Browsing Mathematics by Author "Alqifiary, Qusuay Hatim Eghaar"
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Alqifiary, Qusuay Hatim Eghaar (Beograd , 2015)[more][less]
Abstract: This thesis has been written under the supervision of my mentor Prof. dr. Julka Knezevi cMiljanovi c at the University of Belgrade in the academic year 20142015. The aim of this study is to investigate HyersUlam stability of some types of differential equations, and to study a generalized HyersUlam stability and as well as a special case of the HyersUlam stability problem, which is called the superstability. Therefore, when there is a differential equation, we answer the three main questions: 1 Does this equation have Hyers Ulam stability? 2 What are the conditions under which the differential equation has stability ? 3 What is a HyersUlam constant of the differential equation? The thesis is divided into three chapters. Chapter 1 is divided into 3 sections. In this chapter, we introduce some sufficient conditions under which each solution of the linear differential equation u′′(t) + ( 1 + (t) ) u(t) = 0 is bounded. Apart from this we prove the HyersUlam stability of it and the nonlinear differential equations of the form u′′(t) + F(t; u(t)) = 0, by using the Gronwall lemma and we prove the HyersUlam stability of the secondorder linear differential equations with boundary conditions. In addition to that we establish the superstability of linear differential equations of secondorder and higher order with continuous coefficients and with constant coefficients, respectively. Chapter 2 is divided into 2 sections. In this chapter, by using the Laplace transform method, we prove that the linear differential equation of the nthorder y(n)(t) + nΣ1 k=0 ky(k)(t) = f(t) has the generalized HyersUlam stability. And we prove also the HyersUlam Rassias stability of the secondorder linear differential equations with initial and boundary conditions, as well as linear differential equations of higher order in the form of y(n)(x) + (x)y(x) = 0, with initial conditions. Furthermore, we establish the generalized superstability of differential equations of nthorder with initial conditions and investigate the generalized superstability of differential equations of secondorder in the form of y′′(x)+p(x)y′(x)+q(x)y(x) = 0. Chapter 3 is divided into 2 sections. In this chapter, by applying the xed point alternative method, we give a necessary and sufficient condition in order that the rst order linear Alqi ary Abstract ii system of differential equations z_(t) + A(t)z(t) + B(t) = 0 has the HyersUlam Rassias stability and nd HyersUlam stability constant under those conditions. In addition to that, we apply this result to a secondorder differential equation y (t) + f(t)y_(t) + g(t)y(t) + h(t) = 0. Also, we apply it to differential equations with constant coefficient in the same sense of proofs. And we give a sufficient condition in order that the rst order nonlinear system of differential equations has HyersUlam stability and HyersUlamRassias stability. In addition, we present the relation between practical stability and HyersUlam stability and also Hyers UlamRassias stability. URI: http://hdl.handle.net/123456789/4295 Files in this item: 1
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