This paper is concerned with solving Cauchy problem for parabolic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, numerical convergence analysis of the energy-like minimization method is carried out and leads to an adapted stopping criterion depending on noise rate for the minimization process. Numerical...

A method of solving the eikonal equation, in either flat or curved space-times, with arbitrary Cauchy data, is extended to the case of data given on a characteristic surface. We find a beautiful relationship between the Cauchy and characteristic data for the same solution, namely they are related by a Legendre transformation. From the resulting solutions, we study and describe the wave-front si...

We explore a function theory connected with the principal series representation of SL(2, R) in contrast to standard complex analysis connected with the discrete series. We construct counterparts for the Cauchy integral formula, the Hardy space, the Cauchy-Riemann equation and the Taylor expansion. ∗Supported by grant 3GP03196 of the FWO-Vlaanderen (Fund of Scientific ResearchFlanders), Scientif...

The causality structure of two-dimensional manifolds with degenerate metrics is analysed in terms of global solutions of the massless wave equation. Certain novel features emerge. Despite the absence of a traditional Lorentzian Cauchy surface on manifolds with a Euclidean domain it is possible to uniquely determine a global solution (if it exists), satisfying well de ned matching conditions at ...

In this paper we propose a computational method for solving a Cauchy problem of Laplace’s equation. By using the Green formula, the Cauchy problem is transformed to a moment problem so that numerical computations by using a regularization technique can be achieved. Stability estimation and suitable choice of regularization parameter for the proposed method are also given. For numerical verifica...

The Cauchy problem for harmonic maps from Minkowski space with its standard flat metric to a certain non-constant curvature Lorentzian 2-metric is studied. The target manifold is distinguished by the fact that the Euler–Lagrange equation for the energy functional is Darboux integrable. The time evolution of the Cauchy data is reduced to an ordinary differential equation of Lie type associated t...

We develop a numerical method based on Sine functions to obtain an approximate solution of a one-dimensional Cauchy singular integral equation (CSIE) over an arbitrary, smooth, open arc L of finite length in the complex plane. At the outset, we reduce the CSIE to a Fredholm integral equation of the second kind via a regularization procedure. We then obtain an approximate solution to the Fredhol...

By space-fractional (or L evy-Feller) diiusion processes we mean the processes governed by a generalized diiusion equation which generates all L evy stable probability distributions with index (0 < 2), including the two symmetric most popular laws, Cauchy (= 1) and Gauss (= 2). This generalized equation is obtained from the standard linear diiusion equation by replacing the second-order space d...

We consider a nonlinear partial differential equation for complexvalued functions which is related to the two-dimensional stationary Schrödinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation as, e.g., the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical “one-dimensional” results we discuss n...

We prove existence and uniqueness of strong generalized entropy solution to the Cauchy problem for an infinite dimensional system of Keyfitz-Kranzer type, in which the unknown vector takes its value in an arbitrary Banach space. We study the Cauchy problem for an equation ut + (φ(‖u‖)u)x = 0 (1) with initial condition u(0, x) = u0(x). (2) Here the unknown vector u = u(t, x) is defined in a half...