Solution of an Integro-differential Nonlinear Equation of Volterra Arising of Earthquake Model

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 Nyström method is builded to approximate the solution. The numerical tests show the effectiveness of this type of modeling.


Introduction and Motivation
We are interested in modeling the Earthquake phenomenon that results from movement of tectonic plates. During their movements, tectonic plates may collide along faults. While they pursue their movement, the pressure increases. When this pressure reaches a significant degree, a sudden slip of the plate occurs . The resulting shock trains release of energy in the form of seismic waves which cause the sensation of trembling (see [1,2]).
The Incorporated Research Institutions for Seismology (IRIS) (www.iris.edu) presented a demo ("Earthquake Machine: Basic One block and simple graph animated") to explain the Earthquake model. This demonstration is presented in figure 1 In this work, we will present a model that provid a constant friction. This choice is built on the difficulty in the theoretical and numerical plan. However with this choice our vision still a performance vision. 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 This paper represents a physical foundation for recent works ( [3,4,5]) on this equation and a generalization of the mathematical assumptions required in these papers to ensure the existence and uniqueness of the solution.
We will construct a numerical method based on the vision of Nyström (see [6,7,8,9]) to approach this unique solution, and show the convergence under the local Lipschitz assumptions, which represents a generalization of those assumptions required in [3,4,5,6].
The numerical tests developed in this paper show our good vision on mathematical modeling, analytical and numerical plans.
In the sense of establishing a coherent system and keeping the fundamental aspects of our phenomenon, and to properly present the mechanism of the earthquake machine, we consider a coupled mass spring oscillator model consists of two masses M and m, linked together by a spring of length L at rest, which are both free to move. The resulting motions can be very intriguing, but for simplicity, we will neglect the effects of friction and external forces (The latter is due to the difficulty of theoretical and numerical studies). The system is placed on a smooth horizontal surface.
We denote by F the force applied to the mass m, R the resistance of the mass M . Because each mass is free to move, we apply Newton's second law (see [10]) to each object. Let x(t) denote the displacement of the mass M from its equilibrium position and similarly, let y(t) denote the corresponding displacement for the mass m which has a force F acting on its right side, applying Hooke's law (see [10]), where T is the tension of the spring, ̺ is the stiffness constant and l(t) the variation of the length between the two masses at a moment t such that Applying Newton's second law to these objects, we obtain the system in the state of rest 2) and the system in the state of motion where x ′′ (t) and y ′′ (t) are the acceleration of masses M and m respectively, and H is Heaviside function defined by By using the function δ defined by Solution of an Integro-differential Nonlinear Equation δ(τ ) = 1, τ = 0, 0, τ = 0, the systems (1.2) and (1.3) can be brought together in one system as the following form (1.5) Where x ′ (t) and y ′ (t) are the velocity of masses M and m respectively. As the model represents a simulation of an Earthquake Machine, and in order to be well posed, we put the seismic function u given by (1.6) Hence, the previous system (1.5) can be written as the form We integrate the equations (1.7) and (1.8), replacing the results in (1.9), under the initial conditions we obtain the equation The equation (1.12) represents a discontinuity given by the function δ. Therefore, to solve this problem we will approach the equation (1.12) by another equation through approaching δ using δ ξ of class C 1 (R) defined as Therefore, the equation (1.12) is written as Which also gives, For T > 0 maximum experiment time, the equation (1.14) which is an integro-differential nonlinear equation of Volterra can be written as the following simple and clear formula and The present paper is organized as follow, in section 2, we study our equation in an analytical sense, we show the existence and the uniqueness of the solution using similar methods to those used in [3,4,5,6]. In section 3, we study this equation in a numerical sense using a Nyström method (cf. [3,4,6,8]) to approximate the solution of this equation. Finally, in section 4 we give an experimental tests which show the effectiveness of our model.

Analytical study
We consider the previous equation It is clear that f and K verify the following properties 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 {u n (t)} n∈N , {ϕ n (t)} n∈N . This method is called the Picard method. These sequences are given by Solution of an Integro-differential Nonlinear Equation and which gives us successively and (2.6)

Theorem 2.1. According to the properties (H), and knowing that there exist points
Defining the sequences u n , ϕ n , u ′ n and ϕ ′ n as in (2.2)-(2.5), we can now show by induction, for n ∈ N * that In fact, for n = 1, we have u 1 (t) = f (t) and u ′ 1 (t) = f ′ (t), then so that (a 1 ) and (b 1 ) are satisfied. Also, (2.10) (c 1 ) and (d 1 ) holds. The property (P 1 ) is satisfied. Now, assuming that (P n ) is satisfied, we prove the property (P n+1 ). Then on the other hand, we have and Also, from (2.9), (2.12) and (2.14) we obtain this completes the inductive argument.
Since (P n ) is obviously true for all n, this bound makes it obvious that the sequence {u n } n∈N converges uniformly to u ∈ C 1 (0, δ 1 ), and we can write To prove that u satisfies the original equation (1.16), we put
To show that u(t) is the only continuous solution, suppose that there exists another solutionũ(t) ∈ C 1 (0, δ 1 ) of (1.16). Then, for all t ∈ [0, δ 1 ] Since u(t) andũ(t) are both continuous in [0, δ 1 ], ∃C > 0 such that, substituting this into (2.17) By repeating the argument n times, we are led to for any n hence, we conclude that u(t) =ũ(t) for all t ∈ [0, δ 1 ]. Now, for t ∈ [δ 1 , δ 2 ], we write the equation as and u 0 (s) is the solution obtained in the first step, but (2.18) is just the same Volterra equation with an origin shifted from 0 to δ 1 . So, we can apply the same basic steps. We define It is clear that u ∈ C 1 (0, δ 2 ) is the unique solution of (1.16) over [0, δ 2 ]. 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 C 1 (0, T ).

Numerical study
In this part, we use a numerical method based on the numerical integration to approximate the solution of the equation. This technique is called Nyström method, which is studied in [3,4,6] For N ∈ N * , we construct a subdivision of the interval [0, T ], We denote U n ≈ u(t n ) and V n ≈ u ′ (t n ), by applying the Nyström method for the approximation of the integrals that appear in our equation, using the quadratic formula of the numerical integration on the equations (1.16) and where, w i are real, such that it exits W > 0, ∀N ∈ N * , max 0≤j≤N |w j | ≤ W.
We obtain the following system Solution of an Integro-differential Nonlinear Equation

System study
The next theorem shows the existence and uniqueness of the solution of the system (3.1)-(3.4), under the properties (H)

Theorem 3.2. For h sufficiently small, the system (3.1)-(3.4) has a unique solution.
Proof. It is clear that we can obtain the solution of (3.3) by the recurrence formula, it remains to prove the existence of the solution of the equation (3.4). For all n ≥ 1, we define For all X, Y ∈ R, we have Consequently, ψ n is a contraction for h small enough. Using Banach's fixed point, we get the result.

Error analysis
In this part, we will show that the numerical method constructed previously converges to the exact solution of the equation. For N ∈ N * , we define for 0 ≤ n ≤ N , The method converges if lim h→0 ( max 0≤n≤N ε n ) = 0.
We define local consistency error by  Proof. For n ≥ 1.

Experimental results
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 Nyström approximation method proved in the previous section.
To perform the calculus, we need to clarify some parameters that intervene in our equation ( The results presented in these graphs are compatible with the mechanism of the earthquake machine, and they show the efficiency of our proceeding and the good vision for this phenomenon modeling. From different values of parameters that we have put, it can be seen that the velocity x ′ (t) can take a negative values in some moments. This latter is due to the theoretical approach of replacing the function δ with another smoother function δ ξ defined in (1.13), in addition to the error spread of the numerical approximation using the Nyström method.
The reader at this point can ask the following question: What happens when ξ approaches zeros? The answer is 'nothing !', because the curves of u, x and x ′ keep the same shape if we replace δ ξ by δ in our program, as it is entailed in the figures 2, 3 and 4 . Since the studied equation remains well constructed and admits a unique solution, although theoretically we can not treat the case ξ = 0 using Picard's successive method. In addition, the numerical method is also convergent in this case. We explore another analytical and numerical track to study our equation when ξ = 0.
During its movement, the mass M jumps into different distances over time, as shown in the figures 4. This jump reflects the null values of speed recorded in our results which proves our good vision of approximation of the physical phenomenon.
The following table shows that each time the discretization N is increased , the minimum of the velocity converges to 0.

Conclusion
The main purpose of the current paper has been to construct a modeling of an earthquake machine mechanism, and to study the existence and uniqueness of the solution of an integro-differential nonlinear Volterra equation that results from this modeling . Developed numerical tests show the effectiveness of this modeling of the problem as well as Nyström method used to approximate the solution of this equation.
Currently, we are studying the case where the resistance is not constant. This situation is more realistic and will give better results, but gives rise to new mathematical challenges.