Related papers: \QTR{group}{Exact solutions of nonlinear equations…
We present a 'calculator' for constructing a homogeneous approximation of nonlinear control systems, which is based on the algebraic approach developed by the authors in their previous papers. This approach mainly uses linear algebraic and…
In this work the longitudinal wave equation through magneto-electro-elastic circular rod is studied analytically. Painlev\'e test is performed to check integrability of the equation. Exact solitary wave solutions are found by homogeneous…
This paper presents an innovative approach, the Adaptive Orthogonal Basis Method, tailored for computing multiple solutions to differential equations characterized by polynomial nonlinearities. Departing from conventional practices of…
Selection of 25 examples from extensive nontrivial families for different types of nonlinear PDEs and their formal general solutions are given. The main goal here is to show on examples the types of solvable PDEs and what their general…
In this paper we established a class of optimal fourth-order methods which is obtained by existing third-order method for solving nonlinear equations for simple roots by using weight functions. Some physical examples are given to illustrate…
The equilibration of a finite Bose system is modelled using a gradient expansion of the collision integral that leads to a nonlinear transport equation. For constant transport coefficients, it is solved in closed form through a nonlinear…
Whether integrable, partially integrable or nonintegrable, nonlinear partial differential equations (PDEs) can be handled from scratch with essentially the same toolbox, when one looks for analytic solutions in closed form. The basic tool…
For the system of second order quasilinear parabolic equations the problem of reducing them to the equations of diffusion type is considered. In non-degenerate case an effective algorithm for solving this problem is suggested.
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical…
We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the…
The method of harmonic balance (HB) is a spectrally accurate method used to obtain periodic steady state solutions to dynamical systems subjected to periodic perturbations. We adapt HB to solve for the stress response of the Giesekus model…
The completeness of solutions of homogeneous as well as non-homogeneous unsteady Stokes equations are examined. A necessary and sufficient condition for a divergence-free vector to represent the velocity field of a possible unsteady Stokes…
We introduce a new method for studying stochastic homogenization of elliptic equations in nondivergence form. The main application is an algebraic error estimate, asserting that deviations from the homogenized limit are at most proportional…
Direct algebraic method of obtaining exact solutions to nonlinear PDE's is applied to certain set of nonlinear nonlocal evolutionary equations, including nonlinear telegraph equation, hyperbolic generalization of Burgers equation and some…
A functional differential equation related to the logistic equation is studied by a combination of numerical and perturbation methods. Parameter regions are identified where the solution to the nonlinear problem is approximated well by…
Initially introduced in the framework of quantum control, the so-called "monotonic algorithms" have demonstrated excellent numerical performance when dealing with bilinear optimal control problems. This paper presents a unified formulation…
The method of nonlinear realizations is a convenient tool for building dynamical realizations of a Lie group, which relies solely upon structure relations of the corresponding Lie algebra. The goal of this work is to discuss advantages and…
This work investigates the application of the Newton's method for the numerical solution of a nonlinear boundary value problem formulated through an ordinary differential equation (ODE). Nonlinear ODEs arise in various mathematical modeling…
We present a short review of the evolution of the methodology of the Method of simplest equation for obtaining exact particular solutions of nonlinear partial differential equations (NPDEs) and the recent extension of a version of this…
A modified method of functional constraints is used to construct the exact solutions of nonlinear equations of reaction-diffusion type with delay and which are associated with variable coefficients. This study considers a most generalized…