “Energy bounds for some nonstandard problems in partial differential equations,” M.S. - Master of Science, Middle East Technical University, 2005.