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In most introductory numerical analysis textbooks, the treatment of a single nonlinear equation often consists of a collection of all-purpose methods that frequently do not work or are inefficient. These textbooks neglect to teach the…
There is no general existence theorem for solutions for nonlinear difference equations, so we must prove the existence of solutions in accordance with models one by one. In our work, we found theorems for the existence of analytic solutions…
In this paper we present a new approach to proving some exponential inequalities involving the sinc function. Power series expansions are used to generate new polynomial inequalities that are sufficient to prove the given exponential…
This paper examines a number of extrapolation and acceleration methods, and introduces a few modifications of the standard Shanks transformation that deal with general sequences. One of the goals of the paper is to lay out a general…
A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of…
Maximization of submodular functions under various constraints is a fundamental problem that has been studied extensively. A powerful technique that has emerged and has been shown to be extremely effective for such problems is the…
We provide new exact Taylor's series with fixed coefficients and without the remainder. We demonstrate the usefulness of this contribution by using it to obtain very simple solutions to (non-linear) PDEs. We also apply the method to the…
Using an explicit Euler substitution it was obtained a system of differential equations, which can be used to find the solution of time-dependent 1-dimentional Schr\H{o}dinger equation for a general form of the time-dependent potential.
In this paper, we study about existence and non-existence of finite order transcendental entire solutions of the certain non-linear differential-difference equations. We also study about conjectures posed by Rong et al. and Chen et al.
A new method of numerical solution for partial differential equations is proposed. The method is based on a fast matrix multiplication algorithm. Two-dimensional Poison equation is used for comparison of the proposed method with…
A method to find exact solutions to nonlinear Schr\"odinger equation, defined on a line and on a plane, is found by connecting it with second order linear ordinary differential equation. The connection is essentially made using Riccati…
We discuss an extension of the modified method of simplest equation for obtaining exact analytical solutions of nonlinear partial differential equations. The extension includes the possibility for use of: (i) more than one simplest…
The authors proposed a general way to find particular solutions for overdetermined systems of PDEs previously, where the number of equations is greater than the number of unknown functions. In this paper, we propose an algorithm for finding…
The issue of inheriting periodicity of an exact solution of a dynamic system by a difference scheme is considered. It is shown that some difference schemes (midpoint scheme, Kahan scheme) in some special cases provide approximate solutions…
We construct asymptotic expansions for ordinary differential equations with highly oscillatory forcing terms, focussing on the case of multiple, non-commensurate frequencies. We derive an asymptotic expansion in inverse powers of the…
We derive approximate expressions for the dispersion relation of the nonlinear Klein-Gordon equation in the case of strong nonlinearities using a method based on the Linear Delta Expansion. All the results obtained in this article are fully…
A novel explicit method to model Lorentz linear dispersive media with finite difference method are presented. The method shows an explicit method without any modification to the Leap-Frogging scheme. The polarizations of the Lorentz media…
Using exhaustion method and finite differences a new method to solve system of partial differential equations and is presented. This method allows design algorithm to solve linear and nonlinear systems in irregular domains. Applying this…
Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations, but is better described by fractional diffusion models. The nonlocal nature of the fractional diffusion operators makes substantially…
We extend the piecewise orthogonal collocation method to computing periodic solutions of coupled renewal and delay differential equations. Through a rigorous error analysis, we prove convergence of the relevant finite-element method and…