时间分数阶扩散方程柯西问题的迭代正则化方法.pdf
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1、应用数学MATHEMATICA APPLICATA2023,36(4):1007-1024Iterative Regularization Method for the CauchyProblem of Time-Fractional Diffusion EquationLV Yong(吕拥),ZHANG Hongwu(张宏武)(School of Mathematics and Information Science,North Minzu University,Yinchuan 750021,China)Abstract:We consider a Cauchy problem of th
2、e time-fractional diffusion equation,whichis seriously ill-posed.This paper constructs an iterative regularization method based onFourier truncation to overcome the ill-posedness of considered problem.And then,underthe a-prior and a-posterior selection rules of regularization parameter,the convergen
3、ceestimates of the proposed method are derived.Finally,we verify the effectiveness ofour method by doing some numerical experiments.The corresponding numerical resultsshow that the proposed method is stable and feasible in solving the Cauchy problem oftime-fractional diffusion equation.Key words:Cau
4、chy problem;Time-fractional diffusion equation;Iteration regularizationmethod;Convergence estimate;Numerical simulationCLC Number:O175.24;O175.26AMS(2010)Subject Classification:35R25;47A25;35R30;65N20Document code:AArticle ID:1001-9847(2023)04-1007-181.IntroductionThe time-fractional diffusion equat
5、ion is deduced by replacing the first-order time deriva-tive with the derivative of fractional order,and which has the important applications in de-scribing the various anomalous diffusion phenomena.In the past few decades,the forwardproblems for time-fractional diffusion equation have been studied
6、extensively.13In recentyears,driven by practical applications,the inverse problems of this equation have attractedwide attention.The research contents mainly include the parameter identification problem,inverse initial value problem(final value problem or backward problem in time),Cauchyproblem,side
7、ways problem(inverse heat conduction problem),inverse source problem,in-verse boundary condition problem,and so on.This paper considers the Cauchy problem of time-fractional diffusion equationY(x,t)t Yxx(x,t)=0,0 x 0,Y(0,t)=(t),t 0,Yx(0,t)=(t),t 0,Y(x,0)=0,0 x L,(1.1)Received date:2022-10-30Foundati
8、on item:Supported by the NSF of Ningxia(2022AAC03234),the NSF of China(11761004),the Construction Project of First-Class Disciplines in Ningxia Higher Education(NXYLXK2017B09)andthe Postgraduate Innovation Project of North Minzu University(YCX22094)Correspondence author:ZHANG Hongwu,male,Han,Gansu,a
9、ssociate professor,major in inverseproblems of partial differential equation.1008MATHEMATICA APPLICATA2023whereYtdenotes the Caputo fractional derivative of the function Y(x,t)defined in 4 byYt=1(1 )t0Y(x,s)sds(t s),0 1,(1.2)Yt=Y(x,t)t,=1,(1.3)and()is the Gamma function.The physical background and m
10、otivation for the problem(1.1)can be described as below.In practical scientific research,the solute concentration of the pollution in soil at internal pointand one end x=L are not available to be measured since the internal point is inaccessible orthe end of x=L is far away,but the solute concentrat
11、ion and diffusion flux can be measuredat another end x=0.Our task is to determine the unknown solute concentration Y(x,t)(0 x L)from the measured solute concentration(t)and diffusion flux(t)on theaccessible end x=0.Here,denotes the bound of measured error,and the noisy datum(t),(t)satisfy?(t)(t)?L2+
12、?(t)(t)?L2.(1.4)There are two main difficulties in solving the above problems.(i)Problem(1.1)is ill-posedin the sense that the solution does not depend continuously on the noisy datum(t),(t),sosome regularized methods are required to overcome its ill-posedness and recover the stabilityof the solutio
13、n.(ii)The Caputo derivative in time-fractional diffusion equation possesses theproperties of weak singularity and non-locality,this can lead to some difficulties in the aspectof numerical computation,then in order to solve problem(1.1),one needs to design a stablediscretization scheme to complete th
14、e numerical computation and simulation.In order to re-cover the stability of the solution and obtain the stable numerical solution for Cauchy problemof time-fractional diffusion equation,recently some works have been published,in which somemeaningful regularized methods and numerical algorithms have
15、 been proposed.For example,spectral regularization and finite difference method5,convolution type regularization andfinite difference method6,modified Tikhonov regularization and finite difference method7,Tikhonov regularization and boundary element method8,modified Tikhonov regularizationand conjug
16、ate gradient algorithm9,kernel-based approximation method10,truncation-typeregularization11,etc.In the present paper,we propose an iterative regularization method based on Fouriertruncation to recover the stability of solution for the problem(1.1).And then,the a-prior anda-posterior convergence resu
17、lts for this method are given and proven.Finally,in considerationof the properties of weak singularity and non-locality of Caputo derivative,we solve a directproblem to construct the exact datum by adopting an unconditionally stable finite differencescheme proposed in 2,and use the fast discrete Fou
18、rier transform and inverse transform tocompute the regularized solution.Meanwhile,we verify the calculation effect of our methodby doing some numerical experiments.Numerical results show that this method works wellin doing with the problem(1.1).On other references for the similar iteration methods,w
19、ecan see 12-16,etc.The rest of this article is arranged as below.Section 2 gives a mathematical descriptionfor the considered problem.Section 3 describes the ill-posedness of Cauchy problem and theconstruction procedure of iterative(or regularized)method.Section 4 derives the a-prior andNo.4LV Yong,
20、et al.:Iterative Regularization Method for the Cauchy Problem1009a-posterior convergence results of regularization method.Section 5 verifies the calculationeffect of our method by doing some numerical experiments.Some conclusions and furtherdiscussion are shown in Section 6.2.Mathematical Formulatio
21、n of the Cauchy ProblemLet f L2(R),Fourier transform and inverse Fourier transform are defined asbf():=12f(y)eiydy,R,(2.1)f(y):=12bf()eiyd,y R.(2.2)Since we shall work with Fourier transform with respect to the variable t,we extend thedomain of appearing functions with respect to t by defining them
22、to be zero for t (,0).According to the principle of linear superposition,as in 7,problem(1.1)can be splitinto the following two Cauchy problems:u(x,t)t uxx(x,t)=0,0 x 0,u(0,t)=(t),t 0,ux(0,t)=0,t 0,u(x,0)=0,0 x L,(2.3)v(x,t)t vxx(x,t)=0,0 x 0,v(0,t)=0,t 0,vx(0,t)=(t),t 0,v(x,0)=0,0 x 0 such that the
23、 solutions of(2.3)and(2.4)satisfy the a-priori boundmaxu(L,)p,v(L,)p E,(2.11)1010MATHEMATICA APPLICATA2023where,u(L,)p,v(L,)pare both Sobolev space Hp-norm defined byu(L,)p=(1+2)p|b u(L,)|2d)1/2,(2.12)andv(L,)p=(1+2)p|b v(L,)|2d)1/2.(2.13)3.The Iteration Regularization MethodFrom(2.5)and(2.6),we not
24、e that,for fixed 0 x L the values of|cosh(x()|and?sinh(x()()?tend to infinity as|.Then,problems(2.3)and(2.4)are both notstable(the solutions do not depend continuously on the measured datum).We point out thatlim()0b v(x,)=xb(),therefore the expression(2.6)has its meaning when =0.Inorder to recover t
25、he continuous dependency of solutions of(2.3)and(2.4),below we constructthe regularization solutions of two problems.The iteration regularization method for Problem(2.3)Take the initial guess as zero,if utilizing the original Landweber iteration method17,weknow that the iteration solution(regulariza
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