Nonlinear Right-hand Side Research Articles

Positive Fitted Finite Volume Method for Semilinear Parabolic Systems on Unbounded Domain

This work deals with a semilinear system of parabolic partial differential equations (PDEs) on an unbounded domain, related to environmental pollution modeling. Although we study a one-dimensional sub-model of a vertical advection–diffusion, the results can be extended in each direction for any number of spatial dimensions and different boundary conditions. The transformation of the independent variable is applied to convert the nonlinear problem into a finite interval, which can be selected in advance. We investigate the positivity of the solution of the new, degenerated parabolic system with a non-standard nonlinear right-hand side. Then, we design a fitted finite volume difference discretization in space and prove the non-negativity of the solution. The full discretization is obtained by implicit–explicit time stepping, taking into account the sign of the coefficients in the nonlinear term so as to preserve the non-negativity of the numerical solution and to avoid the iteration process. The method is realized on adaptive graded spatial meshes to attain second-order of accuracy in space. Some results from computations are presented.

On the singular limit problem in nonlocal balance laws: Applications to nonlocal lane-changing traffic flow models

We present a convergence result from nonlocal to local behavior for a system of nonlocal balance laws. The velocity field of the underlying conservation laws is diagonal. In contrast, the coupling to the remaining balance laws involves a nonlinear right-hand side that depends on the solution, nonlocal term, and other factors. The nonlocal operator integrates the density around a specific spatial point, which introduces nonlocality into the problem. Inspired by multi-lane traffic flow modeling and lane-changing, the nonlocal kernel is discontinuous and only looks downstream. In this paper, we prove the convergence of the system to the local entropy solutions when the nonlocal operator (chosen to be of an exponential type for simplicity) converges to a Dirac distribution. Numerical illustrations that support the main results are also presented.

A non-local scalar field problem: existence, multiplicity and asymptotic behavior

The main purpose of this work is to study a non-local scalar field equation in bounded domains. More precisely we consider the semi-linear fractional elliptic problem: 0 \\hbox{ in } \\Omega, \\hbox{ and } u= 0 \\hbox{ in }\\mathbb{R}^N\\setminus \\Omega,\\] ]]> ( − Δ ) s u = λ u p − 1 − u m − 1 in Ω , u > 0 in Ω , and u = 0 in R N ∖ Ω , where Ω is a regular bounded domain of R N , m>p>2 and 0 $ ]]> λ > 0 . We discuss the existence, non-existence, and multiplicity of solutions, for the largest possible range of the parameters λ , p , m . Optimal results are obtained in the sub-critical and critical regimes, and partial results are obtained in the super-critical case. Furthermore, a careful asymptotic analysis when m tends to ∞ will be provided and we identify the limiting profiles as solutions to a free boundary type problem with a nonlinear right-hand side.

Anisotropic $ (\vec{p}, \vec{q}) $-Laplacian problems with superlinear nonlinearities

<abstract><p>In this paper we consider a class of anisotropic $ (\vec{p}, \vec{q}) $-Laplacian problems with nonlinear right-hand sides that are superlinear at $ \pm\infty $. We prove the existence of two nontrivial weak solutions to this kind of problem by applying an abstract critical point theorem under very general assumptions on the data without supposing the Ambrosetti-Rabinowitz condition.</p></abstract>

Stability analysis of an implicit fractional integro-differential equation via integral boundary conditions

The primary objective of this research study is to analyze a boundary problem involving Caputo fractional integro-differential equations. The focus is on a differential equation with a nonlinear right-hand side composed of two terms. The stability analysis of a fractional integro-differential equation is presented using Ulam's concept. Furthermore, this research study establishes the correlation between the stated problem and the Volterra integral equation. The investigation proceeds by utilizing the renowned Banach and Krasnoselskii's fixed point theorems to explore the existence and uniqueness of solutions for the problem. Additionally, to provide tangible evidence of the abstract findings, two illustrative examples are presented.

Cauchy Problem for a Singularly Perturbed Delay Equation

In the study, the Cauchy problem for a singularly perturbed first-order ordinary differential equation is considered, with, generally speaking, a nonlinear right-hand side that depends not only on the desired function but also on this same function taken with a time delay. The problem under consideration is singularly perturbed due to the presence of a small parameter in front of the time derivative. For such problems, solutions that possess a large gradient in the vicinity of the initial time moment and in the vicinity of the moment equal to the delay time are typical. The aim of the work is to construct an asymptotic approximation and to prove the existence of a smooth solution to the problem.

Sufficient Solvability Conditions for the Problem of Pursuit under an Impulse Action

The article considers a linear differential pursuit game under the condition that an integral constraint is imposed on the evader’s control and the pursuer uses an impulse control. These impulse actions on the object are made at predetermined time points, and the corresponding control is represented using the Dirac delta function. The article studies linear conflicts described by a system of ordinary differential equations whose trajectories have jumps at certain times. The terminal set is represented by a cylinder in an n-dimensional Euclidean space. The problem is solved using the resolving function method. The fact that the lower bound is reached is proved using the theory of support functions. As a result, the quasi-strategy is replaced by an almost stroboscopic strategy and a method for building this strategy is presented. An example of a nonlinear right-hand side is given.

About the analytical solution for the advertising model of two competing firms

Introduction. The criterion for the success of an advertising campaign is the maximum profit from sales, taking into account the costs of its implementation, while the sale of the same type of goods is sales occur in a competitive environment. The article examines a model for predicting mass sales of two similar products depending on the tactics of an advertising campaign. First of all, the distribution of funds between its separate types is considered: expenses for advertising paper products, banners and advertising in electronic media (EMM).Materials and methods. The model is formulated in the form of a Cauchy problem for a system of two ordinary differential equations with nonlinear right-hand sides, taking into account: the total number of potential solvent buyers of the first and second goods; the intensity of the advertising campaign, mainly through EMM, the positive impact on sales of the interaction of those who have already bought the first or second type of goods with potential buyers, as well as informal (at the level of buyers) anti-advertising.The results of the study. A solution is given for the case of constant coefficients determined by the above factors for the corresponding Cauchy problem in closed form.Discussion and conclusions. The results obtained can be used to replay model situations of advertising organization in order to determine the conditions for extracting the greatest profit from sales minus advertising costs.

The nonconforming virtual element method for semilinear elliptic problems

The nonconforming virtual element method (VEM) for the semilinear elliptic problem is developed in this paper. The nonlinear right-hand side is approximated by using the L2 projection. The optimal convergence of the nonconforming VEM in the broken H1 norm is proved. Finally, some numerical experiments are carried out to support the theoretical results.

Solution of the initial-value problem of first-order nonlinear recursions with homogeneous right-hand sides

In this paper, we identify systems of an arbitrary number N of first-order recursions with nonlinear right-hand sides homogeneous of arbitrary degree M, which feature simple explicit solutions. A novelty of these findings is to consider systems of recursions characterized by constraints involving their parameters and/or their initial data. Simple extensions of these findings to first-order recursions with nonhomogeneous right-hand sides are also provided.

Nonexistence of local conservation laws for generalized Swift–Hohenberg equation

We prove that the generalized Swift–Hohenberg equation with nonlinear right-hand side, a natural generalization of the Swift–Hohenberg equation arising in physics, chemistry and biology and describing inter alia pattern formation, has no nontrivial local conservation laws.

On solvability of one class of third order differential equations

UDC 517.9 One-dimensional mixed problem for one class of third order partial differential equation with nonlinear right-hand side is considered. The concept of generalized solution for this problem is introduced. By the Fourier method, the problem of existence and uniqueness of generalized solution for this problem is reduced to the problem of solvability of the countable system of nonlinear integro-differential equations. Using Bellman's inequality, the uniqueness of generalized solution is proved. Under some conditions on initial functions and the right-hand side of the equation, the existence theorem for the generalized solution is proved using the method of successive approximations.

Fractional Kirchhoff-type problems with exponential growth without the Ambrosetti–Rabinowitz condition

Abstract The main aim of this paper is to investigate the existence of nontrivial solutions for a class of fractional Kirchhoff-type problems with right-hand side nonlinearity which is subcritical or critical exponential growth (subcritical polynomial growth) at infinity. However, it need not satisfy the Ambrosetti–Rabinowitz (AR) condition. Some existence results of nontrivial solutions are established via Mountain Pass Theorem combined with the fractional Moser–Trudinger inequality.

On the Existence of Periodic and Bounded Solutions for Functional Differential Equations of Pointwise Type with a Strongly Nonlinear Right-Hand Side

Solutions of functional differential equation of pointwise type (FDEPT) are in one-to-one correspondence with the traveling-wave type solutions for the canonically induced infinite-dimensional ordinary differential equation and vice versa. In particular, such infinite-dimensional ordinary differential equations are finite difference analogues of equations of mathematical physics. An important class of traveling-wave type solutions is made up of periodic and bounded traveling-wave type solutions. On the other hand, an important class of such systems is systems with strongly nonlinear potentials (polynomial potentials), for which periodic and bounded traveling wave solutions are studied. Such a problem is equivalent to the study of periodic and bounded solutions of the induced FDEPT to which the present work is devoted.

Research Summary on Implementing Functional Patterns by Synthesizing Inverse Functions

In this research summary we present our recent work on implementing functional patterns with inverse functions in the lazy functional-logic programming language Curry. Our goal is the synthesis of the inverse of any given function in Curry itself. The setting of a functional-logic language especially allows the inversion of non-injective functions. In general, inverse computation is a non-trivial problem in lazy programming languages due to their non-strict semantics. We are so far able to directly derive the inverse function for a limited class of functions, namely those consisting of rules that do not involve both extra variables and non-linear right-hand sides. Because the synthesized definitions are based on standard code, known optimizations techniques can be applied to them. For all other functions we can still provide an inverse function by using non-strict unification.

Existence and uniqueness of elliptic systems with double phase operators and convection terms

In this paper we study quasilinear elliptic systems driven by so-called double phase operators and nonlinear right-hand sides depending on the gradients of the solutions. Based on the surjectivity result for pseudomonotone operators we prove the existence of at least one weak solution of such systems. Furthermore, under some additional conditions on the data, the uniqueness of weak solutions is shown.

Numerical Study of Nonlinear Oscillations in a Clock Frequency MEMS-Generator

Under consideration is some mathematical model of a clock frequency generator, a device of the MEMS class (microelectromechanical systems). We numerically study the solution of the corresponding second-order ordinary differential equation with nonlinear right-hand side and show that there is a region of the model parameters in which the bounded solutions tend to a stable limit cycle in the phase plane and, therefore, the periodic oscillations are stable with respect to the external perturbations. To determine the boundary of the region, we use the parameter continuation method of the solution of the boundary value problem defining the limit cycle. The model leads to the numerical identification of the region of generator operability.

Asymptotics of the Solution of a Singularly Perturbed Second-Order Delay Differential Equation

We consider a singularly perturbed boundary value problem for a second-order ordinary differential equation with nonlinear right-hand side containing functions of delayed argument. We prove the existence of a solution with a transition layer that has a more sophisticated structure than the ones studied before and construct a uniform asymptotic approximation to this solution with respect to a small parameter. Vasil’eva’s method is used when constructing the asymptotic approximation, while the existence theorem is proved by combining the matching method and the asymptotic differential inequality method. Conditions for the existence of a solution with monotone internal transition and boundary layers are stated. An example illustrating the class of problems studied here is given.

Accounting for local thermal and hydraulic parameters of water fouling development in plate heat exchanger

Abstract The complex phenomenon of fouling when heating water solutions can significantly hamper the performance of heat exchangers. The fouling process is considered in an application for a plate heat exchanger (PHE) with enhanced heat transfer that proved to have much lower fouling tendencies than conventional shell and tube heat exchangers. To eliminate the drawbacks of the dimensional fouling model forms the dimensionless its form is developed. It is based on the equation for transport and chemical reaction fouling mechanism initially proposed for other types of fouling media. Thermo-hydraulic mathematical model of PHE under fouling conditions accounting for the distribution of local process parameters along heat transfer surface is presented. It enables to predict not only thermal performance of PHE, but also pressure losses. The mathematical model consists of the system of differential equations with the nonlinear right-hand side. Its solution is implemented with software for the personal computer. The model application is demonstrated with two practical examples. It confirms models' validity and its acceptable accuracy for practical calculations of PHE in industry and also the possibility of proposed dimensionless model application for different fouling substances with the similar types of fouling mechanism.

On Some Inverse Problems for First Order Operator-Differential Equations

We study solvability of the inverse problems of recovering an unknown function on the nonlinear right-hand side of a first order operator-differential equation in some Banach space. The equation is furnished with the Cauchy data, and the overdetermination condition is the value of some operator at a solution. The existence and uniqueness theorems local in time are established.