Quasilinear Differential Systems Research Articles

Derivation of a quasi-linear second-order elliptic-parabolic model for the efficiency of silicon solar cells

In this note, we propose a mathematical model for the efficiency of a silicon solar cell. The model is rigorously derived from continuity and transport assumptions, along with the Poisson equations to describe the behavior of charge carriers in a semiconductor. The photovoltaic phenomenon was thoroughly studied to develop the model, and some parameters that play a crucial role in its efficiency have been identified. Those parameters include the diffusion, the mobility and the absorption coefficients, the permittivity, the regions width and the minority carrier lifetime. The mathematical model obtained herein is a quasi-linear second-order elliptic-parabolic partial differential equation system with strong coupling and interface. The model is defined over a spatial domain composed of three regions with different electrical properties. The fundamental physical laws of the photovoltaic effect are employed here to develop the mathematical model and its boundary conditions.

Positive radial solutions for Dirichlet problem of quasilinear differential system with mean curvature operator in Minkowski space

In this paper, we are considered with the Dirichlet problem of quasilinear differential system, involving the mean curvature operator in Minkowski space $$\begin{aligned} {\mathcal {M}}(w)=\text {div}\left( \frac{\nabla w}{\sqrt{1-|\nabla w|^2}}\right) , \end{aligned}$$ in a ball in $${\mathbb {R}}^N$$ . Using global bifurcation technique, we obtain the existence of an unbounded branch of positive radial solutions, which is unbounded in positive $$\lambda $$ -direction.

Multiplicity of positive radial solutions for systems with mean curvature operator in Minkowski space

In this paper, we are considered with the Dirichlet problem of quasilinear differential system with mean curvature operator in Minkowski space $ \mathcal{M}(w): = \text{div}\Big(\frac{\nabla w}{\sqrt{1-|\nabla w|^2}}\Big), $ in a ball in $ \mathbb{R}^N $. In particular, we deal with this system with Lane-Emden type nonlinearities in a superlinear case, by using the Leggett-Williams' fixed point theorem, we obtain the existence of three positive radial solutions.

ON THE EXISTENCE OF A CONDITIONALLY PERIODIC SOLUTION OF A QUASILINEAR SYSTEM DIFFERENTIAL EQUATION IN THE CRITICAL CASE

In the theory of nonlinear oscillations one often encounters conditionally periodic oscillations resulting from the superposition of several oscillations with frequencies incommensurable with each other. When finding a solution to a resonant quasilinear differential system in the form of a conditionally periodic function, the problem of a small denominator arises. Consequently, the proof of the existence and even more the construction of such a solution is not an easy task. In this article, drawing on the work of V.I. Arnold, I. Moser, and other researchers proved the existence and constructed a conditionally periodic solution of a second-order quasilinear differential system in the critical case. Accelerated convergence method by N.N. Bogolyubova, Yu.A. Mitropolsky, A.M. Samoylenko. The result can be applied to construct a conditionally periodic solution of specific differential systems.

Perturbations of planar quasilinear differential systems

The quasilinear differential systemx′=ax+b|y|p⁎−2y+k(t,x,y),y′=c|x|p−2x+dy+l(t,x,y) is considered, where a, b, c and d are real constants with b2+c2>0, p and p⁎ are positive numbers with (1/p)+(1/p⁎)=1, and k and l are continuous for t≥t0 and small x2+y2. When p=2, this system is reduced to the linear perturbed system. It is shown that the behavior of solutions near the origin (0,0) is very similar to the behavior of solutions to the unperturbed system, that is, the system with k≡l≡0, near (0,0), provided k and l are small in some sense. It is emphasized that this system can not be linearized at (0,0) when p≠2, because the Jacobian matrix can not be defined at (0,0). Our result will be applicable to study radial solutions of the quasilinear elliptic equation with the differential operator r−(γ−1)(rα|u′|β−au′)′, which includes p-Laplacian and k-Hessian.

Local Normal Forms of Autonomous Quasi-Linear Constrained Differential Systems

The paper presents a study of impasse (singular) points of autonomous quasi-linear constrained differential systems, also called differential-algebraic equations. The interest in such systems is motivated by their applications in various problems of pure and applied mathematics, including control theory, biology, and electric engineering. Local normal forms of such systems in a neighborhood of their impasse points are established.

On a Quasi-Linear Partial Differential Algebraic System of Equations

A mixed problem for a quasi-linear first-order partial differential algebraic system of equations of index $$(1,0)$$ with two independent variables is considered. An existence theorem for this problem is proved using the method of characteristics and an iterative method.

Stability of a Spline Collocation Difference Scheme for a Quasi-Linear Differential Algebraic System of First-Order Partial Differential Equations

A quasi-linear differential algebraic system of partial differential equations with a special structure of the pencil of Jacobian matrices of small index is considered. A nonlinear spline collocation difference scheme of high approximation order is constructed for the system by approximating the required solution by a spline of arbitrary in each independent variable. It is proved by the simple iteration method that the nonlinear difference scheme has a solution that is uniformly bounded in the grid space. Numerical results are demonstrated using a test example.

Controllability of fractional impulsive quasilinear differential systems with state dependent delay

In this paper, we study some controllability results for an abstract fractional impulsive quasilinear integro-differential system with state-dependent delay. We will use the concepts of fractional calculus, measure of noncompactness and abstract phase space to come up with a new set of sufficient conditions for controllability by using Monch’s fixed point theorem. Finally, an application is given to illustrate the obtained results.

О существовании условно-периодического решения одной квазилинейной дифференциальной системы в критическом случае

О существовании условно-периодического решения одной квазилинейной дифференциальной системы в критическом случае

Elliptic Euler–Poisson–Darboux equation, critical points and integrable systems

The structure and properties of families of critical points for classes of functions obeying the elliptic Euler–Poisson–Darboux equation E(1/2, 1/2) are studied. General variational and differential equations governing the dependence of critical points in variational (deformation) parameters are found. Explicit examples of the corresponding integrable quasi-linear differential systems and hierarchies are presented. There are the extended dispersionless Toda/nonlinear Schrödinger hierarchies, the ‘inverse’ hierarchy and equations associated with the real-analytic Eisenstein series among them. The specific bi-Hamiltonian structure of these equations is also discussed.

On one special case of the existence of solutions of quasilinear differential systems represented by Fourier series with slowly varying parameters

For a quasilinear differential system of second order, whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we obtain conditions for the existence of a particular solution of analogous structure in one special case.

On some resonance cases in quasilinear differential systems with slowly varying parameters

For a quasilinear differential system of the second order whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we find features of existence of a partial solution of analogous structure with slowly varying coefficients and frequency in the case of purely imaginary eigenvalues of the matrix of the linear part that satisfy a certain resonance relation.

On solutions of a quasilinear differential second-order system represented by Fourier series with slowly varying parameters in some critical cases

We study a quasilinear second-order differential system whose coefficients are represented in the form of absolutely and uniformly convergent Fourier series with slowly varying coefficients and the frequency. The conditions of existence of a particular solution of a similar structure are obtained in the case where the eigenvalues of the matrix of the linear part of the system are equal to zero.

Lyapunov-type inequality for a class of quasilinear systems

We give a generalization of the Lyapunov-type inequality for the second-order quasilinear differential equations given in De Nápoli and Pinasco (2006) [9] to a class of quasilinear differential systems of dimension n. The generalization is carried out in such a way that more general nonlinearities are allowed in our system than those in [9]. The results of this paper generalize some earlier results on this topic.

Elliptic and parabolic inequalities with point singularities on the boundary

It is shown that various quasilinear elliptic and parabolic differential inequalities and systems of such inequalities defined on bounded domains, and which have point singularities on the boundary do not have solutions. The method of nonlinear capacity is used in the proof. Examples show that the conditions obtained by this method cannot be improved in the class of problems under consideration.Bibliography: 14 titles.

Bifurcation results to some quasilinear differential equation systems

The main purpose of this paper is to present a new global bifurcation approach to the studies of certain quasilinear differential equation systems.

Numerical solution of a quasilinear algebraic differential system by the spline collocation method

In this paper, a quasilinear algebraic differential system is considered. To solve the system numerically, a spline collocation method is used. A convergence theorem for numerical processes is proved. Results of numerical experiments are presented.

Classical solutions of quasilinear functional differential systems on the Haar pyramid

Classical solutions of quasilinear functional differential systems on the Haar pyramid

The solvability for a class of singularly perturbed quasi-linear differential system

The solvability for a class of singularly perturbed Robin problem of quasi-linear differential system is considered. Using the boundary layer corrective method the formal asymptotic solution is constructed. And using the theory of differential inequality the uniform validity of the asymptotic expansions for solution is proved.