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Determination of a control function in three-dimensional parabolic equations

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  • Dehghan, Mehdi

Abstract

This study presents numerical schemes for solving two three-dimensional parabolic inverse problems. These schemes are developed for indentifying the parameter p(t) which satisfy ut=uxx+uyy+uzz+p(t)u+φ, in R×(0,T], u(x,y,z,0)=f(x,y,z),(x,y,z)∈R=[0,1]3. It is assumed that u is known on the boundary of R and subject to the integral overspecification over a portion of the spatial domain ∫01∫01∫01u(x,y,z,t)dxdydz=E(t), 0≤t≤T, or to the overspecification at a point in the spatial domain u(x0,y0,z0,t)=E(t), 0≤t≤T, where E(t) is known and (x0,y0,z0) is a given point of R. These schemes are considered for determining the control parameter which produces, at any given time, a desired energy distribution in the spacial domain, or a desired temperature distribution at a given point in the spacial domain. A generalization of the well-known, explicit Euler finite difference technique is used to compute the solution. This method has second-order accuracy with respect to the space variables. The results of numerical experiments are presented and the accuracy and the central processor (CPU) times needed are reported.

Suggested Citation

  • Dehghan, Mehdi, 2003. "Determination of a control function in three-dimensional parabolic equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 61(2), pages 89-100.
  • Handle: RePEc:eee:matcom:v:61:y:2003:i:2:p:89-100
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    Citations

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    Cited by:

    1. Tatari, Mehdi & Dehghan, Mehdi, 2007. "He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation," Chaos, Solitons & Fractals, Elsevier, vol. 33(2), pages 671-677.
    2. Shivanian, Elyas & Jafarabadi, Ahmad, 2018. "An inverse problem of identifying the control function in two and three-dimensional parabolic equations through the spectral meshless radial point interpolation," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 82-101.
    3. Dehghan, Mehdi & Tatari, Mehdi, 2008. "Identifying an unknown function in a parabolic equation with overspecified data via He’s variational iteration method," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 157-166.
    4. Deng, Zui-Cha & Yu, Jian-Ning & Yang, Liu, 2008. "Identifying the coefficient of first-order in parabolic equation from final measurement data," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 77(4), pages 421-435.
    5. Yang, Liu & Dehghan, Mehdi & Yu, Jian-Ning & Luo, Guan-Wei, 2011. "Inverse problem of time-dependent heat sources numerical reconstruction," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 81(8), pages 1656-1672.
    6. Yang, Liu & Deng, Zui-Cha & Yu, Jian-Ning & Luo, Guan-Wei, 2009. "Optimization method for the inverse problem of reconstructing the source term in a parabolic equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(2), pages 314-326.

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