Numerical methods for solving inverse problems for time fractional diffusion equation with variable coefficient
Numerical methods for solving inverse problems for time fractional diffusion equation with variable coefficient
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DOI：
10.1515/jiip.2009.028
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
A. Bondarenko;D. Ivaschenko
中科院分区：
文献类型：
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作者：
A. Bondarenko;D. Ivaschenko
摘要 我们考虑用可变广义扩散系数 q(x) 求解时间分数扩散方程 (TFDE) 反问题的数值方法。反问题是根据TFDE解的附加信息求出q(x)和时间导数的阶数α。通过积分插值法构造加权差分格式，并发展广义因式分解方法；给出了差分格式的稳定性分析结果，并研究了正演问题的数值解的性质。变系数TFDE的反问题被表述为残差函数最小化问题，并讨论了相应残差函数的性质。使用用于残差函数最小化的 Levenberg-Marquardt 算法并给出了数值结果。
Abstract We consider numerical methods for solving inverse problems for time fractional diffusion equation (TFDE) with the variable generalized diffusion coefficient q(x). Inverse problems are to find q(x) and the order α of the time derivative according to an additional information about a solution of TFDE. The weighted difference scheme is constructed via integro-interpolation method and the generalized factorization method is developed; results of stability analysis of the difference scheme are presented and properties of numerical solutions of forward problems are investigated. Inverse problems for TFDE with the variable coefficient are formulated as residual function minimization problems and properties of corresponding residual functions are discussed. The Levenberg–Marquardt algorithm for residual function minimization is used and numerical results are presented.