Explicit Formula For Solution Research Articles

Grouped Transformations and Regularization in High-Dimensional Explainable ANOVA Approximation

In this paper we propose a tool for high-dimensional approximation based on trigonometric polynomials where we allow only low-dimensional interactions of variables. In a general high-dimensional setting, it is already possible to deal with special sampling sets such as sparse grids or rank-1 lattices. This requires black-box access to the function, i.e., the ability to evaluate it at any point. Here, we focus on scattered data points and grouped frequency index sets along the dimensions. From there we propose a fast matrix-vector multiplication, the grouped Fourier transform, for high-dimensional grouped index sets. Those transformations can be used in the application of the previously introduced method of approximating functions with low superposition dimension based on the analysis of variance (ANOVA) decomposition where there is a one-to-one correspondence from the ANOVA terms to our proposed groups. The method is able to dynamically detected important sets of ANOVA terms in the approximation. In this paper, we consider the involved least-squares problem and add different forms of regularization: Classical Tikhonov-regularization, namely, regularized least squares and the technique of group lasso, which promotes sparsity in the groups. As for the latter, there are no explicit solution formulas which is why we applied the fast iterative shrinking-thresholding algorithm to obtain the minimizer. Moreover, we discuss the possibility of incorporating smoothness information into the least-squares problem. Numerical experiments in under-, overdetermined, and noisy settings indicate the applicability of our algorithms. While we consider periodic functions, the idea can be directly generalized to non-periodic functions as well.

Vector multi-pole solutions in the [formula omitted]-coupled Hirota equation

In this paper, we study the vector multi-pole (MP) solutions of the r-coupled Hirota equation (RCHE). First of all, based on the Darboux transformation and the N-soliton solution, we derive the explicit formulas of the arbitrary-order vector MP solutions. Then, through the balance between exponential and algebraic terms, we give all asymptotic solitons in the vector double- and triple-pole solutions via an asymptotic analysis method. Furthermore, we analyze the soliton interactions in vector MP solutions. By comparing with the scalar Hirota equation, we find that the RCHE has richer collision properties: the position shift of each component soliton grows logarithmically with r, the asymptotic solitons have the same shape in each component, and their amplitudes are proportional among the components.

Some exponential diophantine equations II: The equationx2–Dy2=kzfor evenk

AbstractLetDbe a nonsquare integer, and letkbe an integer with |k| ≥ 1 and gcd(D,k) = 1. In the partIof this paper, using some properties on the representation of integers by binary quadratic primitive forms with discriminant 4D, M.-H. Le gave a series of explicit formulas for the integer solutions (x,y,z) of the equationx2–Dy2=kz, gcd(x,y) = 1,z> 0 for 2 ∤kor |k| is a power of 2. In this part, we give similar results for the other cases ofk.

Boundary behavior for the heat equation on the half‐line

The initial‐boundary value problem for the heat equation on with nonzero Dirichlet boundary data is studied rigorously, with emphasis on the behavior of the solution along the boundary of the spatiotemporal domain, that is, in the limits or , including the case . Our results specify how the regularity of the solution at the boundary is dictated by the level of smoothness of the relevant initial and boundary data. To the best of our knowledge, for unbounded domains like the half‐line , such results are not available in the literature. The corresponding analysis for the derivatives of the solution is also performed and yields additional compatibility conditions on the initial and boundary data which guarantee the extension of the solution to a function up to the boundary, that is, on . The analysis relies on the explicit solution formula provided by the unified transform, also known as the Fokas method, whose rigorous derivation is presented. The main advantage of this formula, namely, its uniform convergence at the boundary, largely reduces the proof of our results to fundamental tools from real and complex analysis such as Lebesgue's dominated convergence theorem and Cauchy's theorem. Moreover, thanks to uniform convergence, the unified transform solution formula can actually be directly evaluated at the boundary to yield the prescribed initial and boundary data.

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.

Functional differentiation of integral operators of special form and some questions of the inverse interpolation

This article is devoted to the problem of operator interpolation and functional differentiation. Some information about the variational derivatives and explicit formulas for the exact solutions of the simplest equations containing the first variational derivatives of the required functional are given. For functionals defined on sets of functions and square matrices, various interpolating polynomials of the Hermitе type with nodes of the second multiplicity, which contain the first variational derivatives of the interpolated operator, are constructed. The presented solutions of the Hermitе interpolation problems are based on the algebraic Chebyshev system of functions. For analytic functions with an argument from a set of square matrices, explicit formulas for antiderivatives of functionals are obtained. The solution of some differential equations with integral operators of a special form and the first variational derivatives is found. The problem of the inverse interpolation of functions and operators is considered. Explicit schemes for constructing inverse functions and functionals, including the case of functions of a matrix variable, obtained using certain well-known results of interpolation theory, are demonstrated. Data representation is illustrated by a number of examples.

Inverse, ill-posed problems, essentially different solutions and explicit formulas for solutions

A request for an inverse problem, as well as for an incorrect problem produces tens of millions of answers in the Internet. In the past few decades, hundreds of international conferences on these topics have been held annually. Problems of this kind are quite involved, and their numerical analysis requires the development of special methods and numerical algorithms. Explicit formulas provide the main tool for testing these methods and numerical algorithms. The Cauchy problem for an elliptic equation is a classical ill-posed problem, which serves as a model for many inverse and incorrect problems. In the present paper we give a numerically realizable explicit formula for solving the Cauchy problem in a two-dimensional domain for a general second-order linear elliptic equation with analytic coefficients and the Cauchy analytic data on the analytic boundary.

On the steady state of continuous-time stochastic opinion dynamics with power-law confidence

AbstractThis paper introduces a non-linear and continuous-time opinion dynamics model with additive noise and state-dependent interaction rates between agents. The model features interaction rates which are proportional to a negative power of the opinion distances. We establish a non-local partial differential equation for the distribution of opinion distances and use Mellin transforms to provide an explicit formula for the stationary solution of the latter, when it exists. Our approach leads to new qualitative and quantitative results on this type of dynamics. To the best of our knowledge these Mellin transform results are the first quantitative results on the equilibria of opinion dynamics with distance-dependent interaction rates. The closed-form expressions for this class of dynamics are obtained for the two-agent case. However, the results can be used in mean-field models featuring several agents whose interaction rates depend on the empirical average of their opinions. The technique also applies to linear dynamics, namely with a constant interaction rate, on an interaction graph.

Direct and inverse scattering problems for the first‐order discrete system associated with the derivative NLS system

AbstractThe direct and inverse scattering problems are analyzed for a first‐order discrete system associated with the semi‐discrete version of the derivative nonlinear Schrödinger (NLS) system. The Jost solutions, the scattering coefficients, the bound‐state dependency and norming constants are investigated and related to the corresponding quantities for two particular discrete linear systems associated with the semi‐discrete version of the NLS system. The bound‐state data set with any multiplicities is described in an elegant manner in terms of a pair of constant matrix triplets. Several methods are presented to solve the inverse problem to recover the potential values in the first‐order discrete system. One of these methods uses a newly derived, standard discrete Marchenko system using as input the scattering data directly coming from the first‐order discrete system. This new Marchenko method is presented in a way that it is generalizable to other first‐order systems both in the discrete and continuous cases for which a Marchenko system and a Marchenko theory are not yet available. Finally, using the time‐evolved scattering data set, the inverse scattering transform is applied on the corresponding semi‐discrete derivative NLS system, and in the reflectionless case certain explicit solution formulas are presented in closed form expressed in terms of the two matrix triplets.

Well-posedness of the initial-boundary value problem for the Hirota equation on the half line

In this paper we study the regularity properties of Hirota equation on the right half line with data of low regularity. In particular, using an explicit solution formula of the initial and boundary value problem and the restricted norm method, we prove the local existence, uniqueness, and continuous dependence on initial data in Xs,b spaces. Moreover, we obtain the global existence and that the nonlinearity of Hirota equation on the half line is smoother than the initial data.

The Explicit Formula for Solution of Anomalous Diffusion Equation in the Multi-Dimensional Space

This paper intends on obtaining the explicit solution of $$n$$ -dimensional anomalous diffusion equation in the infinite domain with non-zero initial condition and vanishing condition at infinity. It is shown that this equation can be derived from the parabolic integro-differential equation with memory in which the kernel is $$t^{-\alpha}E_{1-\alpha,1-\alpha}(-t^{1-\alpha})$$ , $$\alpha\in(0,1),$$ where $$E_{\alpha,\beta}$$ is the Mittag-Liffler function. Based on Laplace and Fourier transforms the properties of the Fox H-function and convolution theorem, explicit solution for anomalous diffusion equation is obtained.

On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions

<p style='text-indent:20px;'>In this paper we consider the Schrödinger equation with nonlinear derivative term. Our goal is to initiate the study of this equation with non vanishing boundary conditions. We obtain the local well posedness for the Cauchy problem on Zhidkov spaces <inline-formula><tex-math id="M1">\begin{document}$ X^k( \mathbb{R}) $\end{document}</tex-math></inline-formula> and in <inline-formula><tex-math id="M2">\begin{document}$ \phi+H^k( \mathbb{R}) $\end{document}</tex-math></inline-formula>. Moreover, we prove the existence of conservation laws by using localizing functions. Finally, we give explicit formulas for stationary solutions on Zhidkov spaces.</p>

Numerical Study of Stochastic Leontief-Type Model with Impulses

ABSTRACT In this article, a stochastic Leontief model with impulses has been studied, which is represented by a system of stochastic differential-algebraic equations, in both sides of the rectangular constant numerical matrices that form a singular pencil. The system has been considered in terms of the current velocity of the solution, which is a direct analog of the physical velocity of deterministic processes. The proposed approach in this work does not impose restrictions on the size and the form of the matrices included in the Leontief-type system. The Kroenke-Weierstrass transformation of the pencil was conducted by the coefficient matrices to the canonical form has been used to simplify the study of equations. This study also involves two methods: Firstly, using a stochastic differential equation, this was followed by using the so-called mean derivatives of Nelson random processes to describe the solutions of this equation. The distinguishing feature of the work proposed an approach based on the convergence of the theoretical results to the exact one. The findings show that explicit formulas for solutions and solvability conditions are obtained, and for a subsystem resolved with respect to the symmetric derivative. The theorem of existence of solutions for the system under consideration has been proved under certain conditions on the coefficients of the system. Conducting computational experiments on the model confirming the effectiveness of the proposed approach. Error, maximum and minimum of the singular values of matrices.

Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane

In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for the Helmholtz operator on the plane. This family is parameterized by function K(w) which depends on the space dimension. In this paper, based on the results of previous works, the better results can be obtained by choosing the function K(w).

Optimization of additive chemotherapy combinations for an in vitro cell cycle model with constant drug exposures

Proliferation of an in vitro population of cancer cells is described by a linear cell cycle model with n states, subject to provocation with m chemotherapeutic compounds. Minimization of a linear combination of constant drug exposures is considered, with stability of the system used as a constraint to ensure a stable or shrinking cell population. The main result concerns the identification of redundant compounds, and an explicit solution formula for the case where all exposures are nonzero. The orthogonal case, where each drug acts on a single and different stage of the cell cycle, leads to a version of the classic inequality between the arithmetic and geometric means. Moreover, it is shown how the general case can be solved by converting it to the orthogonal case using a linear invertible transformation. The results are illustrated with two examples corresponding to combination treatment with two and three compounds, respectively.

On the nonlinear Schrödinger equation with a boundary condition involving a time derivative of the field

We study the nonlinear Schrödinger equation on the half-line with a new boundary condition presented by Zambon. This new boundary involves a time derivative of the field and was already shown by Zambon to be integrable. In this paper we re-establish the integrability of such a boundary both by using the Sklyanin’s formalism and by using the tool of Bäcklund transformations. Moreover, we present a method to derive explicit formulae for multi-soliton solutions of the boundary problem by virtue of the Darboux transformation method in conjunction with a boundary dressing technique.

Solution of a transport equation with discontinuous coefficients

Abstract In this article, we study initial and initial-boundary value problems for a non-strictly hyperbolic system whose characteristic speed is not smooth and takes values in { - 1 , 0 , 1 } {\{-1,0,1\}} . We construct an explicit formula for the weak solution. We also study the interaction of waves and the large time asymptotic behavior of a solution for the case when the initial data is periodic with zero mean over the period and also for the case when the initial data has compact support.

Initial value problem for the nonconservative zero-pressure gas dynamics system

In this article, we study initial value problem for the zero-pressure gas dynamics system in non conservative form and the associated adhesion approximation. We use adhesion approximation and modi-ed adhesion approximation in the construction of weak asymptotic solution. First we prove a general existence result for the adhesion model for the initial velocity component in $$$H^s \mbox{ for } s$$$ > $$$ \frac{n}{2} + 1$$$ and the initial data for the density component being a $$$C^1$$$ function. Using this, we construct weak asymptotic solution for the system with initial velocity in $$$L^2 \cap L^{\infty}$$$ and the initial density being a bounded Borel measure. Then we make a detailed analysis of the explicit formula for the weak asymptotic solution and generalized solution for the plane-wave type initial data.

Implicit linear difference equations over a non-Archi-medean ring

Over any field an implicit linear difference equation one can reduce to the usual explicit one, which has infinitely many solutions ~ one for each initial value. It is interesting to consider an implicit difference equation over any ring, because the case of implicit equation over a ring is a significantly different from the case of explicit one. The previous results on the difference equations over rings mostly concern to the ring of integers and to the low order equations. In the present article the high order implicit difference equations over some other classes of rings, particularly, ring of polynomials, are studied. To study the difference equation over the ring of integer the idea of considering p-adic integers ~ the completion of the ring of integers with respect to the non-Archimedean p-adic valuation was useful. To find a solution of such an equation over the ring of polynomials it is naturally to consider the same construction for this ring: the ring of formal power series is a completion of the ring of polynomials with respect to a non-Archimedean valuation. The ring of formal power series and the ring of p-adic integers both are the particular cases of the valuation rings with respect to the non-Archimedean valuations of some fields: field of Laurent series and field of p-adic rational numbers respectively. In this article the implicit linear difference equation over a valuation ring of an arbitrary field with the characteristic zero and non-Archimedean valuation are studied. The sufficient conditions for the uniqueness and existence of a solution are formulated. The explicit formula for the unique solution is given, it has a form of sum of the series, converging with respect to the non-Archimedean valuation. Difference equation corresponds to an infinite system of linear equations. It is proved that in a case the implicit difference equation has a unique solution, it can be found using Cramer rules. Also in the article some results facilitating the finding the polynomial solution of the equation are given.

Well-posedness of the initial-boundary value problem for the Schrödinger-Boussinesq system

In the paper, we establish the local well-posedness of the Schödinger-Boussinesq system on the half line with data of low regularity. The proof is based on the explicit solution formula of the linear boundary problem and the restricted norm method. Besides, we prove that the nonlinearity is smoother than the initial data. Our result match the known result on the full line in [9].