Right-hand Side Function Research Articles

$C^\infty$ Local solutions of elliptical $2-$Hessian equation in $\mathbb{R}^3$

In this work, we study the existence of $C^{\infty}$ local solutions to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case that the right hand side function $f$ possibly vanishes, changes the sign, is positively or negatively defined. We also give the convexities of solutions which are related with the annulation or the sign of right-hand side function $f$. The associated linearized operator are uniformly elliptic.

Solving systems of IVPs with discontinuous derivatives—Numerical experiments

We deal in this paper with solving IVPs with singular right-hand side. We consider two recent algorithms of Kacewicz and Przybyłowicz for solving systems of IVPs with right-hand side functions which are globally Lipschitz continuous and piecewise r-smooth with piecewise Hölder rth partial derivatives with Hölder exponent ρ∈(0,1]. The singularity hypersurface is defined by the zeros of an unknown event function. We run several numerical experiments to verify the theoretical results of Kacewicz and Przybyłowicz. Our tests confirm that the bounds on the error O(n−(r+ρ)) can be achieved with O(n) function evaluations, where n is a number of discretization points.

On a problem for a system of two the second order differential equations via the theory of vector fields

We consider Dirichlet boundary value problem for systems of two second-order differential equations with nonlinear continuous and bounded functions in right-hand sides. We prove the existence of a nontrivial solution to the problem comparing behaviors of solutions of auxiliary Cauchy problems at zero solution and at infinity.

Stability analysis for optimal control problems governed by semilinear evolution equation

In this paper, the stability of solutions of optimal control for the distributed parameter system governed by a semilinear evolution equation with compact control set in the space \(L^{1}(0,T;E)\) is discussed. The stability results for optimal control problems with respect to the right-hand side functions are obtained by the theory of set-valued mapping and the definition of essential solutions for optimal control problems.

Structure of Solutions for Continuous Linear Programs with Constant Coefficients

We consider continuous linear programs over a continuous finite time horizon $T$, with linear cost coefficient functions, linear right-hand side functions, and a constant coefficient matrix, as well as their symmetric dual. We search for optimal solutions in the space of measures or of functions of bounded variation. These models generalize the separated continuous linear programming models and their various duals, as formulated in the past by Anderson, by Pullan, and by Weiss. In a recent paper, we have shown that under a Slater-type condition, these problems possess optimal strongly dual solutions. In this paper, we give a detailed description of optimal solutions and define a combinatorial analogue to basic solutions of standard LP. We also show that feasibility implies existence of strongly dual optimal solutions without requiring the Slater condition. We present several examples to illustrate the richness and complexity of these solutions.

Existence and stability analysis for nonlinear optimal control problems with $1$-mean equicontinuous controls

In this paper, the existence and stability of solutions of nonlinear optimal control problems with $1$-mean equicontinuous controls are discussed. In particular, a new existence theorem is obtained without convexity assumption. We investigate the stability of the optimal control problem with respect to the right-hand side functions, which is important in computational methods for optimal control problems when the function is approximated by a new function. Due to lack of uniqueness of solutions for an optimal control problem, the stability results for a class of optimal control problems with the measurable admissible control set is given based on the theory of set-valued mappings and the definition of essential solutions for optimal control problems. We show that the optimal control problems, whose solutions are all essential, form a dense residual set, and so every optimal control problem can be closely approximated arbitrarily by an essential optimal control problem.

Randomized and quantum complexity of nonlinear two-point BVPs

We deal with the complexity of nonlinear BVPs with nonlinear two-point boundary conditions. We consider the randomized and quantum models of computation. We assume that the right-hand side function is r times differentiable with all derivatives bounded by a constant. We show that the ε-complexity is roughly of order ε-1/(r+1/2) in the randomized setting, and ε-1/(r+1) in the quantum setting. We compare our results with known results in the deterministic setting. The speed-up of the randomized computations with respect to the deterministic computations is by 1/(r(2r+1)) in the exponent of 1/ε, and the speed-up of the quantum computations by 1/(r(r+1)) in the exponent.

Complexity of the derivative-free solution of systems of IVPs with unknown singularity hypersurface

It is well-known that classical integration schemes designed for regular systems of IVPs become inefficient for piecewise regular problems. To avoid a significant increase of the cost of obtaining an ε-approximation for piecewise regular problems, special adaptive algorithms are usually designed. Constructing such algorithms is often based on the knowledge of a ‘switching function’ whose zeros define the singularity hypersurface, or it relies on certain heuristic arguments. In this paper we analyze the worst-case ε-complexity of globally continuous and piecewise regular problems with unknown smooth switching function. We consider problems with right-hand side functions from the class Fr,ϱ which consists of piecewise r-smooth functions with Hölder r-th partial derivatives with the exponent ϱ∈(0,1]. In contrast to our previous work, we restrict ourselves to information defined only by values of the right-hand side function (computation of partial derivatives is not allowed). We design an algorithm that is based on the computation of Lagrange interpolation polynomials to locate singularity points and to approximate a function of a single variable at intermediate stages. A rigorous analysis (independent of heuristic arguments) of its error and cost is given. It turns out that both the error and the cost of the algorithm are of the same order as for regular problems. That is, no reduction of the order is observed when passing through singularities, and the cost of the algorithm remains at the same level as for regular problems. In the class Fr,ϱ the algorithm has the worst-case error O(m−(r+ϱ)) (in the Chebyshev norm) and the total cost O(m), where m is the number of discretization points. This leads to the conclusion that the worst-case ε-complexity of the considered problem in Fr,ϱ is Θ(ε−1/(r+ϱ)), and the minimal cost is achieved by the defined algorithm that only needs function evaluations. The auxiliary algorithm that locates singularities and approximates a piecewise regular function of a single variable is also original, and it may be of interest per se.

Multidimensional ARKN methods for general oscillatory second-order initial value problems

Based on B-series theory, the order conditions of the multidimensional ARKN methods are presented for the general multi-frequency and multidimensional oscillatory second-order initial value problems by Wu et al. (2009). These multidimensional ARKN methods exactly integrate the multi-frequency and multidimensional unperturbed oscillators. In this paper, we pay attention to the analysis of the concrete multidimensional ARKN methods for the general multi-frequency oscillatory second-order initial value problems whose right-hand side functions depend on bothyandy′ (the class of physical problems which fall within its scope is broader). Numerical experiments are carried out to show that the new multidimensional ARKN methods are more efficient compared with some well-known methods for dealing with the oscillatory problems in the scientific literature.

Регулярність розв'язків крайових задач для дифузійно-хвильового рівняння з узагальненими функціями в правих частинах

Доведено однозначну розв'язність першої крайової задачі для рівняння$$u^{(\beta)}_t-a(t)u_{xx}=F(x,t), \;\;\; (x,t) \in(0,l)\times (0,T]$$з дробовою похідною $u^{(\beta)}_t$Рімана-Ліувілля порядку $\beta\in (0,2)$, додатним гладким коефіцієнтом $a(t)$, узагальненими функціями в правих частинахта та встановлено деякі достатні умови регулярності його розв'язку за змінною $t$.

Singular solutions with exponential growth to Protter’s problems

Four-dimensional boundary value problems which were formulated by Proter for the nonhomogeneous wave equation are studied. They can be considered as multidimensional versions of the Darboux problems in ℝ2. Protter’s problem is not well posed in the frame of classical solvability. On the other hand, it is known that the unique generalized solution may have a strong power-type singularity at one boundary point. This singularity is isolated at the vertex of the characteristic cone and does not propagate along the cone. Some known results suggest that the solution may have at most exponential growth. We construct an infinitely smooth right-hand side function such that the corresponding generalized solution to Protter’s problem has an exponential singularity.

Protter-Morawetz multidimensional problems

About 50 years ago M.H. Protter introduced boundary value problems that are multidimensional analogues of the classical plane Morawetz problems for equations of mixed hyperbolic-elliptic type that model transonic fluid flows. Up to now there are no general existence results for the Protter-Morawetz multidimensional problems, and an understanding of the situation is not at hand. At the same time, Protter also formulated boundary value problems in the hyperbolic part of the domain—the nonhomogeneous wave equation is studied in a (3+1)-D domain bounded by two characteristic cones and a non-characteristic ball. These problems could be considered as multidimensional variants of the Darboux problem in ℝ2. In the frame of classical solvability the hyperbolic Protter problem is not Fredholm, because it has an infinite-dimensional cokernel. On the other hand, it is known that the unique generalized solution of a Protter problem may have a strong power-type singularity even for some very smooth right-hand side functions. This singularity is isolated at the vertex O of the boundary light cone and does not propagate along the characteristic cone. In the general case of smooth right-hand side function, some necessary and sufficient conditions for the existence of a bounded solution are given and a priori estimates for the solution are found. The semi-Fredholm solvability of the problem is proved.

Dynamically consistent numerical methods for general productive–destructive systems

The dynamic consistency of a class of non-standard finite-difference schemes is analysed for general 2D and 3D productive–destructive systems (PDS). Based on those results a methodology for construction of positive and elementary stable non-standard numerical methods is developed. The numerical techniques are based on a non-local modelling of the right-hand side function and a non-standard treatment of the time derivative. This discretization approach leads to significant qualitative improvements in the behaviour of the numerical solutions. The explicit form of the proposed new schemes makes them a computationally effective tool in simulations of the dynamics of systems of biological, chemical and physical interactions that are naturally modelled by PDS. Applications to several specific biological systems are presented.

The determination of heat sources in two dimensional inverse steady heat problems by means of the method of fundamental solutions

This article deals with the inverse determination of heat sources in steady 2-D heat conduction problems. The problem is described by Poisson's equation in which the right-hand side function is unknown. The identification of the strength of a heat source is given by using the boundary conditions for the temperature and heat flux at chosen points on the boundary. For the solution of the inverse problem of identification of the heat source, the method of fundamental solutions with radial basis functions and the Tikhonov regularization technique is proposed. Accurate results have been obtained for six test problems where the analytical solutions are available.

A Fourth-Order Boundary Value Problem with One-Sided Nagumo Condition

The aim of this paper is to study a fourth-order separated boundary value problem with the right-hand side function satisfying one-sided Nagumo-type condition. By making a series of a priori estimates and applying lower and upper functions techniques and Leray-Schauder degree theory, the authors obtain the existence and location result of solutions to the problem.

On the complexity of a two-point boundary value problem in different settings

We study the complexity of a two-point boundary value problem. We concentrate on the linear problem of order k with separated boundary conditions. Right-hand side functions are assumed to be r times differentiable with all derivatives bounded by a constant. We consider three models of computation: deterministic with standard and linear information, randomized and quantum. In each setting, we construct an algorithm for solving the problem, which allows us to establish upper complexity bounds. In the deterministic setting, we show that the use of linear information gives us a speed-up of at least one order of magnitude compared with the standard information. For randomized algorithms, we show that the speed-up over standard deterministic algorithms is by 1/2 in the exponent. For quantum algorithms, we can achieve a speed-up by one order of magnitude. We also provide lower complexity bounds. They match upper bounds in the deterministic setting with the standard information, and almost match upper bounds in the randomized and quantum settings. In the deterministic setting with the linear information, a gap still remains between the upper and lower complexity bounds.

Euler-Cauchy Using Undetermined Coefficients

SummaryThe Euler-Cauchy equation is often the first higher order differential equation with variable coefficients introduced in an undergraduate differential equations course. Putting a non-homogeneous Euler-Cauchy equation on an exam in such a course, I was surprised when some of my students decided to apply the method of undetermined coefficients, supposedly guaranteed to work only for constant-coefficient equations, and obtained the correct answer! It turns out that a particular solution to this equation has a form similar to that of standard undetermined coefficients, if the right-hand side function is of a certain type. Thus the Euler-Cauchy equations can be solved without using variation of parameters or a substitution transforming the equation to a constant-coefficient equation.

Sensitivity Analysis in Linear Semi-Infinite Programming via Partitions

This paper provides sufficient conditions for the optimal value function of a given linear semi-infinite programming (LSIP) problem to depend linearly on the size of the perturbations, when these perturbations involve either the cost coefficients or the right-hand side function or both, and they are sufficiently small. Two kinds of partitions are considered. The first concerns the effective domain of the optimal value as a function of the cost coefficients and consists of maximal regions on which this value function is linear. The second class of partitions considered in this paper concerns the index set of the constraints through a suitable extension of the concept of optimal partition from ordinary to LSIP. These partitions provide convex sets, in particular, segments, on which the optimal value is a linear function of the size of the perturbations, for the three types of perturbations considered in this paper.

Stable zero duality gaps in convex programming: Complete dual characterisations with applications to semidefinite programs

The zero duality gap that underpins the duality theory is one of the central ingredients in optimisation. In convex programming, it means that the optimal values of a given convex program and its associated dual program are equal. It allows, in particular, the development of efficient numerical schemes. However, the zero duality gap property does not always hold even for finite-dimensional problems and it frequently fails for problems with non-polyhedral constraints such as the ones in semidefinite programming problems. Over the years, various criteria have been developed ensuring zero duality gaps for convex programming problems. In the present work, we take a broader view of the zero duality gap property by allowing it to hold for each choice of linear perturbation of the objective function of the given problem. Globalising the property in this way permits us to obtain complete geometric dual characterisations of a stable zero duality gap in terms of epigraphs and conjugate functions. For convex semidefinite programs, we establish necessary and sufficient dual conditions for stable zero duality gaps, as well as for a universal zero duality gap in the sense that the zero duality gap property holds for each choice of constraint right-hand side and convex objective function. Zero duality gap results for second-order cone programming problems are also given. Our approach makes use of elegant conjugate analysis and Fenchel's duality.

Multiple global bifurcation branches for nonlinear Picard problems

In this paper we prove the global bifurcation theorem for the nonlinear Picard problem. The right-hand side function ' is a Caratheodory map, not differentiable at zero, but behaving in the neighbourhood of zero as specified in details below. We prove that in some interval [a,b] � R the Leray-Schauder degree changes, hence there exists the global bifurcation branch. Later, by means of some approximation techniques, we prove that there exist at least two such branches.