Quadratic Differential Equations Research Articles

Computational Technology for Shell Models of Magnetohydrodynamic Turbulence Constructing

The paper discusses the computational technology for constructing one type of small-scale magnetohydrodynamic turbulence models – shell models. Any such model is a system of ordinary quadratic nonlinear differential equations with constant coefficients. Each phase variable is interpreted in absolute value as a measure of the intensity of one of the fields of the turbulent system in a certain range of spatial scales (scale shell). The equations of any shell model must have several quadratic invariants, which are analogues of conservation laws in ideal magnetohydrodynamics. The derivation of the model equations consists in obtaining such expressions for constant coefficients for which the predetermined quadratic expressions will indeed be invariants. Derivation of these expressions «manually» is quite cumbersome and the likelihood of errors in formula transformations is high. This is especially true for non-local models in which large-scale shells that are distant in size can interact. The novelty and originality of the work lie in the fact that the authors proposed a computational technology that allows one to automate the process of deriving equations for shell models. The technology was implemented using computer algebra methods, which made it possible to obtain parametric classes of models in which the invariance of given quadratic forms is carried out absolutely accurately – in formula form. The determination of the parameter values in the resulting parametric class of models is further carried out by agreement with the measures of the interaction of shells in the model with the probabilities of their interaction in a real physical system. The idea of the described technology and its implementation belong to the authors. Some of its elements were published by the authors earlier, but in this work, for the first time, its systematic description is given for models with complex phase variables and agreement of measures of interaction of shells with probabilities. There have been no similar works by other authors previously. The technology allows you to quickly and accurately generate equations for new non-local turbulence shell models and can be useful to researchers involved in modeling turbulent systems.

On formal series solutions to 4th-order quadratic homogeneous differential equations and their convergence

Abstract It is known that all τ functions of the Painlevé equations satisfy the fourth-order quadratic differential equation. Among them, for the III, V, and VI equations, it is possible to express the formal series solutions combinatorially by using conformal blocks. In this paper, we show the convergence of the formal series, including the solutions of more general equations. The convergence of the conformal block function also follows in the case c = 1 by the absolute convergence of τ series, since it is a partial sum of the τ series. We also characterized the form of a homogeneous quadratic equation with a series solution similar to the tau functions of the Painlevé equations.

Bifurcations in planar, quadratic mass-action networks with few reactions and low molecularity

In this paper we study bifurcations in mass-action networks with two chemical species and reactant complexes of molecularity no more than two. We refer to these as planar, quadratic networks as they give rise to (at most) quadratic differential equations on the nonnegative quadrant of the plane. Our aim is to study bifurcations in networks in this class with the fewest possible reactions, and the lowest possible product molecularity. We fully characterise generic bifurcations of positive equilibria in such networks with up to four reactions, and product molecularity no higher than three. In these networks we find fold, Andronov–Hopf, Bogdanov–Takens and Bautin bifurcations, and prove the non-occurrence of any other generic bifurcations of positive equilibria. In addition, we present a number of results which go beyond planar, quadratic networks. For example, we show that mass-action networks without conservation laws admit no bifurcations of codimension greater than m-2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m-2$$\\end{document}, where m is the number of reactions; we fully characterise quadratic, rank-one mass-action networks admitting fold bifurcations; and we write down some necessary conditions for Andronov–Hopf and cusp bifurcations in mass-action networks. Finally, we draw connections with a number of previous results in the literature on nontrivial dynamics, bifurcations, and inheritance in mass-action networks.

Extensibility of Solutions of Nonautonomous Systems of Quadratic Differential Equations and Their Application in Optimal Control Problems

Extensibility of Solutions of Nonautonomous Systems of Quadratic Differential Equations and Their Application in Optimal Control Problems

Diagonally quadratic BSDE with oblique reflection and optimal switching

The present paper is devoted to the study of diagonally quadratic backward stochastic differential equation with oblique reflection. Using a penalization approach, we show the existence of a solution by providing some delicate a priori estimates. We further obtain the uniqueness by verifying the first component of the solution is indeed the value of a switching problem for quadratic BSDEs. Moreover, we provide an extension for the solvability and apply our results to study a risk-sensitive switching problem for functional stochastic differential equations.

Representation of solutions to quadratic 2BSDEs with unbounded terminal values

Second order backward stochastic differential equations (2BSDEs, for short) are one of useful tools in solving stochastic control problems with model uncertainty. In this paper, we prove a representation formula for quadratic 2BSDEs with an unbounded terminal value under a convex assumption on the generator. Because of the unboundedness of the terminal value, we are unable to use some fine properties of BMO martingales, which are often employed in the literature to deal with bounded solutions to quadratic backward stochastic differential equations. Instead, we utilize the θ-technique. We also prove an existence result under an additional assumption that the terminal value is of uniformly continuous.

Weak solutions to coupled quadratic forward backward stochastic differential equations and Sobolev solutions to their related partial differential equations

We establish existence and uniqueness of a weak (in law sense) solution to coupled forward–backward stochastic differential equations (FBSDEs), with possibly discontinuous diffusion coefficient. We assume that the coefficient of the backward component is of quadratic growth. We first prove the existence of a Sobolev solution for the related partial differential equation (PDE) in the Sobolev space , by using compactness arguments. Next, we use Itô–Krylov's formula to get the existence of weak solution to our considered FBSDE. For the uniqueness part, we first establish the weak uniqueness of the FBSDEs then deduce the uniqueness of its related PDE by using the Feynmann–Kac formula.

Efficient Numerical Method for Solving a Quadratic Riccati Differential Equation

This study presents families of the fourth-order Runge–Kutta methods for solving a quadratic Riccati differential equation. From these families, the England version is more efficient than other fourth-order Runge–Kutta methods and practically well-suited for solving initial value problems in general and quadratic Riccati differential equation in particular. The stability analysis of the present method is well-established. In order to verify the accuracy, we compared the numerical solutions obtained using the England version of fourth-order Runge–Kutta method with the recently published works reported in the literature. Several counter examples are solved using the present methods to demonstrate their reliability and efficiency.

Form of solutions to quadratic trinomial partial differential equations with two complex variables

This article describes the from of entire solutions to quadratic trinomial partial differential equations (PDEs). By applying the Nevanlinna theory and the characteristic equation of PDEs, we extend some of the results obtained in [24]. We also provide examples that illustrate our results. For more information see https://ejde.math.txstate.edu/Volumes/2024/23/abstr.html

Bir Direnç ve İki Kapasitörlü Lineer Bir RC Devresi için Cauchy Probleminin Fourier Serisi Cinsinden Çözümü

As it is known, solutions obtained in terms of Fourier series are widely used for more detailed analysis of mathematical models defined for different engineering problems. In this study, for the mathematical analysis of transient events in active RC based linear electronic circuits, the Cauchy problem is defined and solved in terms of Fourier series. For this reason, first of all, the mathematical model needed for the mathematical definition of the investigated problem was created. Based on the defined mathematical model, a quadratic differential equation for the linear RC electric circuit is obtained and the voltage and current changes are analysed for simple special cases using the Fourier series. As a result, mathematical modelling of transient processes occurring in linear RC circuits and mathematical problems defined as the Cauchy problem for such circuits have been solved analytically. It is thought that the results of the research will contribute theoretically and practically to the solution of the problems that arise in the study and design automation of different electronic circuits containing nonlinear circuit elements.

Combined Newton-Gradient Method for Constrained Root-Finding in Chemical Reaction Networks

In this work, we present a fast, globally convergent, iterative algorithm for computing the asymptotically stable states of nonlinear large-scale systems of quadratic autonomous ordinary differential equations (ODE) modeling, e.g., the dynamic of complex chemical reaction networks. Toward this aim, we reformulate the problem as a box-constrained optimization problem where the roots of a set of nonlinear equations need to be determined. Then, we propose to use a projected Newton’s approach combined with a gradient descent algorithm so that every limit point of the sequence generated by the overall algorithm is a stationary point. More importantly, we suggest replacing the standard orthogonal projector with a novel operator that ensures the final solution to satisfy the box constraints while lowering the probability that the intermediate points reached at each iteration belong to the boundary of the box where the Jacobian of the objective function may be singular. The effectiveness of the proposed approach is shown in a practical scenario concerning a chemical reaction network modeling the signaling network of colorectal cancer cells. Specifically, in this scenario the proposed algorithm is proved to be faster and more accurate than a classical dynamical approach where the asymptotically stable states are computed as the limit points of the flux of the Cauchy problem associated with the ODE system.

Using aromas to search for preserved measures and integrals in Kahan’s method

The numerical method of Kahan applied to quadratic differential equations is known to often generate integrable maps in low dimensions and can in more general situations exhibit preserved measures and integrals. Computerized methods based on discrete Darboux polynomials have recently been used for finding these measures and integrals. However, if the differential system contains many parameters, this approach can lead to highly complex results that can be difficult to interpret and analyse. But this complexity can in some cases be substantially reduced by using aromatic series. These are a mathematical tool introduced independently by Chartier and Murua and by Iserles, Quispel and Tse. We develop an algorithm for this purpose and derive some necessary conditions for the Kahan map to have preserved measures and integrals expressible in terms of aromatic functions. An important reason for the success of this method lies in the equivariance of the map from vector fields to their aromatic functions. We demonstrate the algorithm on a number of examples showing a great reduction in complexity compared to what had been obtained by a fixed basis such as monomials.

The five-point bootstrap

We study five-point correlation functions of scalar operators in d-dimensional conformal field theories. We develop a new approach to computing the five-point conformal blocks for exchanged primary operators of arbitrary spin by introducing a generalization of radial coordinates, using an appropriate ansatz, and perturbatively solving two quadratic Casimir differential equations. We then study five-point correlators 〈σσϵσσ〉 in the critical 3d Ising model. We truncate the operator product expansions (OPEs) in the correlator by including a finite number of primary operators with conformal dimension below a cutoff ∆ ⩽ ∆cutoff. We then compute several OPE coefficients involving ϵ and two spinning operators by demanding that the truncated correlator approximately satisfies the crossing relation.

Malliavin Calculus and Its Application to Robust Optimal Investment for an Insider

In the theory of portfolio selection, there are few methods that effectively address the combined challenge of insider information and model uncertainty, despite numerous methods proposed for each individually. This paper studies the problem of the robust optimal investment for an insider under model uncertainty. To address this, we extend the Itô formula for forward integrals by Malliavin calculus, and use it to establish an implicit anticipating stochastic differential game model for the robust optimal investment. Since traditional stochastic control theory proves inadequate for solving anticipating control problems, we introduce a new approach. First, we employ the variational method to convert the original problem into a nonanticipative stochastic differential game problem. Then we use the stochastic maximum principle to derive the Hamiltonian system governing the robust optimal investment. In cases where the insider information filtration is of the initial enlargement type, we derive the closed-form expression for the investment by using the white noise theory when the insider is ’small’. When the insider is ’large’, we articulate a quadratic backward stochastic differential equation characterization of the investment. We present the numerical result and conduct an economic analysis of the optimal strategy across various scenarios.

New solitary wave solutions of the Korteweg-de Vries (KdV) equation by new version of the trial equation method

New solitary wave solutions for the Korteweg-de Vries (KdV) equation by a new version of the trial equation method are attained. Proper transformation reduces the Korteweg-de Vries (KdV) equation to a quadratic ordinary differential equation that is fully integrated using the new version trial equation approach. The family of solitary wave solutions of the reduced equation ensures a combined expression for the Korteweg-de Vries (KdV) equation, which contains exact solutions derived in recent years using different integration methods. The analytic solution of the reduced equation permits to find exact solutions for the Korteweg-de Vries (KdV) equation, providing a variety of new solitary wave solutions that have not been reported before.

Mean escape time of switched Riccati differential equations

Riccati differential equations are a class of first-order quadratic ordinary differential equations and have various applications in systems and control theory. In this study, we analyzed a switched Riccati differential equation driven by a Poisson-like stochastic signal. We specifically focused on computing the mean escape time of the switched Riccati differential equation. The contribution of this study is twofold. We first show that, under the assumption that the subsystems described as deterministic Riccati differential equations escape in finite time regardless of their initial state, the mean escape time of the switched Riccati differential equation admits a power series expression. To further expand the applicability of this result, we then present an approximate formula to compute the escape time of deterministic Riccati differential equations. Numerical simulations were performed to illustrate the obtained results.

Malliavin Regularity of Non-Markovian Quadratic BSDEs and Their Numerical Schemes

We study both Malliavin regularity and numerical approximation schemes for a class of quadratic backward stochastic differential equations (QBSDEs for short) in cases where the terminal data need not be a function of a forward diffusion. By using the connection between the QBSDE under study and some backward stochastic differential equations (BSDEs) with global Lipschitz coefficients, we firstly prove Lq, (q≥2) existence and uniqueness results for QBSDE. Secondly, the Lp-Hölder continuity of the solutions is established for (q>4 and 2≤p<q2). Then, we analyze some numerical schemes for our systems and establish their rates of convergence. Moreover, our results are illustrated with three examples.

Analytical and Physical Investigation on Source Resistance in InxGa1−xAs Quantum-Well High-Electron-Mobility Transistors

We present a fully analytical model and physical investigation on the source resistance (RS) in InxGa1-xAs quantum-well high-electron mobility transistors based on a three-layer TLM system. The RS model in this work was derived by solving the coupled quadratic differential equations for each current component with appropriate boundary conditions, requiring only six physical and geometrical parameters, including ohmic contact resistivity (ρc), barrier tunneling resistivity (ρbarrier), sheet resistances of the cap and channel regions (Rsh_cap and Rsh_ch), side-recessed length (Lside) and gate-to-source length (Lgs). To extract each model parameter, we fabricated two different TLM structures, such as cap-TLM and recessed-TLM. The developed RS model in this work was in excellent agreement with the RS values measured from the two TLM devices and previously reported short-Lg HEMT devices. The findings in this work revealed that barrier tunneling resistivity already played a critical role in reducing the value of RS in state-of-the-art HEMTs. Unless the barrier tunneling resistivity is reduced considerably, innovative engineering on the ohmic contact characteristics and gate-to-source spacing would only marginally improve the device performance.

Exactly Solvable Quadratic Differential Equation Systems Through Generalized Inversion

We study the autonomous systems of quadratic differential equations of the form {dot{x}}_i(t)={textbf{x}}(t)^T {textbf{A}}_i {textbf{x}}(t) + {textbf{v}}_i^T {textbf{x}}(t) with {textbf{x}}(t) = (x_1(t),x_2(t), ldots ,x_i(t),dots ) which, in general, cannot be solved exactly. In the present paper, we introduce a subclass of analytically solvable quadratic systems, whose solution is realized through a multi-dimensional generalization of the inversion which transforms a quadratic system into a linear one. We provide a constructive algorithm which, on one hand, decides whether the system of differential equations is analytically solvable with the inversion transformation and, on the other hand, provides the solution. The presented results apply for arbitrary, finite number of variables.

Control problem for quadratic parabolic differential equations with sparse sensor sets of finite volume or anisotropically decaying density

We prove observability and null-controllability for quadratic parabolic differential equations. The sensor set is allowed to be sparse and have finite volume if the generator has trivial singular space S. In the case of generators with singular space S ≠ {0} the sensor set is permitted to decay in directions determined by S. The proof is based on dissipation estimates for the quadratic differential operator with respect to spectral projections of partial harmonic oscillators and corresponding uncertainty relations.