基于积分观测条件反演抛物型方程的辐射系数
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摘要
研究了一类基于积分观测条件重构二阶非散度抛物型方程的辐射系数的反问题,这里的辐射系数仅依赖于空间变量.首先利用Cauchy不等式与Gronwall不等式得到正问题解的先验估计式;然后将原问题转化为非线性算子方程,基于Schauder不动点定理,证明了反问题解的存在性;最后基于正问题解的一些先验估计式和附加条件,得到了反问题解唯一的充分条件.
This paper investigates an inverse problem of using integral observation condition to reconstruct the radiative coefficient in second order nondivergence parabolic equations,and the radiative coefficient is only space dependent.Firstly,a-priori estimates of the direct problem are obtained by using Cauchy inequality and Gronwall inequality.Secondly,the original problem is transformed into a non-linear operator equation, and the existence is proved based on Schauder fixed point theorem.Finally,based on some a-priori estimates of the direct problem and additional conditions,a sufficient condition for the uniqueness of the solution is obtained.
This paper investigates an inverse problem of using integral observation condition to reconstruct the radiative coefficient in second order nondivergence parabolic equations,and the radiative coefficient is only space dependent.Firstly,a-priori estimates of the direct problem are obtained by using Cauchy inequality and Gronwall inequality.Secondly,the original problem is transformed into a non-linear operator equation, and the existence is proved based on Schauder fixed point theorem.Finally,based on some a-priori estimates of the direct problem and additional conditions,a sufficient condition for the uniqueness of the solution is obtained.
引文
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[7] 蔡超.一类Kolmogorov型方程的系数反演问题[J].山东大学学报(理学版),2016,51(4):127-134.
[8] 张泰年,李照兴.一类退化抛物型方程反问题的收敛性分析[J].山东大学学报(理学版),2017,52(8):35-42.
[9] CHEN Q,LIU J.Solving an inverse parabolic problem by optimization from final measurement data[J].Journal of Computational & Applied Mathematics,2006,193(1):183-203.
[10] KRUZKOV S N.Quasilinear parabolic equations and systems with two independent variables[J].Trudy Sem Petrovsk,1985,5(5):217-272.
[11] HUZYK N.Inverse problem of determining the coefficients in a degenerate parabolic equation[J].Electronic Journal of Differential Equations,2014,21:87-101.
[12] FAZELI H,MIRZAEI M.Shape identification problems on detecting of defects in a solid body using inverse heat conduction approach[J].Journal of Mechanical Science Technology,2012,26(11):3681-3690.
[13] 任建龙,曾剑,甄苇苇.数值重构二阶抛物型方程的一阶项系数[J].兰州交通大学学报,2018,37(4):99-104.
[14] 李照兴,张泰年,蔡成松.一个抛物型方程不适定问题的全变差正则化方法[J].兰州交通大学学报,2016,35(3):142-147.
[15] SU J,NETO A J S.Heat source estimation with the conjugate gradient method in inverse linear diffusive problems[J].Journal of the Brazilian Society of Mechanical Sciences & Engineering,2001,23(3):321-334.
[2] KAMYNIN V L.Inverse problem of finding the coefficient of a lower derivative in a parabolic equation on the plane[J].Differential Equations,2012,48(2):214-223.
[3] KAMYNIN V L.On the unique solvability of an inverse problem for parabolic equations under a final overdetermination condition[J].Mathematical Notes,2003,73(1/2):202-211.
[4] YANG L,DEHGHAN M,YU J N,et al.Inverse problem of time-dependent heat sources numerical reconstruction[J].Mathematics & Computers in Simulation,2011,81(8):1656-1672.
[5] WEI T,WANG J G.A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation[J].Applied Numerical Mathematics,2014,78(4):95-111.
[6] SOlOV′EV V V.Coefficient control in a semilinear equation of parabolic type[J].Computational Mathematics and Modeling,1994,5(1):31-35.
[7] 蔡超.一类Kolmogorov型方程的系数反演问题[J].山东大学学报(理学版),2016,51(4):127-134.
[8] 张泰年,李照兴.一类退化抛物型方程反问题的收敛性分析[J].山东大学学报(理学版),2017,52(8):35-42.
[9] CHEN Q,LIU J.Solving an inverse parabolic problem by optimization from final measurement data[J].Journal of Computational & Applied Mathematics,2006,193(1):183-203.
[10] KRUZKOV S N.Quasilinear parabolic equations and systems with two independent variables[J].Trudy Sem Petrovsk,1985,5(5):217-272.
[11] HUZYK N.Inverse problem of determining the coefficients in a degenerate parabolic equation[J].Electronic Journal of Differential Equations,2014,21:87-101.
[12] FAZELI H,MIRZAEI M.Shape identification problems on detecting of defects in a solid body using inverse heat conduction approach[J].Journal of Mechanical Science Technology,2012,26(11):3681-3690.
[13] 任建龙,曾剑,甄苇苇.数值重构二阶抛物型方程的一阶项系数[J].兰州交通大学学报,2018,37(4):99-104.
[14] 李照兴,张泰年,蔡成松.一个抛物型方程不适定问题的全变差正则化方法[J].兰州交通大学学报,2016,35(3):142-147.
[15] SU J,NETO A J S.Heat source estimation with the conjugate gradient method in inverse linear diffusive problems[J].Journal of the Brazilian Society of Mechanical Sciences & Engineering,2001,23(3):321-334.