Browsing by Author "Tchoukouegno Ngnotchouye, Jean Medard."
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Item Conservation laws models in networks and multiscale flow optimization.(2011) Tchoukouegno Ngnotchouye, Jean Medard.; Banda, Mapundi K.; Sibanda, Precious.The flow of fluids in a network is of practical importance in gas, oil and water transport for industrial and domestic use. When the flow dynamics are understood, one may be interested in the control of the flow formulated as follows: given some fluid properties at a final time, can one determine the initial flow properties that lead to the desired flow properties? In this thesis, we first consider the flow of a multiphase gas, described by the drift flux model, in a network of pipes and that of water, modeled by the shallow water equations, in a network of rivers. These two models are systems of partial differential equations of first order generally referred to as systems of conservation laws. In particular, our contribution in this regard can be summed up as follows: For the drift-flux model, we consider the flow in a network of pipes seen mathematically as an oriented graph. We solve the standard Riemann problem and prove a well posedness result for the Riemann problem at a junction. This result is obtained using coupling conditions that describe the dynamics at the intersection of the pipes. Moreover, we present numerical results for standard pipes junctions. The numerical results and the analytical results are in agreement. This is an extension for multiphase flows of some known results for single phase flows. Thereafter, the shallow water equations are considered as a model for the flow of water in a network of canals. We analyze coupling conditions at the confluence of rivers, precisely the conservation of mass and the equality of water height at the intersection, and implement these results for some classical river confluences. We also consider the case of pooled stepped chutes, a geometry frequently utilized by dams to spill floodwater. Here we consider an approach different from the engineering community in the sense that we resolve the dynamics by solving a Riemann problem at the dam for the shallow water equations with some suitable coupling conditions. Secondly, we consider an optimization problem constrained by the Euler equations with a flow-matching objective function. Differently from the existing approaches to this problem, we consider a linear approximation of the flow equation in the form of the microscopic Lattice Boltzmann Equations (LBE). We derive an adjoint calculus and the optimality conditions from the microscopic LBE. Using multiscale analysis, we obtain an equivalent macroscopic result at the hydrodynamic limit. Our numerical results demonstrate the ability of our method to solve challenging problems in fluid mechanics.Item Nonclassical solutions of hyperbolic conservation laws.Agbavon, Koffi Messan.; Tchoukouegno Ngnotchouye, Jean Medard.This dissertation studies the nonclassical shock waves which appears as limits of certain type diffusive-dispersive regularisation to hyperbolic of conservation laws. Such shocks occur very often when the ux function lacks the convexity especially when the initial conditions for Riemann problem belong to different region of convexity. They have negative entropy dissipation. They do not verify the classical Oleinik entropy criterion. The cubic function is taken as a ux function. The existence and uniqueness of such shock waves are studied. They are constructed as limits of traveling-wave solutions for diffusive-dispersive regularisation. A kinetic relation is introduced to choose a unique nonclassical solution to the Riemann problem. The numerical simulations are investigated using a transport-equilibrium scheme to enable computing the nonclassical solution at the discrete level of kinetic function. The method is composed of an equilibrium step containing the kinetic relation at any nonclassical shock and a transport step advancing the discontinuity with time.Item Open-loop nash and stackelberg equilibrium for non-cooperative differential games with pure state constraints.(2017) Joseph, Kristina.; Tchoukouegno Ngnotchouye, Jean Medard.This dissertation deals with the study of non-cooperative di erential games with pure state constraints. We propose a rst order optimality result for the open-loop Nash and Stackelberg equilibriums solutions of an N-player di erential game with pure state constraints. A numerical method to solve for an open-loop Nash equilibrium is discussed. Numerical examples are included to illustrate this method.Item Pricing and hedging of defaultable claims in discontinuous market.(2017) Okoye-Ogbalu, Izuchukwu.; Tchoukouegno Ngnotchouye, Jean Medard.; Pamen, Olivier Menoukeu.Credit risk has become one of the highest-pro le risk facing participants in the nancial markets. In this dissertation, we study the pricing and hedging of defaultable claim in a discontinuous market. Here, we present the pricing of credit default swap under stochastic intensity within the set up of a generic reduced form credit risk model. In this context, we present di erent approaches to pricing and hedging of defaultable claim in a discontinuous market and then pro er results concerning the trading of credit default swap. We rst assume that the default intensity is deterministic and the rate of interest is equal to zero. We derive a closed-form solution for replicating strategy for an arbitrary non-dividend paying defaultable claim. We then extend the established results under deterministic intensity to the case of stochastic intensity, where the objective is to hedge both default (jump) risk and the spread (volatility) risk.