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For one class of nonlinear wave equations with a small parameter, an initial-boundary value problem with zero boundary conditions is considered. The solution of such a problem is constructed with using series with recurrently calculated coefficients in two ways. In the first case, the method of special series is considered, which is based on the choice of some functions (basic functions), by the powers of these functions the solution of the original problem is presented into a series with recurrently calculated coefficients. In the other case to represent solutions of the problem a combination of Fourier and small parameter methods is used. It is shown that both proposed constructions of series with recurrently calculated coefficients converge to the solution of the initial- boundary value problem on a finite time interval. [ABSTRACT FROM AUTHOR]

We investigate a specific ill-posed nonlinear inverse problem that arise on the financial markets. Precisely, as a benchmark problem, we consider the simultaneous recovery of implied volatility and interest rate functions over a finite interval corresponding to call and put prices for idealized continuous European vanilla options over the same maturity. In order to find these time-dependent functions, we minimize the cost functional which is the sum of the squared error between the market values and the computed values, obtained by implicit difference scheme for the Black–Scholes model. Furthermore, results of numerical experiments are discussed. [ABSTRACT FROM AUTHOR]

In this paper we propose a numerical method for solving regime-switching problem with power utility function. The model is a system of parabolic equations with non-linear gradient term and weakly coupled by exponential non-linearity. First, we approximate the gradient term, involving an auxiliary non-linear function, then we construct implicit-explicit finite difference discretization. We prove discrete comparison principle and on this base we establish minimum principle, i.e. we prove that the numerical solution has the same lower bound as the exact solution. Convergence result in maximal norm is obtained. Numerical results, confirming the theoretical statements are provided. [ABSTRACT FROM AUTHOR]

In this paper, we find approximate solution of the nonlinear Volterra-Fredholm fuzzy integro-differential equation (NVF-FIDE) by using Adomian decomposition method (ADM). We convert this equation to a nonlinear system of Volterra-Fredholm integro-differential equations in crisp case and find an approximate solution of this system. Hence, we obtain an approximation for the fuzzy solution of the nonlinear Volterra-Fredholm fuzzy integro-differential equation. A numerical example is given to demonstrate the validity and applicability of the proposed technique. [ABSTRACT FROM AUTHOR]

We display new extended Strichartz estimates for the solutions to the heat equation with pure power mass-energy inter-critical nonlinearity. As an application, local well-posedness in some Lebesgue spaces assuming initial data with fractional Sobolev regularity. [ABSTRACT FROM AUTHOR]

We prove unconditional well-posedness in the energy space for the solution to the heat equation with pure power mass-energy intercritical nonlinearity, both in focusing and defocusing cases. The main tools used are new nonlinear extended Strichartz estimates. [ABSTRACT FROM AUTHOR]

We perform a new inequality regarding the operator L+ = - Δ + 1- β(Q²) - 2β′(Q²) concerning the ground states Q associated to the nonlinear Schrödinger equations (NLS), ... with d ≥ 1. We present also some results straightforward consequence of an application of such an inequality. [ABSTRACT FROM AUTHOR]

We perform new Morawetz identities, interaction Morawetz identities and their corresponding inequalities for the system of N nonlinear defocusing Schrodinger equations (NLS), with N ≥ 2, ... Here, for all ... and .... are coupling parameters, moreover we require that the nonlinearity parameter p is both L²(Rd × T)-supercritical and H¹(Rd × T)-subcritical (or at most H¹(Rd × T)-critical). [ABSTRACT FROM AUTHOR]