The solution y(x) of an IVP for a linear ODE with analytic coefficient functions is represented as a power series. A high–order Taylor polynomial is used for an approximate numerical solution. The Taylor remainder series is rigorously estimated by some geometric series. The method has been implemented and tested on a computer. Guaranteed enclosures are achieved by taking into account all roundo...

In this paper, Numerical solution of Differential-Algebraic equations (DAEs) with index-3 is considered by Pade approximation. We applied this method to two examples. First differential-algebraic equations (DAEs) with index-3 has been converted to power series by one-dimensional differential transformation, Then the numerical solution of equation was put into Pade series form. Thus we obtained ...

In this paper we study nonlocal functionals whose kernels are homogeneous generalized functions. We also use such functionals to solve the Kortewegde Vries (KdV), the nonlinear Schrödinger (NLS) and the Davey-Stewartson (DS) equations. The solution of certain integrable equations in terms of formal power series was obtained in [4], [5]. In these papers the solution was expressed in a formal pow...

It is shown how the Python library Sympy can be used to compute symbolically the coefficients of the power series solution of the Lane-Emden equation (LEE). Sympy is an open source Python library for symbolic mathematics. The power series solutions are compared to the numerically computed solutions using matplotlib. The results of a run time measurement of the implemented algorithm are discusse...

Abstract. We propose to look at (a simplified version of) Heisenberg’s fundamental field equation (see [2]) as a relativistic quantum field theory with a fundamental length, as introduced in [1] and give a solution in terms of Wick power series of free fields which converge in the sense of ultrahyperfunctions but not in the sense of distributions. The solution of this model has been prepared in...

We present a self-contained formalism for calculating the background solution, the linearized solutions, and a class of generalized Frobenius solutions to a system of scale invariant differential equations. We first cast the scale invariant model into its equidimensional and autonomous forms, find its fixed points, and then obtain power-law background solutions. After linearizing about these fi...

In this paper, an analytical method is developed to obtain the solution for the one dimensional transient thermal and mechanical stresses in a hollow cylinder made of functionally graded material (FGM) and piezoelectric layers. The FGM properties are assumed to depend on the variable r and they are expressed as power functions of r but the Poisson’s ratio is assumed to be constant. Transient te...

In the present paper, an exact solution for transient response of an infinitely long functionally graded thick-walled cylinder subjected to dynamic pressures at the boundary surfaces is presented for arbitrary initial conditions. The cylinder is assumed to have a plane-strain condition and the dynamic pressures are assumed to be imposed uniformly and axis...

For the quantum integer [n]q = 1+q+q + · · ·+q there is a natural polynomial multiplication such that [mn]q = [m]q ⊗q [n]q . This multiplication is described by the functional equation fmn(q) = fm(q)fn(q ), defined on a given sequence F = {fn(q)} ∞ n=1 of polynomials such that fn(0) = 1 for all n. If F = {fn(q)} ∞ n=1 is a solution of the functional equation, then there exists a formal power se...

We give a necessary condition for algebraicity of finite modules over the ring of formal power series. This condition is given in terms of local zero estimates. In fact we show that this condition is also sufficient when the module is a ring with some additional properties. To prove this result we show an effective Weierstrass Division Theorem and an effective solution to the Ideal Membership P...