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Determination of a control function in three-dimensional parabolic equations by Legendre pseudospectral method. (English) Zbl 1252.65161
Summary: A Legendre pseudospectral method is proposed for solving approximately an inverse problem of determining an unknown control parameter $$p(t)$$ which is the coefficient of the solution $$u(x, y, z, t)$$ in a diffusion equation in a three-dimensional region. The diffusion equation is to be solved subject to suitably prescribed initial-boundary conditions. The presence of the unknown coefficient $$p(t)$$ requires an extra condition. This extra condition considered as the integral overspecification over the spacial domain.
For discretizing the problem, after homogenization of the boundary conditions, we apply the Legendre pseudospectral method in a matrix based manner. As a result, a system of nonlinear differential algebraic equations is generated. Then, by using a suitable transformation, the problem is converted to a homogeneous time varying system of linear ordinary differential equations. Also, a pseudospectral method for the efficient solution of the resulting system of ordinary differential equations is proposed. The solution of this system gives an approximation to values of $$u$$ and $$p$$. The matrix based structure of the present method makes it easy to implement. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed computational procedure.

##### MSC:
 65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs 35K20 Initial-boundary value problems for second-order parabolic equations 35R30 Inverse problems for PDEs 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
##### Software:
Differentiation Matrix Suite
Full Text:
##### References:
 [1] Cannon, A class of nonlinear nonclassical parabolic equations, J Differential Equations 79 pp 266– (1989) · Zbl 0702.35120 · doi:10.1016/0022-0396(89)90103-4 [2] Cannon, Numerical solutions of some parabolic inverse problems, Numer Methods Partial Differential Eq 6 pp 177– (1990) · Zbl 0709.65105 · doi:10.1002/num.1690060207 [3] Cannon, Determination of parameter p(t) in HĂ¶lder classes for some semilinear parabolic equations, Inverse Problems 4 pp 595– (1988) · Zbl 0688.35104 · doi:10.1088/0266-5611/4/3/005 [4] Cannon, An inverse problem of finding a parameter in a semi-linear heat equation, J Math Anal Appl 145 pp 470– (1990) · Zbl 0727.35137 · doi:10.1016/0022-247X(90)90414-B [5] Cannon, Diffusion subject to the specification of mass, J Math Anal Appl 115 pp 517– (1986) · Zbl 0602.35048 · doi:10.1016/0022-247X(86)90012-0 [6] Cannon, Determination of source parameter in parabolic equations, Meccanica 27 pp 85– (1992) · Zbl 0767.35105 · doi:10.1007/BF00420586 [7] Day, Extensions of a property of the heat equation to linear thermoelasticity and other theories, Quart Appl Math 40 pp 319– (1982) · Zbl 0502.73007 · doi:10.1090/qam/678203 [8] Dehghan, Parameter determination in a partial differential equation from the overspecified data, Math Comput Model 41 pp 196– (2005) · Zbl 1080.35174 · doi:10.1016/j.mcm.2004.07.010 [9] MacBain, Existence and uniqueness properties for the one-dimensional magnetotellurics inversion problem, J Math Phys 27 pp 645– (1986) · doi:10.1063/1.527219 [10] MacBain, Inversion theory for a parameterized diffusion problem, SIAM J Appl Math 47 pp 1386– (1987) · Zbl 0664.35075 · doi:10.1137/0147091 [11] Prilepko, Determination of the parameter of an evolution equation and inverse problems of mathematical physics I, Differentsial’nye Uravneniya 21 pp 119– (1985) · Zbl 0571.35052 [12] Prilepko, On the solvability of inverse boundary value problems for the determination of the coefficient preceding the lower derivative in a parabolic equation, Differentsial’nye Uravneniya 23 pp 136– (1987) [13] Rundell, Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data, Appl Anal 10 pp 231– (1980) · Zbl 0454.35045 · doi:10.1080/00036818008839304 [14] Dehghan, A tau method for the one-dimensional parabolic inverse problem subject to temperature overspecification, Comput Math Appl 52 pp 933– (2006) · Zbl 1125.65340 · doi:10.1016/j.camwa.2006.04.017 [15] Saadatmandi, The Legendre-tau technique for the determination of a source parameter in a semilinear parabolic equation, Math Probl Eng pp 11– (2006) · Zbl 1200.65077 [16] Dehghan, An inverse problem of finding a source parameter in a semilinear parabolic equation, Appl Math Model 25 pp 743– (2001) · Zbl 0995.65098 · doi:10.1016/S0307-904X(01)00010-5 [17] Dehghan, Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions, Math Comput Model 44 pp 1160– (2006) · Zbl 1137.65408 · doi:10.1016/j.mcm.2006.04.003 [18] Wang, A finite-difference solution to an inverse problem for determining a control function in a parabolic partial differential equation, Inverse Probl 5 pp 631– (1989) · Zbl 0683.65106 · doi:10.1088/0266-5611/5/4/013 [19] Dehghan, Determination of a control parameter in the two-dimensional diffusion equation, Appl Numer Math 37 pp 489– (2001) · Zbl 0982.65103 · doi:10.1016/S0168-9274(00)00057-X [20] Dehghan, Fourth-order techniques for identifying a control parameter in the parabolic equations, Int J Eng Sci 40 pp 433– (2002) · Zbl 1211.65120 · doi:10.1016/S0020-7225(01)00066-0 [21] Ang, The determination of a control parameter in a two-dimensional diffusion equation using a dual-reciprocity boundary element method, Int J Comput Math 80 pp 65– (2003) · Zbl 1013.65105 · doi:10.1080/00207160304662 [22] Dehghan, Determination of a control function in three-dimensional parabolic equations, Math Comput Simul 61 pp 89– (2003) · Zbl 1014.65097 · doi:10.1016/S0378-4754(01)00434-7 [23] Dehghan, Efficient techniques for the second-order parabolic equation subject to nonlocal specifications, Appl Numer Math 52 pp 39– (2005) · Zbl 1063.65079 · doi:10.1016/j.apnum.2004.02.002 [24] Dehghan, Implicit collocation technique for heat equation with non-classic initial condition, Int J Nonlinear Sci Numer Simul 7 pp 447– (2006) · Zbl 06942230 · doi:10.1515/IJNSNS.2006.7.4.461 [25] Dehghan, Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer Methods Partial Differential Eq 21 pp 611– (2005) · Zbl 1069.65104 · doi:10.1002/num.20055 [26] Dehghan, A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer Methods Partial Differential Eq 22 pp 220– (2006) · Zbl 1084.65099 · doi:10.1002/num.20071 [27] Dehghan, The one-dimensional heat equation subject to a boundary integral specification, Chaos Soliton Fract 32 pp 661– (2007) · Zbl 1139.35352 · doi:10.1016/j.chaos.2005.11.010 [28] Trefethen, Software-Environments-Tools. 10, SIAM, Society for Industrial and Applied Mathematics (2000) [29] Fahroo, Second look at approximating differential inclusions, J Guidance Control Dyn 24 pp 131– (2001) · doi:10.2514/2.4686 [30] Fahroo, Direct trajectory optimization by a Chebyshev pseudospectral method, J Guidance Control Dyn 25 pp 160– (2002) · doi:10.2514/2.4862 [31] Josselyn, Rapid verification method for the trajectory optimization of reentry vehicles, J Guidance Control Dyn 26 pp 505– (2003) · doi:10.2514/2.5074 [32] Ross, The Proceedings of the 2002 IEEE Mediterranean Conference (2002) [33] Elnagar, The pseudospectral Legendre method for discretizing optimal control problems, IEEE Trans Autom Control 40 pp 1793– (1995) · Zbl 0863.49016 · doi:10.1109/9.467672 [34] Shamsi, Recovering a time-dependent coefficient in a parabolic equation from overspecified boundary data using the pseudospectral Legendre method, Numer Methods Partial Differential Eq 23 pp 196– (2007) · Zbl 1107.65085 · doi:10.1002/num.20174 [35] Dehghan, Numerical solution of two-dimensional parabolic equation subject to nonstandard boundary specifications using the pseudospectral Legendre method, Numer Methods Partial Differential Eq 22 pp 1255– (2006) · Zbl 1108.65101 · doi:10.1002/num.20150 [36] Lakestani, A new technique for solution of a parabolic inverse problem, Kybernetes 37 pp 352– (2008) · Zbl 1179.49038 · doi:10.1108/03684920810851230 [37] Dehghan, Method of lines solutions of the parabolic inverse problem with an overspecification at a point, Numer Algorithm 50 pp 417– (2009) · Zbl 1162.65048 · doi:10.1007/s11075-008-9234-3 [38] Gottlieb, Spectral methods for partial differential equations (Hampton, Va., 1982) pp 1– (1984) [39] Canuto, Spectral methods in fluid dynamics, Springer series in computational physics (1991) [40] Gautschi, Numerical analysis, An introduction (1997) · Zbl 0877.65001 [41] Welfert, Generation of pseudospectral differentiation matrices I, SIAM J Numer Anal 34 pp 1640– (1997) · Zbl 0889.65013 · doi:10.1137/S0036142993295545 [42] Baltensperger, Improving the accuracy of the matrix differentiation method for arbitrary collocation points, Appl Numer Math 33 pp 143– (2000) · Zbl 0964.65021 · doi:10.1016/S0168-9274(99)00077-X [43] Costa, On the computation of high order pseudospectral derivatives, Appl Numer Math 33 pp 151– (2000) · Zbl 0964.65020 · doi:10.1016/S0168-9274(99)00078-1 [44] Weideman, A MATLAB differentiation matrix suite, ACM Trans Math Softw 26 pp 465– (2000) · doi:10.1145/365723.365727 [45] Baltensperger, Spectral differencing with a twist, SIAM J Sci Comput 24 pp 1465– (2003) · Zbl 1034.65016 · doi:10.1137/S1064827501388182
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