Ellabib, Abdellatif Nachaoui, Abdeljalil Ousaadane, Abdessamad
Published in
Inverse Problems

In this paper, we are interested in solving a Cauchy inverse problem in linear elasticity. For this, we propose a new method based on Robin conditions on the inaccessible boundary, then we study the convergence and regularizing property of the proposed algorithm. We use the finite element method for the discretization of our problem. Further, we tr...

Ghoshal, Shyam Sundar Junca, Stéphane Parmar, Akash

This article deals with the regularity of the entropy solutions of scalar conservation laws with discontinuous flux. It is well-known [Adimurthi et al., Comm. Pure Appl. Math. 2011] that the entropy solution for such equation does not admit BV regularity in general, even when the initial data belongs to $BV$. Due to this phenomenon fractional $BV^s...

Sin, Chung-Sik
Published in
Fractional Calculus and Applied Analysis

In the present work, we consider the Cauchy problem for the time fractional diffusion equation involving the general Caputo-type differential operator proposed by Kochubei [11]. First, the existence, the positivity and the long time behavior of solutions of the diffusion equation without source term are established by using the Fourier analysis tec...

Chapko, Roman Johansson, B. Tomas Vlasiuk, Mariia
Published in
International Journal of Applied and Computational Mathematics

The elastostatic Cauchy problem of fracture mechanics is studied in a two-dimensional bounded domain containing a crack. Given Cauchy data on the boundary of the domain, the displacement and normal stress (traction) are reconstructed on the crack. The reconstruction is done by reducing the original problem, via the elastostatic potential, to a syst...

Xuan, Pham Truong
Published in
Classical and Quantum Gravity

In this paper, we study the Cauchy and Goursat problems of the spin-n/2 zero rest-mass equations on Minkowski spacetime by using the conformal geometric method. In our strategy, we prove the wellposedness of the Cauchy problem in Einstein’s cylinder. Then we establish pointwise decays of the fields and prove the energy equalities of the conformal f...

Chatzakou, Marianna Ruzhansky, Michael Tokmagambetov, Niyaz

In this paper we consider a fractional wave equation for hypoelliptic operators with a singular mass term depending on the spacial variable and prove that it has a very weak solution. Such analysis can be conveniently realised in the setting of graded Lie groups. The uniqueness of the very weak solution, and the consistency with the classical solut...

Kowalczyk, Jan

This thesis examines the parareal method, which is on of the first parallel-in-time integration methods. First, we present an overwiev of sequential methods for solving initial value problems, focusing on Forward Eulers's method and 4th order Runge-Kutta method. Then we describe parareal method, its algorithm and its convergence theorems. Then we d...

Bourdarias, Christian Choudhury, Anupam Pal Guelmame, Billel Junca, Stéphane

Strictly hyperbolic triangular systems with a decoupled nonlinear conservation law and a coupled ``linear'' transport equation with a discontinuous velocity are known to create measure solutions for the initial value problem. Adding a uniform strictly hyperbolic assumption on such systems we are able to obtain bounded solutions in $L^\infty$ under ...

Chatzakou, Marianna Ruzhansky, Michael Tokmagambetov, Niyaz

In this paper we consider the heat equation with a strongly singular potential and show that it has a very weak solution. Our analysis is devoted to general hypoelliptic operators and is developed in the setting of graded Lie groups. The current work continues and extends the work (Altybay et al. in Appl. Math. Comput. 399:126006, 2021), where the ...

Nachaoui, Abdeljalil Salih, Hero W.

This paper discusses the recovering of both Dirichlet and Neumann data on some part of the domainboundary, starting from the knowledge of these data on another part of the boundary for a family of quasi-linearinverse problems. The nonlinear problem is reduced to a linear Cauchy problem for the Laplace equation coupledwith a sequence of nonlinear sc...