Moving Boundary Problem And Its Applications Based On Two Practical Backgrounds

Since Stefan began to study the moving boundary problem in the late nineteenth century,we have obtained a lot of significative achievements under the efforts of scholars from all over the world in the past one hundred or more ycars,but its application potential is still unlimited.This paper focuses on the study of moving boundary problem and its applications based on two practical backgrounds.It is composed of three chapters which are correlative and inde-pendent to each other.In the first chapter,it just gives a brief introduction to some preliminary knowledge.In the second chapter,the heat conduction prob-lem with phase change in the spherical coordinate system is considered and the solutions are presented in the way of numerical as well as analytical.In the third ehapter.fraetional moving boundary problem for release of high poly-mer from planar polymer matrix is considered and the numerical solutions are presented.For the preliminary knowledge in the first chapter,a brief introduction of the mathematical theory,tools and methods which we will use in the sub-sequent two chapters are covered,so is the introduction of the background of moving boundary problem.In section§1.1,the historical development of moving boundary problem and some characteristics of it are given.In section§1.2,we introduce the history,dcvelopmcnt of the fractional calculus,especially about the basic definitions and main properties of our familiar fractional opera-tors,such as,Riemann-Liouville fractional operator and Caputo fractional op-erator.In section§1.3,we give an introduction to the error function.including its definition and properties.the most important property which we will use in the second chapter is the differential of error function.In section§1.4,we briefly introduce applications of the fractional calculus in science and engi-neering,such as anomalous diffusion phenomenon and power law phenomenon can be described by fractional differential equation.In addition,fractal dynam-ics.biomedical fields,materials science more or less involve the theory of frac-tional calculus recently.In chapter2,we consider the heat conduction problem with phase change in the spherical coordinate system,the model of formation of ice in the spherical coordinate system is established.The phase change problem actually belongs to moving boundary problem.In section§2.1,we introduce the academic research situation about moving boundary problem,for example which kinds of methods researchers always employed to copy with moving boundary problem.In section§2.2,the model is presented and by dimensionless processing,we have by defining a new dependent variable v(r,t)=ru(r,t), we can transform the spherical coordinates into the rectangular coordinates.Then,introducing the new intermediate variable0=1-r/t, which makes partial differential equation into ordinary differential equation.Finally,we get whereas the value of R(t) is acquired by solving the below ordinary differential equation In section§2.3.We discuss the change of value of parameter rv to influence formation of ice,following by drawing related figures to illustrate the results.In section2.4,the conclusions of this chapter are presented.In chapter3,we focus on fractional moving boundary problem for release of high polymer from planar polymer matrix.In section§3.1,we introduce the frae-tional calculus applied in nou Newtonian fluid and anomalous diffusion.In see-tion§3.2,the model of the above problem is established,following by dimension-less processing and making the transformations ζ=ζ/S(τ) and θ(ξ,τ)=φ(ζ,τ) which the region0<ζ <S(τ) can be transformed into the region0<ζ <1. Then,we have (?)θ=ζ/S(τ)dS/(τ)/dτ(?)θ/(?)ζ+(?)(ξ,τ)/Sα(τ)CODαζθ(ξ,τ),0<ξ<1,1<α≤2. θ(0,τ)=0, ξ=0. θ(1,τ)=1, ξ=1. ηdS(τ)/dτ=(?)(ξ,τ)1/S(α-1)(τ)C0Dα-1ξ0(ξ,τ), ξ S(0)=0, τ=0. we can solve the above equations by the numerical method and the values of0(ζ,τ) and S(τ) for small time are needed which can be acquired by experiment or others.In section§3.3.we take fractional moving boundary problem for drug release from polymer matrix as an example.it is clearly seen that the numerical results obtained by using the discrete scheme described above are consistent with Li’s results and the discrete scheme is convergent.Furthermore,we illus-trate the influence of the different values of α to values of the concentration distribution0(ξ,τ) and the position of moving boundary S’(τ).In section3.4,the conclusions of this chapter are presented.