scholarly journals Convergence of an implicit scheme for diagonal non-conservative hyperbolic systems

2020 ◽  
Author(s):  
Ahmad EL HAJJ ◽  
Rachida Boudjerada ◽  
Aya Oussayli
Keyword(s):  
Space Dimension ◽  
Jacobian Matrix ◽  
Hyperbolic Systems ◽  
Implicit Scheme ◽  
Lipschitz Continuous ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Lipschitz Estimate ◽  
Global Existence And Uniqueness ◽  
Convergence Of The Scheme

In this paper, we consider diagonal non-conservative hyperbolic systems in one space dimension with monotone and large Lipschitz continuous data. Under a certain nonnegativity condition on the Jacobian matrix of the velocity of the system, global existence and uniqueness results of a Lipschitz solution for this system, which is not necessarily strictly hyperbolic, was already proven. We propose a natural implicit scheme satisfiying a similar Lipschitz estimate at the discrete level. This property allows us to prove the convergence of the scheme without assuming it strictly hyperbolic.

Nonlinear Riemann—Hilbert problems with Lipschitz continuous boundary condition

2000 ◽  
Vol 130 (4) ◽  
pp. 793-800 ◽  
Author(s):  
E. Wegert ◽  
M. A. Efendiev
Keyword(s):  
Boundary Condition ◽  
Existence And Uniqueness ◽  
Integral Operators ◽  
Norm Inequality ◽  
Lebesgue Spaces ◽  
Lipschitz Continuous ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Continuous Restriction ◽  
Continuous Boundary

Using a norm inequality for singular integral operators in pairs of weighted Lebesgue spaces we are able to prove existence and uniqueness results for solutions of nonlinear RiemannHilbert problems with non-compact Lipschitz continuous restriction curves.

scholarly journals UNIQUENESS RESULTS FOR DIAGONAL HYPERBOLIC SYSTEMS WITH LARGE AND MONOTONE DATA

2013 ◽  
Vol 10 (03) ◽  
pp. 461-494 ◽  
Author(s):  
AHMAD EL HAJJ ◽  
RÉGIS MONNEAU
Keyword(s):  
Global Existence ◽  
Uniqueness Theorem ◽  
Existence And Uniqueness ◽  
Hyperbolic Systems ◽  
Spatial Dimension ◽  
Existence And Uniqueness Theorem ◽  
Uniqueness Of Solutions ◽  
Continuous Solutions ◽  
Uniqueness Results ◽  
Global Existence And Uniqueness

We study the uniqueness of solutions to diagonal hyperbolic systems in one spatial dimension and we present two uniqueness results. First, we establish a global existence and uniqueness theorem for continuous solutions to strictly hyperbolic systems. Second, we establish a global existence and uniqueness theorem for Lipschitz continuous solutions to hyperbolic systems that need not be strictly hyperbolic. Furthermore, an application is presented for one-dimensional flows in isentropic gas dynamics.

scholarly journals A Stochastic Model for Cancer Metastasis: Branching Stochastic Process with Settlement

2018 ◽  
Author(s):  
Christoph Frei ◽  
Thomas Hillen ◽  
Adam Rhodes
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Stochastic Model ◽  
Cancer Metastasis ◽  
Metastatic Cancer ◽  
Distributed Delay ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Long Time ◽  
Global Existence And Uniqueness

We introduce a new stochastic model for metastatic growth, which takes the form of a branching stochastic process with settlement. The moving particles are interpreted as clusters of cancer cells while stationary particles correspond to micro-tumors and metastases. The analysis of expected particle location, their locational variance, the furthest particle distribution, and the extinction probability leads to a common type of differential equation, namely, a non-local integro-differential equation with distributed delay. We prove global existence and uniqueness results for this type of equation. The solutions’ asymptotic behavior for long time is characterized by an explicit index, a metastatic reproduction number R0: metastases spread for R0 > 1 and become extinct for R0 < 1. Using metastatic data from mouse experiments, we show the suitability of our framework to model metastatic cancer.

scholarly journals A stochastic model for cancer metastasis: branching stochastic process with settlement

2019 ◽  
Vol 37 (2) ◽  
pp. 153-182 ◽  
Author(s):  
Christoph Frei ◽  
Thomas Hillen ◽  
Adam Rhodes
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Stochastic Model ◽  
Cancer Metastasis ◽  
Metastatic Cancer ◽  
Distributed Delay ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Long Time ◽  
Global Existence And Uniqueness

Abstract We introduce a new stochastic model for metastatic growth, which takes the form of a branching stochastic process with settlement. The moving particles are interpreted as clusters of cancer cells, while stationary particles correspond to micro-tumours and metastases. The analysis of expected particle location, their locational variance, the furthest particle distribution and the extinction probability leads to a common type of differential equation, namely, a non-local integro-differential equation with distributed delay. We prove global existence and uniqueness results for this type of equation. The solutions’ asymptotic behaviour for long time is characterized by an explicit index, a metastatic reproduction number $R_0$: metastases spread for $R_{0}&gt;1$ and become extinct for $R_{0}&lt;1$. Using metastatic data from mouse experiments, we show the suitability of our framework to model metastatic cancer.

scholarly journals Multi-term fractional differential equations with nonlocal boundary conditions

2018 ◽  
Vol 16 (1) ◽  
pp. 1519-1536
Author(s):  
Bashir Ahmad ◽  
Najla Alghamdi ◽  
Ahmed Alsaedi ◽  
Sotiris K. Ntouyas
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Nonlocal Boundary ◽  
Nonlocal Boundary Conditions ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Fractional Differential ◽  
The Given

AbstractWe introduce and study a new kind of nonlocal boundary value problems of multi-term fractional differential equations. The existence and uniqueness results for the given problem are obtained by applying standard fixed point theorems. We also construct some examples for demonstrating the application of the main results.

scholarly journals BSDEs with polynomial growth generators

2000 ◽  
Vol 13 (3) ◽  
pp. 207-238 ◽  
Author(s):  
Philippe Briand ◽  
René Carmona
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Polynomial Growth ◽  
Probabilistic Representation ◽  
Existence And Uniqueness Results ◽  
State Variable ◽  
Cahn Equation ◽  
Uniqueness Results ◽  
Terminal Time

In this paper, we give existence and uniqueness results for backward stochastic differential equations when the generator has a polynomial growth in the state variable. We deal with the case of a fixed terminal time, as well as the case of random terminal time. The need for this type of extension of the classical existence and uniqueness results comes from the desire to provide a probabilistic representation of the solutions of semilinear partial differential equations in the spirit of a nonlinear Feynman-Kac formula. Indeed, in many applications of interest, the nonlinearity is polynomial, e.g, the Allen-Cahn equation or the standard nonlinear heat and Schrödinger equations.

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