In the present paper, the analytical solution for an EXACT SECOND ORDER FUZZY INITIAL VALUE problem under generalized Hukuhara differentiability is obtained. First the solution of first ORDER linear FUZZY differential equation under generalized Hukuhara differentiability is investigated using integration factor methods in two cases. The SECOND based on the type of generalized Hukuhara differentiability, the analytical solution of an EXACT SECOND ORDER FUZZY INITIAL VALUE problem is derived in two senses of differentiability. Also, several properties for generalized Hukuhara differentiability are obtained on the topics, such as, the univariate FUZZY chain rules, FUZZY integration by parts, among others. At the end, some illustrative examples are presented and these examples show the behavior of the ‎solutions.

In this paper, we study a class of SECOND-ORDER FUZZY INITIAL VALUE problems that are known as the Cauchy-Euler di erential equations, in the crisp case. This work begins by studying the structure of solution function in the crisp case and providing a requirement space of the generalized di erentiable functions. In sequel, the process of production and construction of the solution formula are discussed, in details. Finally, the obtained formulas are applied and illustrated by solving some examples.

In this paper,  rst the Newton's divided di erence interpolation method based on the gH di erence on FUZZY data is introduced. Then the numerical methods entitled FUZZY Euler and modi ed FUZZY Euler are used to solve FUZZY impulsive INITIAL VALUE problem. Moreover the algorithms for the FUZZY impulsive INITIAL VALUE problem are explained and their local truncation errors are obtained in details. Finally, for more illustration some numerical examples are solved.

In this study, for the numerical solution of general SECOND-ORDER ordinary differential equations (ODEs) that exhibit oscillatory or periodic behavior, fifth- and sixth-ORDER explicit multi-step Runge-Kutta-Nystr¨om (MSGRKN) methods, respectively, are constructed. The parameters of the proposed methods rely on the frequency ω of each problem whose solution is a linear combination of functions {e(iωx), e(−iωx)} or {cos(ωx), sin(ωx)}. The study also includes an analysis of the linear stability of the suggested methods. The numerical results indicate the efficiency of the proposed methods in solving such problems compared to methods with similar characteristics in the literature.

In this paper the solution of a SECOND ORDER linear differential equations with intuitionistic FUZZY boundary VALUE is described. It is discussed for two different cases: coefficient is positive crisp number and coefficient is negative crisp number. Here FUZZY numbers are taken as generalized trapezoidal intutionistic FUZZY numbers (GTrIFNs). Further a numerical example is illustrated.

In this paper, using the hyperbolic uniform spline of ORDER 4 we develop the classes of methods for the numerical solution of SECOND ORDER boundary VALUE problems (2VBP) with Dirichlet, Neumann and Cauchy types boundary conditions. The SECOND derivativeis approximated by the three-point central difference scheme. The approximate results, obtained by the proposed method, confirm the convergence of numerical solutions. Numerical results are given to illustrate the efficiency of our methods.