We comment on the new trend in mathematical physics that consists of obtaining Taylor series for fabricated linear and nonlinear unphysical models by means of homotopy perturbation method (HPM), homotopy analysis method (HAM) and Adomian decomposition method (ADM). As an illustrative example we choose a recent application of the HPM to a dynamic system of anisotropic elasticity.

Joint uncertainty decoding has recently achieved promising results by using front-end uncertainty in the back-end in a mathematically consistent framework. One drawback of the method is that it relies on stereo-data or numerical algorithms, such as DPMC, which have high computational complexity and are difficult to deploy in real applications. We propose a Vector Taylor Series (VTS) approach to...

Let e k (x) denote the k-th partial sum of the Maclaurin series for the exponential function. Define the (n + 1) × (n + 1) Hankel determinant by setting Hn(x) = det[e i+j (x)] 0≤i,j≤n. We give a closed form evaluation of this determinant in terms of the Bessel polynomials using the method of recently introduced γ-operators.

A method is presented (with standard examples) based on an elementary complex integral expression, for developing Frobenius series solutions for second-order linear homogeneous ordinary Fuchs differential equations. The method reduces the task of finding a series solution to the solution, instead, of a system of simple equations in a single variable. The method is straightforward to apply as an...

There are several important observations to make about this definition. (i) The definition says that, for each fixed n, ∑n m=0 amx m becomes a better and better approximation to f(x) as x gets smaller. As x → 0, ∑m=0 amx approaches f(x) faster than x tends to zero. (ii) The definition says nothing about what happens as n → ∞. There is no guarantee that for each fixed x, ∑n m=0 amx m tends to f(...

in this paper a new form of the homptopy perturbation method is used for solving oscillator differential equation, which yields the maclaurin series of the exact solution. nonlinear vibration problems and differential equation oscillations have crucial importance in all areas of science and engineering. these equations equip a significant mathematical model for dynamical systems. the accuracy o...

Let R+ = (0,∞) and let M be the family of all mean values of two numbers in R+ (some examples are the arithmetic, geometric, and harmonic means). Given m1, m2 ∈ M, we say that a function f : R+ → R+ is (m1, m2)-convex if f(m1(x, y)) ≤ m2(f(x), f(y)) for all x, y ∈ R+. The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m1, m2)-convexit...

In this paper a new form of the homptopy perturbation method is used for solving oscillator differential equation, which yields the Maclaurin series of the exact solution. Nonlinear vibration problems and differential equation oscillations have crucial importance in all areas of science and engineering. These equations equip a significant mathematical model for dynamical systems. The accuracy o...

There are several important observations to make about this definition. (i) The definition says that, for each fixed n, ∑n m=0 amx m becomes a better and better approximation to f(x) as x gets smaller. As x → 0, ∑m=0 amx approaches f(x) faster than x tends to zero. (ii) The definition says nothing about what happens as n → ∞. There is no guarantee that for each fixed x, ∑n m=0 amx m tends to f(...

1 Overview We begin by recalling the Rolle’s Theorem.1 Using this result, we shall derive the Lagrange Form of the Taylor’s Remainder Theorem. Subsequently, we shall derive several mathematical inequalities as a corollary of this result. For example, 1. We shall use the Taylor’s Remainder Theorem to upper and lower bound exponential functions using polynomials. 2. We shall use the Taylor’ Remai...