Abstract
In this paper, we study a new type of modeling of an Earthquake phenomenon, a mechanical model of the earthquake process in one-dimension using usual mathematical functions, the latter leads to the study of nonlinear integro-differential equation of Volterra. The existence and the uniqueness of the solution are proved. Using Nystrom method is builded to approximate the solution. The numerical tests show the effectiveness of this type of modeling.
Highlights
Thereby, we study this physical experience, we propose a mathematical modeling that describes the movement of earthquake machine, which is presented in the form of a non-linear integro-differential equation of Volterra of the second kind u(t) =
The numerical tests developed in this paper show our good vision on mathematical modeling, analytical and numerical plans
Our study allows us to solve a rather complicated problem where the numerical results show their efficiency and accuracy. To show this efficiency and precision of the proposed method in this paper, and to illustrate the approximation performance of our modeling of the mechanical model as well as the solving of the integro-differential nonlinear equation of Volterra (1.16), we complete the study with results obtained from an experimental simulation of Earthquake machine, using the Nystrom approximation method proved in the previous section
Summary
The classical method that is used to prove the existence and uniqueness of the solution of the equation (1.16) is based on the construction of two successive sequences {un(t)}n∈N, {φn(t)}n∈N. This method is called the Picard method. < δn = T such that, for 0 ≤ i ≤ n, and for t ∈ [δi, δi+1], the equation (1.16) has a unique continuous solution in C1(0, T ). Φn, u′n and φ′n as in (2.2)-(2.5), we can show by induction, for n ∈ N∗. To prove that u satisfies the original equation (1.16), we put We have u(t) = un(t) + ∆n(t), u′(t) = u′n(t) + ∆′n(t). This argument can be repeated and since there is only a finite number of subintervals in [0, T ], we thereby construct the unique solution in C1(0, T )
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