Asymptotic Analysis Of Three Kinds Of Evolution Equations With Applications

Evolution equations are frequently used to describe wave propagations and diffusions in media.The forward problem of evolution equations aims to establish mathematical model,analyze its well-posedness and do the numerical simulation.While,the inverse problem of evolution equations is to reconstruct the unknown parameters in the system from the measurement information on the solution.This problem has important applications in material design,sonar detection,medical imaging,and environmental monitoring.Asymptotic analysis of evolution equations is an important research area.On the one hand,based on asymptotic expansions of the solution,efficient numerical methods for solving forward problems can be developed.On the other hand,it has wide applications in material design and imaging science.In this thesis,the wave propagation problem in electromagnetic metamaterials,the time-domain acoustic wave scattering problem from multiple small sound-soft obstacle,and the initial and boundary problem of the time-fractional diffusion equation are studied.They are modeled by Maxwell equations,wave equation and time-fractional diffusion equation,respectively.We discuss asymptotic analysis of these equations with applications.First,the wave propagation in pure-time modulated step media is studied.The purespace modulated metamaterials have much developed,but the application of pure-space modulated metamaterials is limited by high fabrication costs and the inability to adapt to complex environments.Hence,it is important to develop metamaterials that can be manipulated over time.Assume that the permittivity has a multiple-step near zero profile in time and uniformly constant in space.Based on the equivalent integral model,we study the asymptotic analysis of the scattered field under certain conditions on the number of steps.As an application of the asymptotic analysis,we design an effective medium that can realize full reflections and full transmissions by the sign of the permittivity.The theoretical results are greatly supported by several numerical examples.Second,the time-domain acoustic scattering problem by multiple small sound-soft obstacles is studied.An invertible retarded linear algebraic system is established based on the retarded boundary integral equation method,and the a priori estimate of the density function in the boundary integral equation is analyzed using the scaling of the retarded potential.Without the periodic assumption on the distribution of obstacles,we derive an asymptotic expansion of the scattered field,as the diameter of the obstacles goes to zero.It says the scattered field can be approximated by a linear combination of point-sources,where the weights are given by constants depending on the shape of each obstacle,and the causal signals can be computed by solving a retarded in time linear algebraic system.Then,the asymptotic analysis is applied to numerical computation of the forward problem and the construction of effective medium.Several numerical examples are presented to support the theoretical results.Finally,the forward and inverse problems for the time-fractional diffusion equation are studied.Based on the boundary integral equation method,a numerical scheme for solving the forward problem of time-fractional diffusion equation is developed.The basic idea is to express the solution as a single-layer or double-layer potential,and use the jump relationship of the potential operator on the boundary to transform the initial boundary value problem of the differential equation into a boundary integral equation.We express the boundary integral operator as a generalized Abel integral operators in time of α/2-1 order,whose kernel function is a time-dependent boundary integral.The kernel function is analyzed and its asymptotical expansion at the initial time is established.We use composite trapezoidal rule to discrete the boundary integral equation in time,where the asymptotic analysis of the kernel function is used to deal with the singularity,and then use the standard rectangular formula for the discretization in space.Then,the validity and accuracy of the discretization scheme are verified by numerical examples.An inverse boundary problem of the time-fractional diffusion equation is to reconstruct the geometric information of the inner cavity of the medium,such as the position,size and shape,from the Cauchy data on the outer boundary.By the maximum principle and unique continuation property,we establish the uniqueness of the inverse problem and formulate it as an ill-posed nonlinear operator equation.Based on Fr(?)chet derivatives,a Newton iterative method with regularization is developed.Using the properties of the diffeomorphism,the Fr(?)chet differentiability of the operator is analyzed.Moreover,the Fr(?)chet derivative can be calculated by Neumann data of the corresponding initial boundary value problem.Finally,the linearized operator equation is solved by the least square method,and the validity of the algorithm is verified numerically.