We propose the multiplicative explicit Euler, multiplicative implicit Euler, and multiplicative Crank-Nicolson algorithms for the numerical solutions of the multiplicative partial differential equation. We also consider the truncation error estimation for the numerical methods. The stability of the algorithms is analyzed by using the matrix form. The result reveals that the proposed numerical methods are effective and convenient.

We consider initial value problems for the nonlinear Klein-Gordon equation in de Sitter spacetime. We use the differential transform method for the solution of the initial value problem. In order to show the accuracy of results for the solutions, we use the variational iteration method with Adomian’s polynomials for the nonlinearity. We show that the methods are effective and useful

In this article, we consider the initial value problem for a system of Klein-Gordon equations in anti-de Sitter spacetime. We assume that the mass matrix is diagonalizable with positive eigenvalues. Under the assumption for the small initial data, we prove the global existence of solutions to the initial value problem. The existence result is derived based on the assumption of power-type nonlinearity.

This paper establishes a weighted Simpson-type identity on the parameter domain associated with quantum integral operators. Using this identity together with Hölder’s inequality and the power mean inequality, we derive new estimates for classes of functions whose associated parameter-domain q-derivatives satisfy h-convexity assumptions. Additional bounds are obtained under boundedness and Lipschitz conditions. Applications to the s-moment of a random variable and to several special means are derived in the classical limit $q\to 1^{-}$.

The semilinear Klein-Gordon equation with initial conditions is studied in de Sitter spacetime. The L∞ decay estimates are derived for the solutions to the linear Klein- Gordon equations with and without source term in de Sitter spacetime. It is also showed the global existence of solutions to the initial value problem with power type non-linear terms for small initial data by using these estimates