- Email: [email protected]
Numerical procedures for determining of an unknown parameter in parabolic equation
Applied Mathematics and Computation 162 (2005) 1219–1226
www.elsevier.com/locate/amc
Numerical procedures for determining of an unknown parameter in parabolic equation Emine Can Baran Department of Physics, University of Kocaeli, Anitpark, 41300 Izmit, Kocaeli, Turkey
Abstract Numerical procedures for the solution of an inverse problem of determining unknown source parameter in a parabolic equation are considered. Two different numerical procedures are studied and their comparison analysis is presented. One of the these procedures obtained by introducing transformation of an unknown function, while the other based on trace functional formulation of the problem. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Parabolic equation; Inverse problem; Unknown source parameter; Finite-difference method
1. Introduction In this paper, we study numerical procedures for the following inverse problem of simultaneously finding unknown coefficients pðtÞ and uðx; tÞ satisfy equation ut ¼ uxx þ qux þ pðtÞu þ f ðx; tÞ;
x 2 ð0; 1Þ; t 2 ð0; T
ð1Þ
with the initial-boundary conditions uðx; 0Þ ¼ uðxÞ;
x 2 ð0; 1Þ;
ð2Þ
uð0; tÞ ¼ l1 ðtÞ;
t 2 ½0; T ;
ð3Þ
uð1; tÞ ¼ l2 ðtÞ;
t 2 ½0; T
ð4Þ
E-mail address: [email protected] (E.C. Baran). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.03.003
1220
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
and the additional specification uðx ; tÞ ¼ EðtÞ;
x 2 ð0; 1Þ; t 2 ð0; 1;
ð5Þ
where f ðx; tÞ, uðxÞ, l1 ðtÞ, l2 ðtÞ and EðtÞ 6¼ 0 are known functions, q is known constant and x is a fixed prescribed interior point in ð0; 1Þ. If u denotes the concentration of a chemical or temperature then p ¼ pðtÞ can be regarded as a source control. If pðtÞ is known then direct initial boundary value problem (1)– (3) has a unique smooth solution uðx; tÞ [1]. The similar inverse problem have been studied by authors in [2,3]. In this work we study two type of numerical procedures for considered problem. One of them is proposed [2–4]. According to this procedure the term pðtÞu in (1) is eliminated by introducing some transformation and system (1)– (4) is written in the canonical suitable form for the finite difference solution. Another procedure to the solution of the same problem is obtained by using trace type functional (TTF) formulation [5].
2. Procedure I (canonical representation) This procedure is as follows: the term pðtÞu in (1) eliminated by introducing the following transformation: Z t q vðx; tÞ ¼ uðx; tÞ exp x
ðpðsÞ q2 =4Þ ds ; ð6Þ 2 0 Z t 2 rðtÞ ¼ exp
ðpðsÞ q =4Þ ds ð7Þ 0
and uðx; tÞ ¼
q vðx; tÞ exp x ; rðtÞ 2
0
pðtÞ ¼
q2 r ðtÞ ;
rðtÞ 4
ð8Þ
then for these new functions and variables the auxiliary problem becomes q vt ¼ vxx þ rðtÞ exp x f ðx; tÞ; 0 < x < 1; 0 < t 6 T ; ð9Þ 2 q vðx; 0Þ ¼ uðxÞ exp x ; 0 < x < 1; ð10Þ 2 vð0; tÞ ¼ l1 ðtÞrðtÞ;
0 < t6T; q vð1; tÞ ¼ l2 ðtÞrðtÞ exp ; 0 < t6T 2
subject to
ð11Þ ð12Þ
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
rðtÞ ¼
q vðx ; tÞ Þ exp x ; EðtÞ 2
x 2 ð0; 1Þ; 0 < t 6 T ;
1221
ð13Þ
which we can obtain by elementary calculations. Problems (9)–(12) can be viewed as a nonlocal parabolic problem with nonlocal boundary conditions. This system can be solved by finite difference method. Existence and uniqueness of solutions to the similar problems has been proved in [6,7]. Let vnj and vn be an approximation to vðxj ; tn Þ and vðx ; tn Þ respectively, s ¼ Dt > 0 and h ¼ Dx > 0 be step length on time and space coordinates, f0 ¼ t0 < t1 < < tM ¼ T g and f0 ¼ x0 < x1 < < xN ¼ 1g where tn ¼ ns; xj ¼ jh denotes a partition of ½0; T and ½0; 1; respectively. Then the finite difference scheme can be used for approximation (9)–(13) as follows. Find vnj such that q vnj vjn 1 vnjþ1 2vnj þ vnj 1 x ; ¼ þ fjn Rðvn 1 Þ exp 2 2 s h j ¼ 1; N 1; n ¼ 1; M; q xj ; j ¼ 0; N ; v0j ¼ uðxj Þ exp 2 vn0 ¼ ln1 Rðvn 1 Þ;
n ¼ 1; M; q vnN ¼ ln2 Rðvn 1 Þ exp ; n ¼ 1; M; 2
ð14Þ ð15Þ ð16Þ ð17Þ
where fjn ¼ f ðxj ; tn Þ, ln1 ¼ l1 ðtn Þ, ln2 ¼ l2 ðtn Þ, q vn 1 exp x Rðvn 1 Þ ¼ 2 Eðtn 1 Þ and for x 2 ðxm ; xmþ1 , vn 1 ¼
xmþ1 x n 1 x xm n 1 vm þ vmþ1 h h
approximate value of vn 1 by the linear interpolation. The scheme (14)–(17) has a second order approximation on space and first order approximation on time. Once vni is known numerically the unknown pair ðu; pðtÞÞ can be calculated through the inverse transformation (8) and (13) via numerical differentiation.
3. Procedure II (TTF formulation) If the functions pair ðu; pÞ solves the inverse problem (1)–(4) then Et ¼ uxx jx¼x þ qux jx¼x þ pðtÞujx¼x þ f ðx ; tÞ
ð18Þ
1222
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
from that pðtÞ ¼ ðEt uxx jx¼x qux jx¼x f ðx ; tÞÞ=EðtÞ:
ð19Þ
Substituting in (1) leads to the following initial boundary problem: Et uxx jx¼x qux jx¼x f ðx ; tÞjx¼x ut ¼ uxx þ qux þ u EðtÞ þ f ðx; tÞ x 2 ð0; 1Þ;
ð20Þ
uðx; 0Þ ¼ uðxÞ;
x 2 ½0; 1;
ð21Þ
uð0; tÞ ¼ l1 ðtÞ;
t 2 ð0; T ;
ð22Þ
uð1; tÞ ¼ l2 ðtÞ;
t 2 ð0; T :
ð23Þ
Such representation is called as trace type functional (TTF) formulation of problem (1)–(4) [5]. From the solution of this system the approximate solution of pðtÞ can be determined by (19). Numerical solution of (20)–(23) is realized by the implicit finite difference scheme which can be written as follows: jþ1 jþ1 jþ1 jþ1 ujþ1 ujþ1
uji uiþ1 2ui þ ui 1 i iþ1 ui 1 ¼ þ kðuj Þujþ1 þ q i s h2 2h jþ1 þ fi ; 1 6 i 6 N 1; 0 6 j 6 M 1;
ð24Þ
u0i ¼ uðxi Þ;
0 6 i 6 N;
ð25Þ
uj0
¼ l1 ðtj Þ;
1 6 j 6 M;
ð26Þ
ujN
¼ l2 ðtj Þ;
1 6 j 6 M;
ð27Þ
where fijþ1 ¼ f ðxi ; tjþ1 Þ j j j uji þ1 uji 1 Ejþ1 Ej ui þ1 2ui þ ui 1 ; tj Þ
f ðx kðuj Þ ¼
q Ejþ1 : i s h2 2h ð28Þ The system (24)–(28) can be solved again by standard numerical solver.
4. Numerical result and discussion In this section we report some results of our numerical calculations using the numerical procedures described in the previous section. If we take a solution uðx; tÞ, coefficient q, source parameter pðtÞ and x as uðx; tÞ ¼ t sin x þ 1, q ¼ 2, pðtÞ ¼ ðt3 þ tÞ expð t2 Þ, x ¼ 0:26 then substituting in (1), it can be seen that the input data and additional condition in (1)–(5) can be as follows f ðx; tÞ ¼ sin xð1 þ tÞ 2t cos x pðtÞðt sin x þ 1Þ, uðxÞ ¼ uðx; 0Þ ¼ 1, l1 ðtÞ ¼ uð0; tÞ ¼ 1, l2 ðtÞ ¼ uð1; tÞ ¼ t sinð1Þ þ 1 and EðtÞ ¼ t sinð0:26Þ þ 1: First
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
1223
examples has been done to control whether expressions used in the procedures I and II are approximation expressions. The numerical results for different time and space steps are given in Figs. 1–3 for procedure I and II, respectively. As seen from the figures, that approximation is improved by increasing the number of nodes and for the sufficiently large number of nodes the agreement between numerical and exact solution becomes uniformly good. Second solution examples have been done to control the sensitivity of procedures to errors. Artificial errors were introduced into the additional specification data by defining functions Eðtn Þ ¼ Eðtn Þð1 þ dðtn ; qÞÞ where dðtn ; qÞ represents the level of relative error in the corresponding piece of data. Two case were considered. (a) dðt; qÞ ¼ d ¼ const, (b) dðt; qÞ is a random function of t uniformly distributed on ð q; qÞ (random errors). Calculation results with grid N M ¼ 300 600 are presented in Figs. 4 and 5, according to cases (a) and (b), respectively. Results with constant errors d ¼ 0:13 and d ¼ 0:03 are given in Fig. 4. In Fig. 5 the results of the case of random errors dðt; 0:001Þ and dðt; 0:002Þ are presented. As seen from the figures that for both procedures by constant errors results are worsening, especially for the procedure II. In the case of random errors
Fig. 1. Numerical and exact solution of pðtÞ (procedure I).
1224
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
Fig. 2. Numerical and exact solution of pðtÞ (procedure I).
Fig. 3. The best approximation is obtained for M ¼ 300, N ¼ 80 (procedure II).
approximation is worsening and there is an approximation in some integral norm.
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
1225
Fig. 4. Results with constant errors (procedure I).
Fig. 5. The results of the case of random errors dðt; 0:001Þ and dðt; 0:002Þ (procedure I).
The set of numerical calculations have been done to compare the sensitivity to random errors of procedures and the results indicated that procedure I is less sensitive to errors than procedure II.
1226
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
The next set of numerical examples have been done for different time and space steps to compare of procedures in terms of stability. Calculations show that stability area of procedure I is wider than procedure II. The narrowness of stability area of procedure II is reflected in the fact that the solution in reformulated problem in this case is recursively depends on itself through the second order derivates, which makes the coefficient extremely sensitive to variation in the solution. Both procedures to solve the problem were tried on different test and the results we observed indicated that procedure I is more stable than procedure II. On the other hand, procedure II is more effective on the solution of some problems and less sensitive to artificial errors than procedure I.
References [1] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type, Nauka, Moscow, 1967. [2] A.G. Fatullayev, E. Can, ’Numerical procedures for determining unknown source parameter in parabolic equation’, Math. Comp. Simulat. 1845 (2000) 1–9. [3] Y. Lin, An inverse problem for a class of quasilinear parabolic equations, SIAM J. Math. Anal. 22 (1991) 146–156. [4] A.G. Fatullayev, Numerical procedure for determination of an known parameter in a parabolic equation, Int. J. Comp. Math. 78 (2001) 103–109. [5] D. Colton, R. Ewing, W. Rundell, Inverse Problems in Partial Differential Equation, SIAM Press, Philadelphia, PA, 1990. [6] J.R. Cannon, Y. Lin, Determination of a parameter pðtÞ in some quasi-linear parabolic differential equations, Inverse Problems 4 (1998) 35–45. [7] J.R. Cannon, Y. Lin, S. Wang, Determination of source parameter in parabolic equations, Meccanica 27 (1992) 85–94.
www.elsevier.com/locate/amc
Numerical procedures for determining of an unknown parameter in parabolic equation Emine Can Baran Department of Physics, University of Kocaeli, Anitpark, 41300 Izmit, Kocaeli, Turkey
Abstract Numerical procedures for the solution of an inverse problem of determining unknown source parameter in a parabolic equation are considered. Two different numerical procedures are studied and their comparison analysis is presented. One of the these procedures obtained by introducing transformation of an unknown function, while the other based on trace functional formulation of the problem. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Parabolic equation; Inverse problem; Unknown source parameter; Finite-difference method
1. Introduction In this paper, we study numerical procedures for the following inverse problem of simultaneously finding unknown coefficients pðtÞ and uðx; tÞ satisfy equation ut ¼ uxx þ qux þ pðtÞu þ f ðx; tÞ;
x 2 ð0; 1Þ; t 2 ð0; T
ð1Þ
with the initial-boundary conditions uðx; 0Þ ¼ uðxÞ;
x 2 ð0; 1Þ;
ð2Þ
uð0; tÞ ¼ l1 ðtÞ;
t 2 ½0; T ;
ð3Þ
uð1; tÞ ¼ l2 ðtÞ;
t 2 ½0; T
ð4Þ
E-mail address: [email protected] (E.C. Baran). 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.03.003
1220
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
and the additional specification uðx ; tÞ ¼ EðtÞ;
x 2 ð0; 1Þ; t 2 ð0; 1;
ð5Þ
where f ðx; tÞ, uðxÞ, l1 ðtÞ, l2 ðtÞ and EðtÞ 6¼ 0 are known functions, q is known constant and x is a fixed prescribed interior point in ð0; 1Þ. If u denotes the concentration of a chemical or temperature then p ¼ pðtÞ can be regarded as a source control. If pðtÞ is known then direct initial boundary value problem (1)– (3) has a unique smooth solution uðx; tÞ [1]. The similar inverse problem have been studied by authors in [2,3]. In this work we study two type of numerical procedures for considered problem. One of them is proposed [2–4]. According to this procedure the term pðtÞu in (1) is eliminated by introducing some transformation and system (1)– (4) is written in the canonical suitable form for the finite difference solution. Another procedure to the solution of the same problem is obtained by using trace type functional (TTF) formulation [5].
2. Procedure I (canonical representation) This procedure is as follows: the term pðtÞu in (1) eliminated by introducing the following transformation: Z t q vðx; tÞ ¼ uðx; tÞ exp x
ðpðsÞ q2 =4Þ ds ; ð6Þ 2 0 Z t 2 rðtÞ ¼ exp
ðpðsÞ q =4Þ ds ð7Þ 0
and uðx; tÞ ¼
q vðx; tÞ exp x ; rðtÞ 2
0
pðtÞ ¼
q2 r ðtÞ ;
rðtÞ 4
ð8Þ
then for these new functions and variables the auxiliary problem becomes q vt ¼ vxx þ rðtÞ exp x f ðx; tÞ; 0 < x < 1; 0 < t 6 T ; ð9Þ 2 q vðx; 0Þ ¼ uðxÞ exp x ; 0 < x < 1; ð10Þ 2 vð0; tÞ ¼ l1 ðtÞrðtÞ;
0 < t6T; q vð1; tÞ ¼ l2 ðtÞrðtÞ exp ; 0 < t6T 2
subject to
ð11Þ ð12Þ
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
rðtÞ ¼
q vðx ; tÞ Þ exp x ; EðtÞ 2
x 2 ð0; 1Þ; 0 < t 6 T ;
1221
ð13Þ
which we can obtain by elementary calculations. Problems (9)–(12) can be viewed as a nonlocal parabolic problem with nonlocal boundary conditions. This system can be solved by finite difference method. Existence and uniqueness of solutions to the similar problems has been proved in [6,7]. Let vnj and vn be an approximation to vðxj ; tn Þ and vðx ; tn Þ respectively, s ¼ Dt > 0 and h ¼ Dx > 0 be step length on time and space coordinates, f0 ¼ t0 < t1 < < tM ¼ T g and f0 ¼ x0 < x1 < < xN ¼ 1g where tn ¼ ns; xj ¼ jh denotes a partition of ½0; T and ½0; 1; respectively. Then the finite difference scheme can be used for approximation (9)–(13) as follows. Find vnj such that q vnj vjn 1 vnjþ1 2vnj þ vnj 1 x ; ¼ þ fjn Rðvn 1 Þ exp 2 2 s h j ¼ 1; N 1; n ¼ 1; M; q xj ; j ¼ 0; N ; v0j ¼ uðxj Þ exp 2 vn0 ¼ ln1 Rðvn 1 Þ;
n ¼ 1; M; q vnN ¼ ln2 Rðvn 1 Þ exp ; n ¼ 1; M; 2
ð14Þ ð15Þ ð16Þ ð17Þ
where fjn ¼ f ðxj ; tn Þ, ln1 ¼ l1 ðtn Þ, ln2 ¼ l2 ðtn Þ, q vn 1 exp x Rðvn 1 Þ ¼ 2 Eðtn 1 Þ and for x 2 ðxm ; xmþ1 , vn 1 ¼
xmþ1 x n 1 x xm n 1 vm þ vmþ1 h h
approximate value of vn 1 by the linear interpolation. The scheme (14)–(17) has a second order approximation on space and first order approximation on time. Once vni is known numerically the unknown pair ðu; pðtÞÞ can be calculated through the inverse transformation (8) and (13) via numerical differentiation.
3. Procedure II (TTF formulation) If the functions pair ðu; pÞ solves the inverse problem (1)–(4) then Et ¼ uxx jx¼x þ qux jx¼x þ pðtÞujx¼x þ f ðx ; tÞ
ð18Þ
1222
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
from that pðtÞ ¼ ðEt uxx jx¼x qux jx¼x f ðx ; tÞÞ=EðtÞ:
ð19Þ
Substituting in (1) leads to the following initial boundary problem: Et uxx jx¼x qux jx¼x f ðx ; tÞjx¼x ut ¼ uxx þ qux þ u EðtÞ þ f ðx; tÞ x 2 ð0; 1Þ;
ð20Þ
uðx; 0Þ ¼ uðxÞ;
x 2 ½0; 1;
ð21Þ
uð0; tÞ ¼ l1 ðtÞ;
t 2 ð0; T ;
ð22Þ
uð1; tÞ ¼ l2 ðtÞ;
t 2 ð0; T :
ð23Þ
Such representation is called as trace type functional (TTF) formulation of problem (1)–(4) [5]. From the solution of this system the approximate solution of pðtÞ can be determined by (19). Numerical solution of (20)–(23) is realized by the implicit finite difference scheme which can be written as follows: jþ1 jþ1 jþ1 jþ1 ujþ1 ujþ1
uji uiþ1 2ui þ ui 1 i iþ1 ui 1 ¼ þ kðuj Þujþ1 þ q i s h2 2h jþ1 þ fi ; 1 6 i 6 N 1; 0 6 j 6 M 1;
ð24Þ
u0i ¼ uðxi Þ;
0 6 i 6 N;
ð25Þ
uj0
¼ l1 ðtj Þ;
1 6 j 6 M;
ð26Þ
ujN
¼ l2 ðtj Þ;
1 6 j 6 M;
ð27Þ
where fijþ1 ¼ f ðxi ; tjþ1 Þ j j j uji þ1 uji 1 Ejþ1 Ej ui þ1 2ui þ ui 1 ; tj Þ
f ðx kðuj Þ ¼
q Ejþ1 : i s h2 2h ð28Þ The system (24)–(28) can be solved again by standard numerical solver.
4. Numerical result and discussion In this section we report some results of our numerical calculations using the numerical procedures described in the previous section. If we take a solution uðx; tÞ, coefficient q, source parameter pðtÞ and x as uðx; tÞ ¼ t sin x þ 1, q ¼ 2, pðtÞ ¼ ðt3 þ tÞ expð t2 Þ, x ¼ 0:26 then substituting in (1), it can be seen that the input data and additional condition in (1)–(5) can be as follows f ðx; tÞ ¼ sin xð1 þ tÞ 2t cos x pðtÞðt sin x þ 1Þ, uðxÞ ¼ uðx; 0Þ ¼ 1, l1 ðtÞ ¼ uð0; tÞ ¼ 1, l2 ðtÞ ¼ uð1; tÞ ¼ t sinð1Þ þ 1 and EðtÞ ¼ t sinð0:26Þ þ 1: First
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
1223
examples has been done to control whether expressions used in the procedures I and II are approximation expressions. The numerical results for different time and space steps are given in Figs. 1–3 for procedure I and II, respectively. As seen from the figures, that approximation is improved by increasing the number of nodes and for the sufficiently large number of nodes the agreement between numerical and exact solution becomes uniformly good. Second solution examples have been done to control the sensitivity of procedures to errors. Artificial errors were introduced into the additional specification data by defining functions Eðtn Þ ¼ Eðtn Þð1 þ dðtn ; qÞÞ where dðtn ; qÞ represents the level of relative error in the corresponding piece of data. Two case were considered. (a) dðt; qÞ ¼ d ¼ const, (b) dðt; qÞ is a random function of t uniformly distributed on ð q; qÞ (random errors). Calculation results with grid N M ¼ 300 600 are presented in Figs. 4 and 5, according to cases (a) and (b), respectively. Results with constant errors d ¼ 0:13 and d ¼ 0:03 are given in Fig. 4. In Fig. 5 the results of the case of random errors dðt; 0:001Þ and dðt; 0:002Þ are presented. As seen from the figures that for both procedures by constant errors results are worsening, especially for the procedure II. In the case of random errors
Fig. 1. Numerical and exact solution of pðtÞ (procedure I).
1224
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
Fig. 2. Numerical and exact solution of pðtÞ (procedure I).
Fig. 3. The best approximation is obtained for M ¼ 300, N ¼ 80 (procedure II).
approximation is worsening and there is an approximation in some integral norm.
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
1225
Fig. 4. Results with constant errors (procedure I).
Fig. 5. The results of the case of random errors dðt; 0:001Þ and dðt; 0:002Þ (procedure I).
The set of numerical calculations have been done to compare the sensitivity to random errors of procedures and the results indicated that procedure I is less sensitive to errors than procedure II.
1226
E.C. Baran / Appl. Math. Comput. 162 (2005) 1219–1226
The next set of numerical examples have been done for different time and space steps to compare of procedures in terms of stability. Calculations show that stability area of procedure I is wider than procedure II. The narrowness of stability area of procedure II is reflected in the fact that the solution in reformulated problem in this case is recursively depends on itself through the second order derivates, which makes the coefficient extremely sensitive to variation in the solution. Both procedures to solve the problem were tried on different test and the results we observed indicated that procedure I is more stable than procedure II. On the other hand, procedure II is more effective on the solution of some problems and less sensitive to artificial errors than procedure I.
References [1] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uralceva, Linear and Quasi-linear Equations of Parabolic Type, Nauka, Moscow, 1967. [2] A.G. Fatullayev, E. Can, ’Numerical procedures for determining unknown source parameter in parabolic equation’, Math. Comp. Simulat. 1845 (2000) 1–9. [3] Y. Lin, An inverse problem for a class of quasilinear parabolic equations, SIAM J. Math. Anal. 22 (1991) 146–156. [4] A.G. Fatullayev, Numerical procedure for determination of an known parameter in a parabolic equation, Int. J. Comp. Math. 78 (2001) 103–109. [5] D. Colton, R. Ewing, W. Rundell, Inverse Problems in Partial Differential Equation, SIAM Press, Philadelphia, PA, 1990. [6] J.R. Cannon, Y. Lin, Determination of a parameter pðtÞ in some quasi-linear parabolic differential equations, Inverse Problems 4 (1998) 35–45. [7] J.R. Cannon, Y. Lin, S. Wang, Determination of source parameter in parabolic equations, Meccanica 27 (1992) 85–94.