The multidimensional Exponential Levy equations are used to describe many stochastic phenomena such as market fluctuations. Unfortunately in practice an exact solution does not exist for these equations. This motivates us to propose a numerical solution for n-dimensional Exponential Levy equations by block pulse functions. We compute the jump integral of each block pulse function and present a Poisson operational matrix. Then we reduce our equation to a linear lower triangular system by constant, Wiener and Poisson operational matrices. Finally using the forward substitution method, we obtain an approximate answer with the convergence rate of O(h). Moreover, we illustrate the accuracy of the proposed method with a 95% confidence interval by some numerical examples.

In this paper, we aim to propose a new hybrid version of the Longstaff and Schwartz algorithm under the Exponential Levy Jump-diffusion model using Random Forest regression. For this purpose, we will build the evolution of the option price according to the number of paths. Further, we will show how this approach numerically depicts the convergence of the option price towards an equilibrium price when the number of simulated trajectories tends to a large number. In the second stage, we will compare this hybrid model with the classical model of the Longstaff and Schwartz algorithm (as a benchmark widely used by practitioners in pricing American options) in terms of computation time, numerical stability and accuracy. At the end of this paper, we will test both approaches on the Microsoft share “MSFT” as an example of a real market.

The collocation method based on the Exponential cubic B-splines (ECB-splines) together with the Crank-Nicolson formula is used to solve nonlinear coupled Burgers' equation (CBE). This method is tested by studying three di, erent problems. The proposed scheme is compared with some existing methods. It produced accurate results with the suitable selection of the free parameter of the ECB-spline function. It produces accurate results. Stability of the fully discretized CBE is investigated by the Von Neumann analysis.

We consider the spin-one DKP equation in the presence of vector Cornell and Exponential interactions which are among successful interactions of particle physics. We obtain the exact analytical solutions of the former and the approximate solutions the latter via the Laplace integral transform method in terms of Hypergeometric functions for arbitrary quantum number in three spatial dimensions.

This work is concerned with the initial boundary value problem for a nonlinear viscoelastic Petrovsky wave equation utt +
, 2u 􀀀,Z t 0 g(t 􀀀,, )
, 2u(,)d,􀀀,
, ut 􀀀,
, utt + utjutjm􀀀, 1 = ujujp􀀀, 1: Under suitable conditions on the relaxation function g, the global ex-istence of solutions is obtained without any relation between m and p. The uniform decay of solutions is proved by adapting the perturbed energy method. For p > m and su, cient conditions on g, an unbound-edness result of solutions is also obtained.