n this paper, at first the  concept of caputo fractionalderivative is generalized on time scales. then the fractional orderdifferential equations are introduced on time scales. finally,sufficient and necessary conditions are presented for the existenceand uniqueness of solution of initial valueproblem including fractional order differential equations.

  In this paper, we use modified Laplace decomposition method to solving initial value problems (IVP) of the second order ordinary differential equations. Theproposed method can be applied to linear and nonlinearproblems    

We prove that the initial value problem (IVP) for the BBM equation is ill-posed for data in H(R), s < 0 in the sense that the flow-map u0 7→ u(t) that associates to initial data u0 the solution u cannot be continuous at the origin from H(R) to even D′(R) at any fixed t > 0 small enough. This result is sharp.

This paper focuses on the derivation of diagonally implicit two-point block backward differentiation formulas DI2BBDF for solving first-order initial value problem IVP with two fixed points. The method approximates the solution at two points simultaneously. The implementation and the stability of the proposed method are also discussed. A performance of the DI2BBDF is compared with the existing ...

In this study, we construct new numerical methods for solving the initial value problem (IVP) in ordinary differential equations based on a symmetrical quadrature integration formula using hybrid functions. The proposed are designed to provide an efficient and accurate solution IVP more suitable problems with non-smooth solutions. key idea behind is combine advantages of traditional methods, su...

in this paper, we use modified laplace decomposition method to solving initial value problems (ivp) of the second order ordinary differential equations. theproposed method can be applied to linear and nonlinearproblems

In the numerical integration of second-order nonlinear boundary value problem (BVP), right condition plays role as a target equation, which is solved either by half-interval method (HIM) or new derivative-free Newton (DFNM) to be presented in paper. With help shape function, we can transform BVP an initial (IVP) for variable. The terminal variable expressed function missing original variable, d...

Since many ordinary differential equations (ODEs) do not have a closed solution, approximating them is an important problem in numerical analysis. This work formalizes a method to approximate solutions of ODEs in Isabelle/HOL. We formalize initial value problems (IVPs) of ODEs and prove the existence of a unique solution, i.e. the Picard-Lindelöf theorem. We introduce general one-step methods f...