zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
On the determination of the right-hand side in a parabolic equation. (English) Zbl 1259.65139
An overdetermined initial boundary value problem of parabolic type is analyzed. The parabolic equation has an unknown right hand side; Dirichlet boundary conditions are considered. In this study, difference schemes of first and second order of accuracy are used for the approximate solution of the previously described problem. Stability estimates for the solution of these schemes are established. Since the values of the constants involved in the stability inequalities may be large, the numerical algorithms are tested on an example.

65M32Inverse problems (IVP of PDE, numerical methods)
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35K20Second order parabolic equations, initial boundary value problems
35R30Inverse problems for PDE
Full Text: DOI
[1] Ashyralyev, A.: On a problem of determining the parameter of a parabolic equation, Ukrainian math. J. 62, No. 9, 1200-1210 (2010) · Zbl 1240.35305
[2] Ashyralyev, A.; Erdogan, A. S.; Demirci, E.: Numerical solution of a one-dimensional parabolic inverse problem, , 654-663 (2009) · Zbl 1180.65117
[3] Ashyralyev, A.; Ozdemir, Y.: Stability of difference schemes for hyperbolic-parabolic equations, Comput. math. Appl. 50, 1443-1476 (2005) · Zbl 1088.65082 · doi:10.1016/j.camwa.2005.01.028
[4] Ashyralyev, A.; Sobolevskii, P. E.: New difference schemes for partial differential equations, (2004) · Zbl 1060.65055
[5] Cannon, J. R.; Lin, Y. L.; Xu, S.: Numerical procedures for the determination of an unknown coefficient in semi-linear parabolic differential equations, Inverse problems 10, 227-243 (1994) · Zbl 0805.65133 · doi:10.1088/0266-5611/10/2/004
[6] Cannon, J. R.; Yin, Hong-Ming: Numerical solutions of some parabolic inverse problems, Numer. methods partial differential equations 2, 177-191 (1990) · Zbl 0709.65105 · doi:10.1002/num.1690060207
[7] Dehghan, M.: Finding a control parameter in one-dimensional parabolic equations, Appl. math. Comput. 135, 491-503 (2003) · Zbl 1026.65079 · doi:10.1016/S0096-3003(02)00063-2
[8] Dehghan, M.: Identification of a time-dependent coefficient in a partial differential equation subject to an extra measurement, Numer. methods partial differential equations 21, 611-622 (2005) · Zbl 1069.65104
[9] Dehghan, M.: Parameter determination in a partial differential equation from the overspecified data, Math. comput. Modelling 41, 196-213 (2005) · Zbl 1080.35174
[10] Dehghan, M.; Shakeri, F.: Method of lines solutions of the parabolic inverse problem with an overspecification at a point, Numer. algorithms 50, 417-437 (2009) · Zbl 1162.65048 · doi:10.1007/s11075-008-9234-3
[11] Dehghan, M.; Tatari, M.: Determination of a control parameter in a one-dimensional parabolic equation using the method of radial basis functions, Math. comput. Modelling 44, 1160-1168 (2006) · Zbl 1137.65408 · doi:10.1016/j.mcm.2006.04.003
[12] Eidelman, Y. S.: The boundary value problem for differential equations with a parameter, Differ. uravn. 14, 1335-1337 (1978)
[13] Eidelman, Yu.S.: An inverse problem for an evolution equation, Math. notes 49, No. 5, 535-540 (1991) · Zbl 0807.34072
[14] Yu.S. Eidelman, Boundary value problems for differential equations with parameters, PhD thesis, Voronezh State University, Voronezh, 1984 (in Russian).
[15] Isakov, V.: Uniqueness and stability in multi-dimensional inverse problems, Inverse problems 9, 579-621 (1993) · Zbl 0924.35195 · doi:10.1088/0266-5611/9/6/001
[16] Iskenderov, A. D.; Tagiev, R. G.: The inverse problem of determining the right-hand sides of evolution equations in Banach space, Vopr. prikl. Mat. kibern., nauchn. Tr. azerb. GoS univ. 1, 51-56 (1979) · Zbl 0438.35062
[17] Lakestani, M.; Dehghan, M.: A new technique for solution of a parabolic inverse problem, Kybernetes 37, 352-364 (2008) · Zbl 1179.49038 · doi:10.1108/03684920810851230
[18] Lakestania, M.; Dehghan, M.: The use of Chebyshev cardinal functions for the solution of a partial differential equation with an unknown time-dependent coefficient subject to an extra measurement, J. comput. Appl. math. 235, 669-678 (2010) · Zbl 1200.65076 · doi:10.1016/j.cam.2010.06.020
[19] Mohebbia, A.; Dehghan, M.: High-order scheme for determination of a control parameter in an inverse problem from the overspecified data, Comput. phys. Comm. 181, 1947-1954 (2010) · Zbl 1219.65102 · doi:10.1016/j.cpc.2010.09.009
[20] Prilepko, A. I.; Orlovskiiý, D. G.; Vasin, I. A.: Methods for solving inverse problems in mathematical physics, (2000) · Zbl 0947.35173
[21] Prilepko, A.; Piskarev, S.; Shaw, S. -Y.: On approximation of inverse problem for abstract parabolic differential equations in Banach spaces, J. inverse ill-posed probl. 15, No. 8, 831-851 (2007) · Zbl 1188.65074 · doi:10.1515/jiip.2007.045
[22] Saadatmand, A.; Dehghan, M.: Computation of two time-dependent coefficients in a parabolic partial differential equation subject to additional specifications, Int. J. Comput. math. 87, 997-1008 (2010) · Zbl 1191.65128 · doi:10.1080/00207160802253958
[23] Samarskii, A. A.; Nikolaev, E. S.: Numerical methods for grid equations, vol. 2: iterative methods, (1989)
[24] Shamsi, M.; Dehghan, M.: Determination of a control function in three-dimensional parabolic equations by Legendre pseudospectral method, Numer. methods partial differential equations 28, 74-93 (2012) · Zbl 1252.65161
[25] Tikhonov, I. V.; Eidelman, Yu.S.: Uniqueness criterion in an inverse problem for an abstract differential equation with nonstationary inhomogeneous term, Math. notes 77, No. 2, 246-262 (2005) · Zbl 1084.34008 · doi:10.1007/s11006-005-0024-0
[26] Yang, Liu; Deng, Zui-Cha; Yu, Jian-Ning; Luo, Guan-Wei: Optimization method for the inverse problem of reconstructing the source term in a parabolic equation, Math. comput. Simulation 80, No. 2, 314-326 (2009) · Zbl 1183.65118 · doi:10.1016/j.matcom.2009.06.031
[27] Yang, Fan; Fu, Chu-Li: A simplified Tikhonov regularization method for determining the heat source, Appl. math. Model. 34, No. 11, 3286-3299 (2010) · Zbl 1201.65177 · doi:10.1016/j.apm.2010.02.020