In this paper, we consider the inverse problem for the Laplace equation in two-dimensions which requires the determination of the location, size and shape of an unknown, or partially unknown, portion @ of the boundary @ of a solution domain R from additional Cauchy data on the remaining portion of the boundary 1⁄4 @ . This problem arises in the study of quantitative non-destructive evaluation o...

In this paper, spectral properties of the Cauchy problem pantograph equation are examined. We establish boundaries parameter interval in which remains a Volterra problem.

This work investigates the electrical impedance tomography (EIT) problem in the case when only one or two pairs of Cauchy data is available, which is known to be very difficult in achieving high reconstruction quality owing to its severely ill-posed nature. We propose a simple and efficient direct sampling method (DSM) to locate inhomogeneous inclusions. A new probing function based on the dipo...

In this article, we first give a modified Schwarz–Pompeiu formula in general sector ring with angle \(\vartheta =\frac{\pi }{\alpha },\ \alpha \ge 1/2\) by proper conformal mappings, and obtain the solution of Schwarz problem for Cauchy–Riemann equation explicit forms. Furthermore, construct some integral operators discuss their properties detail. Finally, virtue these operators, problems an in...

Abstract: A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is prese...

Abstract. We consider the Cauchy problem We consider the Cauchy problem ut = uxx + f(u), x ∈ R, t > 0, u(x, 0) = u0(x), x ∈ R, where f is a locally Lipschitz function on R with f(0) = 0, and u0 is a nonnegative function in C0(R), the space of continuous functions with limits at ±∞ equal to 0. Assuming that the solution u is bounded, we study its large-time behavior from several points of view. ...

This paper is concerned with the Cauchy problem of multi-dimensional incompressible magnetohydrodynamic equations inhomogeneous density and fractional dissipation. It shown that when α+β=1+n2 satisfying 1≤β≤α≤min{3β2,n2,1+n4} max{n4,n+16}<α for n≥3, MHD have a unique global strong solution initial data in some Sobolev spaces without requiring small conditions.

The probabilistic approach is used for constructing special layer methods to solve the Cauchy problem for semilinear parabolic equations with small parameter. Despite their probabilistic nature these methods are nevertheless deterministic. The algorithms are tested by simulating the Burgers equation with small viscosity and the generalized KPP-equation with a small parameter.