Problem For Nonlinear Equation Research Articles

Local Solvability, Blow-up, and Hölder Regularity of Solutions to Some Cauchy Problems for Nonlinear Plasma Wave Equations: II. Potential Theory

Local Solvability, Blow-up, and Hölder Regularity of Solutions to Some Cauchy Problems for Nonlinear Plasma Wave Equations: II. Potential Theory

Dirichlet problem for noncoercive nonlinear elliptic equations with singular drift term in unbounded domains

In this paper, we study a Dirichlet problem for noncoercive nonlinear elliptic equations with first order term in an unbounded domain. We obtain Stampacchia type existence, regularity and uniqueness results, when the singular drift term is controlled through a function in a suitable functional space, strictly containing Lebesgue one. The main tools are a weak maximum principle together with some a priori estimates proved by contradiction.

Approximation of solutions of boundary value problems for integro-differential equations of the neutral type using a spline function method

Boundary value problems for nonlinear integro-differential equations of the neutral type are investigated. A scheme for approximating the boundary value problem solution using cubic splines of defect two is proposed and substantiated. A model example illustrating the proposed approximation scheme is considered.

BLOW-UP PROBLEMS FOR FUJITA-TYPE PARABOLIC SYSTEM INVOLVING TIME-DEPENDENT COEFFICIENTS ON GRAPHS

In this paper, we deal with the blow-up problems for Fujita-type parabolic system involving time-dependent coefficients on graphs. Under appropriate conditions, we prove that the nonnegative solution of the parabolic system blows up in a finite time on finite graphs and locally finite graphs, respectively. The results obtained extend some previous results of [Y. Lin and Y. Wu, Blow-up problems for nonlinear parabolic equations on locally finite graphs, Acta Math. Sci. Ser. B 38(3) (2018) 843–856; Y. Wu, Local existence and blow-up of solutions to Fujita-type equations involving general absorption term on finite graphs, Fractals 30(2) (2022) 2240053].

A Nonlinear Equations Problem Solving Algorithm Based on Exponential Model

A Nonlinear Equations Problem Solving Algorithm Based on Exponential Model

Inverse boundary value problem for nonlinear diffusion equation with non-local time-integral conditions of the second kind2

This work deals with the solvability of inverse boundary value problem with time-dependent unknown coefficient for nonlinear diffusion equation. Definition of classical solution for the considered inverse boundary value problem is introduced. By the Fourier method, the problem is reduced to the system of integral equations. Using the contraction mappings method, the existence and uniqueness of the solution of the system of integral equations are proved. Finally, the existence and uniqueness of the classical solution of original problem are proved.

Existence of solutions to a discrete problems for fourth order nonlinear p-Laplacian via variational method

We consider the boundary value problem for a fourth order nonlinear p-Laplacian difference equation and to prove the existence of at least two nontrivial solutions. Our approach is mainly based on the variational method and critical point theory. One example is included to illustrate the result.

Computation of eigenfunctions of nonlinear boundary-value -transmission problems by developing some approximate techniques

In this study, we investigate a boundary value problem for nonlinear Sturm-Liouville equations with additional transmission conditions at one interior singular point. Known numerical methods are intended for solving initial and boundary value problems without transmission conditions. By modifying the Adomian decomposition method and the differential transform method, we present a new numerical algorithm to compute the eigenvalues and eigenfunctions of the considered boundary-value-transmission problem. Some graphic illustrations of the approximate eigenfunctions are also presented.

Investigation of discrete analogue of one boundary problem of optimal control

In the paper, one boundary value problem of optimal control for discrete two-parameter systems with discrete time is investigated. Such problems are discrete analogues of optimal control problems described by integro-differential partial differential equations of the first order. The initial function is controllable and is defined as the solution of the Cauchy problem for nonlinear ordinary difference equation, the right part of which includes concentrated control. The quality functional presents the sum of two different terms and is a Boltz type functional for the optimal control problem under consideration. The cases of arbitrary, convex and open control area defining the corresponding class of admissible controls are studied. Imposing various natural smoothness conditions on the right-hand sides of the two-dimensional and one-dimensional difference equations under consideration, assuming the convexity of the analogue of the set of admissible velocities of the system under consideration, a special increment of the quality criterion is calculated using a modified functional increment method and, based on its non-negativity along the optimal control, an analogue of the discrete Pontryagin maximum principle is proved. Assuming the convexity of the control area, by linearizing the terms in the functional increment formula and introducing a special variation of the admissible control, an analogue of the linearized maximum condition is proved. In contrast to the continuous case, the linearized maximum principle is not a consequence of the discrete maximum principle and has an independent value as a necessary condition for optimality. In the case of open control area, by introducing a classical variation of the control, the first variation (in the classical sense) of the functional is calculated and established that in the case of open control area, the first variation of the quality criterion equals zero, with its help, an analogue of the Euler equation for the optimal control problem under consideration is obtained. The scheme used in the work also makes it possible to further investigate some cases of degeneration of the obtained necessary conditions of optimality of the first order and to deduce new, constructive necessary conditions of optimality of the second order, allowing to narrow down the set of permissible controls suspicious of optimality.

Initial value problems for nonlinear fractional equations with discontinuous sources

In this paper, we consider initial value problems for nonlinear fractional equations where source functions may discontinuous. We obtain the existence and uniqueness of maximal mild solutions of the problem. We also give some appropriate conditions such that mild solutions of the problem blow-up at a finite time. Furthermore, we discuss the continuous dependence of mild solutions of the problem with respect to fractional order.

Velocity averaging for diffusive transport equations with discontinuous flux

We consider a diffusive transport equation with discontinuous flux and prove the velocity averaging result under non-degeneracy conditions. In order to achieve the result, we introduce a new variant of micro-local defect functionals which are able to ‘recognise’ changes of the type of the equation. As a corollary, we show the existence of a weak solution for the Cauchy problem for non-linear degenerate parabolic equation with discontinuous flux. We also show existence of strong traces at t = 0 $t=0$ for so-called quasi-solutions to degenerate parabolic equations under non-degeneracy conditions on the diffusion term.

Методи розв’язування початкової задачі для нелінійних інтегро-диференціальних рівнянь з оцінкою локальної похибки

One of the modern scientific methods of researching phenomena and pro-cesses is mathematical modeling, which in many cases allows replacing the real process and makes it possible to obtain both a qualitative and a quantitative pic-ture of the process. Since the exact solutions of such models can be found in very individual cases, it is necessary to use approximate methods. In applied mathematics, fractional-rational approximations, which under appropriate con-ditions give a high rate of convergence of algorithms, bilateral and monotonic approximations have become widely used.In this work, using the technique of constructing one-step methods for solv-ing the initial problem for ordinary differential equations and developing the sought solution into a finite continued fraction, a numerical method for solving the Cauchy problem for nonlinear integro-differential equations of the Volterra type is proposed. The values of the parameters at which the nonlinear method of the first and second order of accuracy is obtained are found.Computational formulas are proposed, which at each integration step allow obtaining an upper and lower approximation to the exact solution without additional references to the right-hand side of the integro-differential equation. Calculation formulas, in which the main terms of the local error differ only in sign, form a two-sided method. We take the half-sum of bilateral approximations to the exact solution as the approximate solution at the given integration point, and the absolute value of the half-difference determines the error of the obtained result.The modular nature of the proposed algorithms makes it possible to ob-tain several approximations to the exact solution of the initial problem for the nonlinear integro-differential equation at each point of integration. The comparison of these approximations gives useful information in the matter of choosing the integration step or in assessing the accuracy of the result

The Existence of Solution to Fractional Boundary Value Problem with Riemann-Liouville Type History-State-Based Variable-Order Derivative

The paper is devoted to studying the solutions of boundary value problem for nonlinear fractional differential equation with Riemann-Liouville type history-state-based variable-order derivative. Using Schauder fixed point theorem and Banach fixed point thoerem, we prove the existence and uniqueness of solutions in the Hölder space. Lastly, two examples are given to show the applicability of the existence theorems.

Refined Asymptotic Expansions of Solutions to Fractional Diffusion Equations

In this paper, as an improvement of the paper (Ishige et al. in SIAM J Math Anal 49:2167–2190, 2017), we obtain the higher order asymptotic expansions of the large time behavior of the solution to the Cauchy problem for inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations.

Initial and boundary value problem of fuzzy fractional-order nonlinear Volterra integro-differential equations

Initial and boundary value problem of fuzzy fractional-order nonlinear Volterra integro-differential equations

Inverse problem for Einstein-scalar field equations

The paper introduces a method to solve inverse problems for hyperbolic systems where the leading-order terms are nonlinear. We apply the method to the coupled Einstein-scalar field equations and study the question of whether the structure of space-time can be determined by making active measurements near the world line of an observer. We show that such measurements determine the topological, differential, and conformal structure of the space-time in the optimal chronological diamond-type set containing the world line. In the case when the unknown part of the space-time is vacuum, we can also determine the metric itself. We exploit the nonlinearity of the equation to obtain a rich set of propagating singularities, produced by a nonlinear interaction of singularities that propagate initially as for linear wave equations. This nonlinear effect is then used as a tool to solve the inverse problem for the nonlinear system. The method works even in cases where the corresponding inverse problems for linear equations remain open, and it can potentially be applied to a large class of inverse problems for nonlinear hyperbolic equations encountered in practical imaging problems.

The Lp-regularity properties of nonlocal wave equations and applications

In this paper, the Cauchy problem for linear and nonlinear wave equations is studied. The equation involves abstract operator [Formula: see text] in a Banach space [Formula: see text] and convolution terms. Here, assuming enough smoothness on the initial data and on coefficients, the existence, uniqueness and regularity properties of local and global solutions are established in terms of fractional powers of a given sectorial operator [Formula: see text]. We obtain the regularity properties of a wide class of wave equations by choosing the space [Formula: see text] and the operator [Formula: see text] which appear in the field of physics.

Optimal Location of Exit Doors for Efficient Evacuation of Crowds at Gathering Places

This work deals with the optimal design for the location of the exit doors at meeting places (such as sports centers, public squares, street markets, transport stations, etc.) to guarantee a safer emergency evacuation in events of a sporting, social, entertainment or religious type. This problem is stated as an optimal control problem of nonlinear partial differential equations, where the state system is a reformulation of the Hughes model (coupling the eikonal equation for a density-weighted walking velocity of pedestrians and the continuity equation for conservation of the pedestrian density), the control is the location of the exit doors at the domain boundary (subject to several geometric constraints), and the cost function is related to the evacuation rate. We provide a full numerical algorithm for solving the problem (a finite element technique for the discretization and a gradient-free procedure for the optimization), and show several numerical results for a realistic case.

Corrector results for space-time homogenization of nonlinear diffusion

The present paper concerns a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of corrector results (i.e., strong convergences of solutions with corrector terms) for gradients, diffusion fluxes and time-derivatives without assumptions for smoothness of coefficients. Proofs of the main results are based on the space-time version of the unfolding method, which is deeply concerned with the strong two-scale convergence theory.

Decay estimates of solutions to nonlinear elastic wave equations with viscoelastic terms in the framework of Lp-Sobolev spaces

The Cauchy problem for nonlinear elastic wave equations with viscoelastic damping terms is investigated in the Lp framework. Sharp estimates of the global solutions in t for the higher order derivatives are established. The proof is based on the Gronwall type argument for 1<p<∞, the use of the difference of the propagation speed of the fundamental solutions for p=1 and the bootstrap argument for p=∞, combined with the sharp estimates for the linearized solutions.