Sharif Digital Repository

The controllability problem may be formulated roughly as follows. Consider an evolution system (either described in terms of partial or ordinary differential equations) on which we are allowed to act by means of a suitable choice of the control (the right-hand side of the system, the boundary conditions, etc.). Given a time interval 0 < t < T, and initial and final states, the goal is to determine whether there exists a control driving the given initial data to the given final ones in time T. Now, consider the simplest parabolic equation, namely heat equation and suppose that one could act by appropraite controls on this system. The null controllability problem which is one of the very... 

Considering atmospheric wave propagation as a complex phenomenon, it can be represented as a function of variety of parameters such as: properties of atmosphere, boundary conditions, surface characteristics, source related parameters and etc. the aim of this work is to study the propagation of sound mechanisms in troposphere layer and numerical simulation of sound field in order to investigate the wind effect and turbulence in wave propagation. In this manner, the Green’s function parabolic equation (GFPE) is hired to solve the governing equation for sound propagation in a moving inhomogeneous atmosphere. To achieve this, a code is generated using MATLAB software to predict the long range... 

e solution for fully non-linear equations do not exists in general and for this reason the notion of viscosity solutions has been defined in this context. In this dissertation, a Monte carlo method is introduced for parabolic fully non-linear together with its asymptotics. In the proof of convergence, the method of [6] is used. en the rate of convergence from both sides is introduced: Finally, the error due to estimation of conditional expectation is derived for the estimation whose error is known with respect to sample size  

In this thesis we investigating two models of cancer invasion .First, a general mathematical model of cancer invasion is presented. In this model there are three factors: tumor cell, extracellular matrix and enzyme. The model consists of a parabolic partial differential equation (PDE) describing the evolution of tumor cell density , an ordinary differential equation modeling of extracellular matrix and a parabolic PDE governing the evolution of the matrix degrading enzyme concentration. This model is investigated in two special versions for existence and uniqueness of global solutions. In the first model we neglect the remodeling term, this model is named the chemotaxis-haptotaxis model.... 

There are some situations where we would like to solve a partial differential equation (PDE) in a domain whose boundary is not known a priori; such a problem is called free boundary problem and the boundary is called free boundary. For this kind of problems, aside from standard boundary conditions, an extra condition is imposed at the free boundary and the goal would be finding the free boundary in addition to finding a solution for PDE. One of the classical examples of free boundary problems is the Stefan problem (melting of ice) in which ice and water temperatures are determined from the heat equation and we would be interested in finding the boundary between ice and water. Another famous... 

In this thesis , we consider nonnegative (continuous) weak solutions of the porous medium equation with source , u_t-∆u^m=u^p, and p>m>1 .
Assume that, m>1 and 1< p/m u_t-∆u^m=u^p,xϵR^n ,tϵR
has no nontrivial, bounded radial solutions u≥0 .
In one space-dimensional, the conclusion of the result mentioned above remains true without the assumption of the radial symmetry. The proof is based on the intersection-comparison arguments , zero number argum- ents and a key step is to show the... 

We study linear and semilinear stochastic evolution partial differential equations driven by additive noise. We present a general and flexible framework for representing the infinite dimensional Wiener process, which drives the equation. The equation is discretized in space by a standard piecewise linear finite element method. We show how to obtain error estimates when the truncated expansion is used in the equation. We show that the orthogonal expansion of the finite-dimensional Wiener process, that appears in the discretized problem, can be truncated severely without losing theasymptotic order of the method, provided that the kernel of the covariance operator of the Wiener process is... 

Singular differential equations have a wide range of applications. Hardy singularities, which are connected to inequalities of the same name and have various extensions, are the most well-known singularities. The application of Hardy inequalities in quantum physics and also in the linearization of reaction-diffusion equations in thermodynamics and combustion theory motivates researchers to examine them. Singularities on a domain's boundary are another well-known type of singularity. In the study of fluid mechanics and pseudoplastic flows, these singularities emerge.Differential equations with coefficients or functions that are simply functions belonging to $ L^1 $, or bounded Radon measures,...