Initial-Boundary Value Problems For A Class Of Nonlinear Elastic Beam Eqution

Nonlinear developing equation is a kind of representation of many nonlinear problems in mathematics. Recently, due to the development of mathematic and promotion of physics, chemistry, biology and other mechanics, the study of nonlinear developing equation have been become important subject in the mathematical research field. Since the complexity of forms of nonlinear developing equations, as a result, it is difficult or even impossible to acquire analytic solutions for most non-linear equations. So numerical methods have to be employed or properties of the equation are figured out based of the equation. However one frequently ignores the existence and uniqueness of solutions in the process of seeking numerical solutions. In doing so, rationality for simplifying an finite-dimensional system cannot be ensured; or even worse incorrect conclusions may result in. Consequently of nonlinear developing equations is a prerequisite and theoretic foundation for justifying numerical solutions. In view of this, in this thesis we will make research about the initical boundary value problems of a class of nonlinear developing equation-nonlinear elastic beam eqution by means of the Galerkin method in the Sobolev space, under the initial conditionsu(x,0)=u₀(x) , (u|·)(x,0)=u₁(x) (2)And the boundary conditionsu(0,t) = u(l,t) = u⁽²⁾(0,t)=u⁽²⁾(l,t)=0 (3)Where l>0 ,βis a arbitrary constant, k is a arbitrary positive number.The details will go as follows:Firstly, the current study situation about nonlinear elastic beam eqution is introduced.Secondly, we put forward some important definition and lemma, simultaneity explain some marks.Thirdly, we prove the existence and uniqueness of the weak solution of the problem (1)-(3) under the condtion:σ(s)∈c¹,σ'(s)≥c.Fourthly, we prove the existence of the strongly solution of the problem(1)-(3).