Right-hand Side Function Research Articles

Inverse problem for semilinear wave equation with strong damping

The initial-boundary and the inverse coefficient problems for the semilinear hyperbolic equation with strong damping are considered in this study. The conditions for the existence and uniqueness of solutions in Sobolev spaces to these problems have been established. The inverse problem involves determining the unknown time-dependent parameter in the right-hand side function of the equation using an additional integral type overdetermination condition.

A second-order nonstandard finite difference method for a general Rosenzweig–MacArthur predator–prey model

In this paper, we consider a general Rosenzweig–MacArthur predator–prey model with logistic intrinsic growth of the prey population. We develop the Mickens’ method to construct a dynamically consistent second-order nonstandard finite difference (NSFD) scheme for the general Rosenzweig–MacArthur predator–prey model. The second-order NSFD method is based on a novel nonlocal approximation using right-hand side function weights and nonstandard denominator functions.Through rigorous mathematical analysis, we show that the NSFD method not only preserves two important and prominent dynamical properties of the continuous-time model, namely positivity and asymptotic stability independent of the values of the step size, but also is convergent of order 2. Therefore, it provides a solution to the contradiction between the dynamic consistency and high-order accuracy of NSFD methods.The proposed NSFD method improves positive and elementary stable nonstandard numerical schemes constructed in a previous work of Dimitrov and Kojouharov (2006). Moreover, the present approach can be extended to construct second-order NSFD methods for some classes of nonlinear dynamical systems.Finally, the theoretical insights and advantages of the constructed NSFD scheme are supported by some illustrative numerical simulations.

Ordering stability of Nash equilibria for a class of differential games

Abstract This study is concerned with the stability of Nash equilibria for a class of n n -person noncooperative differential games. More precisely, due to a preorder induced by a convex cone on a real linear normed space, we define a new concept called ordering stability of equilibria against the perturbation of the right-hand side functions of state equations for the differential game. Moreover, using the set-valued analysis theory, we present the sufficient conditions of the ordering stability for such differential games.

A general class of second-order [formula omitted]-stable explicit numerical methods for stiff problems

In this paper, we propose a simple approach to the construction of a general class of L-stable explicit second-order one-step methods for solving stiff problems. These methods are nonlinear and derive from a novel approximation for the right-hand side functions of differential equations inspired by the nonstandard finite difference methodology introduced by Mickens. Through rigorous mathematical analysis, it is proved that the proposed numerical methods are not only explicit and L-stable, but also convergent of order two. Therefore, they are suitable and efficient to solve stiff problems.The proposed numerical methods generalize and improve a nonstandard explicit integration scheme for initial value problems formulated by Ramos (2007). Moreover, the present approach can be extended to construct A-stable and L-stable high-order explicit one-step methods for differential equations.Finally, the theoretical findings and advantages of the developed numerical methods are supported and illustrated by a series of numerical experiments in which stiff problems are considered.

Euler scheme for approximation of solution of nonlinear ODEs under inexact information

We investigate error of the Euler scheme in the case when the right-hand side function of the underlying ODE satisfies nonstandard assumptions such as local one-sided Lipschitz condition and local Hölder continuity. Moreover, we assume two cases in regards to information availability: exact and noisy with respect to the right-hand side function. Optimality analysis of the Euler scheme is also provided. Finally, we present the results of some numerical experiments.

Existence, uniqueness and approximation of solutions to Carathéodory delay differential equations

In this paper we address the existence, uniqueness and approximation of solutions of delay differential equations (DDEs) with Carathéodory type right-hand side functions. We provide construction of the randomized Euler scheme for DDEs and investigate its error. We also report results of numerical experiments.

A Right-Hand Side Function Surrogate Model-Based Method for the Black-Box Dynamic Optimization Problem

Abstract When solving the black-box dynamic optimization problem (BDOP) in the sophisticated dynamic system, the finite difference technique requires significant computational efforts on numerous expensive system simulations to provide approximate numerical Jacobian information for the gradient-based optimizer. To save computational budget, this work introduces a BDOP solving framework based on the right-hand side (RHS) function surrogate model (RHSFSM), in which the RHS derivative functions of the state equation are approximated by the surrogate models, and the Jacobian information is provided by inexpensive estimations of RHSFSM rather than the original time-consuming system evaluations. Meanwhile, the sampling strategies applicable to the construction of RHSFSM are classified into three categories: direct, indirect, and hybrid sampling strategy, and the properties of these strategies are analyzed and compared. Furthermore, to assist the RHSFSM-based BDOP solving framework search for the optimum efficiently, a novel dynamic hybrid sampling strategy is proposed to update RHSFSM sequentially. Finally, two dynamic optimization examples and a co-design example of a horizontal axis wind turbine illustrate that the RHSFSM-based BDOP solving framework integrated with the proposed dynamic hybrid sampling strategy not only solves the BDOP efficiently but also achieves the optimal solution robustly and reliably compared to other sampling strategies.

Undetermined Coefficients Method for Sequential Fractional Differential Equations

The undetermined coefficients method is presented for nonhomogeneous sequential fractional differential equations involving Caputo fractional derivative of order n\alpha where n-1 n\alpha\le\ n and n\in\mathbb{N}. By employing proposed method, a particular solution of the considered equation is obtained. Some details about estimating the particular solution required to apply this method are explained. This method is shown to be particularly effective for nonhomogeneous fractional differential equations when the fractional differential equations involve some specific right-hand side functions.

Solving inverse non-linear fractional differential equations by generalized Chelyshkov wavelets

The purpose of this research is to employ a method involving Chelyshkov wavelets to construct a numerical solution to the inverse problem of determining the right-hand side function of a non-linear fractional differential equation by utilizing over-measured data. The novelty of this research is that this type of inverse problem is studied by Chelyshkov wavelets. Firstly, the problem is reduced into a system of algebraic equations with an unknown right-hand side by means of the orthonormal base of Chelyshkov wavelets. Secondly, by choosing suitable nodes, this system is transformed into a homogenous system of algebraic equations. The solution of the homogenous system allows us to determine the coefficients of the bases vectors for the solution of the non-linear fractional differential equation. In the final step, the right-hand side is obtained by substituting the constructed solution into a non-linear fractional differential equation. The presented examples illustrate that the numerical solution, obtained by this method, is remarkably close to the exact solution.

An Interpolated Galerkin Finite Element Method for the Poisson Equation

We develop a new approach to construct finite element methods to solve the Poisson equation. The idea is to use the pointwise Laplacian as a degree of freedom followed by interpolating the solution at the degree of freedom by the given right-hand side function in the partial differential equation. The finite element solution is then the Galerkin projection in a smaller vector space. This idea is similar to that of interpolating the boundary condition in the standard finite element method. Our approach results in a smaller system of equations and of a better condition number. The number of unknowns on each element is reduced significantly from \((k^2+3k+2)/2\) to 3k for the \(P_k\) (\(k\ge 3\)) finite element. We construct bivariate \(P_2\) conforming and nonconforming, and \(P_k\) (\(k\ge 3\)) conforming interpolated Galerkin finite elements on triangular grids; prove their optimal order of convergence; and confirm our findings by numerical tests.

A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems

In this work, a novel second-order nonstandard finite difference (NSFD) method that preserves simultaneously the positivity and local asymptotic stability of one-dimensional autonomous dynamical systems is introduced and analyzed. This method is based on novel non-local approximations for right-hand side functions of differential equations in combination with nonstandard denominator functions. The obtained results not only resolve the contradiction between the dynamic consistency and high-order accuracy of NSFD methods but also improve and extend some well-known results that have been published recently in Kojouharov et al. (2021); Gupta et al. (2020); Wood and Kojouharov (2015). Furthermore, as a simple but important application, we apply the constructed NSFD method for solving the logistic, sine, cubic, and Monod equations. Consequently, the NSFD schemes constructed in the earlier work (Mickens, 1999) are improved significantly. Finally, we report some numerical experiments to support and illustrate the theoretical assertions as well as advantages of the constructed NSFD method.

Approximation of solutions of DDEs under nonstandard assumptions via Euler scheme

We deal with approximation of solutions of delay differential equations (DDEs) via the classical Euler algorithm. We investigate the pointwise error of the Euler scheme under nonstandard assumptions imposed on the right-hand side function f. Namely, we assume that f is globally of at most linear growth, satisfies globally one-side Lipschitz condition but it is only locally Hölder continuous. We provide a detailed error analysis of the Euler algorithm under such nonstandard regularity conditions. Moreover, we report results of numerical experiments.

Generic uniqueness of saddle point for two-person zero-sum differential games

Abstract The generic uniqueness of saddle point for two-person zero-sum differential games, within the class of open-loop, against the perturbation of the right-hand side function of the control system is investigated. By employing set-valued mapping theory, it is proved that the majority of the two-person zero-sum differential games have unique saddle point in the sense of Baire’s category.

On the randomized Euler schemes for ODEs under inexact information

We analyse errors of randomized explicit and implicit Euler schemes for approximate solving of ordinary differential equations (ODEs). We consider classes of ODEs for which the right-hand side functions satisfy Lipschitz condition globally or only locally. Moreover, we assume that only inexact discrete information, corrupted by some noise, about the right-hand side function is available. Optimality and stability of explicit and implicit randomized Euler algorithms are also investigated. Finally, we report the results of numerical experiments which support our theoretical conclusions.

Note on difference equations with the right-hand side function nonincreasing in each variable

We present an example of a difference equation of arbitrary order, possessing the right-hand side function that is homogeneous to a certain degree and nonincreasing in each variable, which has a unique positive equilibrium, as well as solutions that do not converge to the equilibrium. The example shows that the main result in the paper: O. Moaaz, Dynamics of difference equation x_{n+1}=f(x_{n-l}, x_{n-k}) (Adv. Differ. Equ. 2018:447, 2018), is incorrect.

On Mathematical Aspects of Evolution of Dislocation Density in Metallic Materials

This paper deals with the solution of delay differential equations describing evolution of dislocation density in metallic materials. Hardening, restoration, and recrystallization characterizing the evolution of dislocation populations provide the essential equation of the model. The last term transforms ordinary differential equation (ODE) into delay differential equation (DDE) with strong (in general, H\"older) nonlinearity. We prove upper error bounds for the explicit Euler method, under the assumption that the right-hand side function is H\"older continuous and monotone which allows us to compare accuracy of other numerical methods in our model (e.g. Runge-Kutta), in particular when explicit formulas for solutions are not known. Finally, we test the above results in simulations of real industrial process.

Identification of spacewise dependent right-hand side in two dimensional parabolic equation

In this paper numerical solution of the inverse problem of determining a spacewise dependent right-hand side function in two dimensional parabolic equation is considered. Usually, the right-hand side function dependent on spatial variable is obtained from measured data of the solution at the final time point. Many mathematical modeling problems in the field of physics and engineering will encounter the inverse problems to identify the right-hand terms. When studying an inverse problem of identifying the spacewise dependent right-hand function, iterative methods are often used. We propose a new conjugate gradient method based on the constructed self-adjoint operator of the equation for numerical solution of the function and numerical examples illustrate the efficiency and accuracy.

Identification of stationary source in the anomalous diffusion equation

In this paper, we consider the initial-boundary value problem of determining the stationary right-hand side function in the anomalous diffusion equation with a Caputo fractional derivative with respect to time. The value of the solution of the problem at the final time moment is set as the overdetermination condition. In order to carry out the numerical solution, the iterative conjugate gradient method is used, while at each iteration a direct problem is solved by the finite-difference method using a purely implicit difference scheme. The computational experiment results for the model problem are presented to confirm the efficiency of this new method.

Delay-induced homoclinic bifurcations in modified gradient bistable systems and their relevance to optimization

Nonlinear dynamical systems with time delay are abundant in applications but are notoriously difficult to analyze and predict because delay-induced effects strongly depend on the form of the nonlinearities involved and on the exact way the delay enters the system. We consider a special class of nonlinear systems with delay obtained by taking a gradient dynamical system with a two-well “potential” function and replacing the argument of the right-hand side function with its delayed version. This choice of the system is motivated by the relative ease of its graphical interpretation and by its relevance to a recent approach to use delay in finding the global minimum of a multi-well function. Here, the simplest type of such systems is explored for which we hypothesize and verify the possibility to qualitatively predict the delay-induced effects, such as a chain of homoclinic bifurcations one by one eliminating local attractors and enabling the phase trajectory to spontaneously visit vicinities of all local minima. The key phenomenon here is delay-induced reorganization of manifolds, which cease to serve as barriers between the local minima after homoclinic bifurcations. Despite the general scenario being quite universal in two-well potentials, the homoclinic bifurcation comes in various versions depending on the fine features of the potential. Our results are a pre-requisite for understanding general highly nonlinear multistable systems with delay. They also reveal the mechanisms behind the possible role of delay in optimization.

Fully non-linear degenerate elliptic equations in complex geometry

We derive an a priori real Hessian estimate for solutions of a large family of geometric fully non-linear elliptic equations on compact Hermitian manifolds, which is independent of a lower bound for the right-hand side function. This improves on the estimates of Székelyhidi [57] and additionally applies to elliptic equations with a degenerate right-hand side. As an application, we establish the optimal C1,1 regularity of envelopes of (θ,m)-subharmonic functions on compact Hermitian manifolds.