Estimation of Cauchy data for a first-order nonlinear hyperbolic equation

A method for solving first order fuzzy differential equation

Time Estimates for the Cauchy Problem for a Third-order Hyperbolic Equation

A classical solution is considered for the Cauchy problem: (utt −∆u)t +utt − α∆u = f(x,t), x ∈ R3, t > 0; u(x,0) = f0(x), ut(x,0) = f1(x), and utt(x) = f2(x), x ∈R3, where α= const, 0<α< 1. The above equation governs the propagation of time-dependent acoustic waves in a relaxingmedium. A classical solution of this problem is obtained in the form of convolutions of the right-hand side and the in...

Difference Methods for Nonlinear First-Order Hyperbolic Systems of Equations

Two difference methods for approximating some first-order nonlinear hyperbolic differential equations are considered. The methods apply to problems arising in a number of physical applications. Each of the methods is explicit and can be implemented on a computer easily. It is proved that the methods are first-order convergent in the maximum norm. For one of the methods in order to obtain conver...

Generalized Solutions to Nonlinear First Order Cauchy Problems

The recent significant enrichment, see [14] through [15], of the Order Completion Method for nonlinear systems of PDEs [12] resulted in the global existence of generalized solutions to a large class of such equations. In this paper we consider the existence and regularity of the generalized solutions to a family of nonlinear first order Cauchy problems. The spaces of generalized solutions are o...

Cauchy Problem of Nonlinear Schrödinger Equation with Initial Data

In this paper, we consider in Rn the Cauchy problem for the nonlinear Schrödinger equation with initial data in the Sobolev space W s,p for p < 2. It is well known that this problem is ill posed. However, we show that after a linear transformation by the linear semigroup the problem becomes locally well posed in W s,p for 2n n+1 < p < 2 and s > n(1 − 1 p ). Moreover, we show that in one space d...