DSpace at VNU: Identification of the pollution source of a parabolic equation with the time-dependent heat conduction

Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 RESEARCH Open Access Identification of the pollution source of a parabolic equation with the time-dependent heat conduction Nguyen Huy Tuan1,2* , Dang Duc Trong2 , Ta Hoang Thong3 and Nguyen Dang Minh2 * Correspondence: thnguyen2683@gmail.com Saigon Institute for Computational Science and Technology, Ho Chi Minh City, Vietnam Department of Mathematics, University of Natural Science, Vietnam National University, 227 Nguyen Van Cu, Distric 5, Ho Chi Minh City, Vietnam Full list of author information is available at the end of the article Abstract We consider the problem of identifying the pollution source of a 1D parabolic equation from the initial and the final data The problem is ill posed and regularization is in order Using the quasi-boundary method and the truncation Fourier method, we present two regularization methods Error estimates are given and the methods are illustrated by numerical experiments Introduction In this paper, we consider an inverse problem of identifying a pollution source from data measured at some points in a watershed The pollution source causes water contamination in some region In all industrial countries, groundwater pollution is a serious environmental problem that puts the whole ecosystem, including humans, in jeopardy The quality and quantity of groundwater have much effect on human life and may lead to natural environmental changes (see, e.g., []) As we know, most efforts to find pollutant transport are based on the methodology of mathematics Solute transport in a uniform groundwater flow can be described by the one-dimensional (D) linear parabolic equation ∂ u˜ ∂  u˜ ∂ u˜ –D  +V + Ru˜ = F (x, t), ∂t ∂x ∂x x ∈ ,  < t < T, () where is a spatial domain, u˜ is the solute concentration, V represents the velocity of watershed movement, R denotes the self-purifying function of the watershed, and F (x, t) ˜ t) Putting is a source term causing the pollution function u(x, V V ˜ t) = u(x, t)e D x–( D +R)t , u(x, we can transform the latter equation into ∂ u ∂u – D  = F(x, t), ∂t ∂x V () V where F(x, t) = F (x, t)e– D x+( D +R)t ; we still call it the source function Coming from this relationship between the two equations () and (), in the present paper, we will find a pair ©2014 Tuan et al.; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Page of 15 of functions (u, F) satisfying () subject to the initial and the final conditions u(x, ) = , u(x, T) = g(x), x ∈ (, π), () and the boundary condition u(, t) = u(π, t) =  To consider a more general case, we will replace D in () by a given function a(t) which is defined later This inverse source problem is ill posed Indeed, a solution corresponding to the given data does possibly not exist, and even if the solution exists (uniquely) then it may not depend continuously on the data Because the problem is severely ill posed and difficult, many preassumptions on the form of the heat source are in order In fact, let {ϕn (t)} be a basis in L (, T) Then the function F can be written as ∞ ϕn (t)fn (ξ ) F(ξ , t) = () n= In the simplest case, one reduces this approximation to its first term F(x, t) = ϕ(t)f (x), where the function ϕ is given Source terms of this form frequently appear, for example, as a control term for the parabolic equation In another context, this problem is called the identification of heat source; it has received considerable attention from many researchers in a variety of fields using different methods since  If the pollute source has the form of f = f (u), the inverse source problem was studied in [] In [], the authors considered the heat source as a function of both space and time variables, in the additive or separable forms Many researchers viewed the source as a function of space or time only In [, ], the authors determined the heat source dependent on one variable in a bounded domain by the boundary-element method and the iterative algorithm In [], the authors investigated the heat source which is time-dependent only by the method of a fundamental solution Many authors considered the uniqueness and stability conditions of the determination of the heat source under this separate form In spite of the uniqueness and stability results, the regularization problem for unstable cases is still difficult For a long time, it has been investigated for a heat source which is time-depending only [, , ] or space-depending only [, , –] As regards the regularization method, there are few papers with a strict theoretical analysis of identifying the heat source F(x, t) = ϕ(t)f (x), where ϕ is a given function Trong et al [, ] considered this problem by the Fourier transformation method Recently, when a(t) =  and ϕ(t) = e–λt (λ > ), the problem () describes a heat process of radio isotope decay whose decay rate is λ, which has been considered by Qian and Li [] In [], Hasanov identified the heat source which has the form of F(x, t) = F(x)H(t) of the variable coefficient heat conduction equation ut = (k(x)ux )x + F(x)H(t) using the variational method However, the generalized case with the time-dependent coefficient of u in the main equation is still limited and open In this paper, we consider the following generalized equation: ut – a(t)uxx = F(x, t) () and u satisfies the condition () This kind of equation () has many applications in groundwater pollution It is a simple form of advection-convection, which appears in groundwater Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Page of 15 pollution source identification problems (see []) Such a model is related to the detection of the pollution source causing water contamination in some region The remainder of the paper is divided into three sections In Section , we apply the quasi-boundary value method and truncation method to solve the problem ()-() Then we also estimate the error between an exact solution and the regularization solution with the logarithmic order and Hölder order Finally, some numerical experiments will be given in Section  Identification and regularization for inhomogeneous source depending on time variable Let · , ·, · be the norm and the inner product in L (, π) Let a : [, T] → R be a cont tinuous function on [, T] We set A(t) =  a(s) ds The problem () can be transformed into ⎧ d  ⎪ ⎨ dt u(x, t), sin nx + n a(t) u(x, t), sin nx = ϕ(t) f (x), sin nx , u(x, t), sin nx = , ⎪ ⎩ u(x, T), sin nx = g(x), sin nx  < t < T, () By an elementary calculation, we can solve the ordinary differential equation () to get T  A(T) f (x), sin nx = en  A(t) en – ϕ(t) dt g(x), sin nx  or ∞ T  A(T) en f (x) = n=  A(t) en – ϕ(t) dt gn sin nx, ()   where gn = π g(x), sin nx Note that en A(T) increases rather quickly when n becomes large Thus the exact data function g(x) must satisfy the property that g(x), sin nx decays rapidly But in applications, the input data g(x) can only be measured and never be exact We assume the data functions g (x) ∈ L (, π), ϕ, ϕ ∈ L (, T) to satisfy g –g ≤ , ϕ –ϕ ≤ () and ϕ(t) > C , ϕ (t) > C , t ∈ (, T), where the constant C >  represents a noise level and Lemma  Let s > , X ≥  Then for all  ≤ t ≤ T and  < < , we have ( + X)s ( + e–TX ) ≤ ss e–s  + T –s T ln(/ ) s () ≤ eTX ≤ e () Proof Case  X ∈ [, T ] It is clear to see that ( + X)s ( + e–TX ) ≤ ( + X)s e–TX Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Page of 15  From the inequality ≤ ( es )s ( ln(/ )s , we get ) ( + X)k ( + e–TX ) ≤ ss e–s  ln(/ ) s ≤ ss e–s  + T –s T ln(/ ) s Case  X > T Set e–TX = Y Then we obtain ( + X)s ( + e–TX ) = + Y =  +Y T T – ln( Y ) =  +Y T ln(/ ) T ln(/ ) = s T T – ln( Y ) s s s – ln( ) T – ln( Y ) s  +Y – ln( ) T – ln( Y ) s – ln( ) s  We continue to estimate the term +Y ( T–ln( ) Y) If  < Y ≤  then  < – ln( ) < – ln( Y ), thus  +Y – ln( ) T – ln( Y ) s < , else if Y >  then ln Y >  and ln( Y ) = –TX < – due to the assumption X ∈ ( T , ∞) Therefore, ln Y ( + ln( Y )) ≤  This implies that < – ln – ln < <  + ln Y T – ln( Y ) – ln( Y ) () Hence, in this case, we get  +Y – ln( ) T – ln( Y ) s < ( + ln Y )s = ( + ln Y )s Y – Y () Set g(Y ) = ( + ln Y )s Y – for Y > e– Taking the derivative of this function, we get g (Y ) = ( + ln Y )s– Y – (s –  – ln Y ) () The function g has a maximum at the point Y , so that g (Y ) =  This implies that Y = es– Therefore sup( + ln Y )s Y – ≤ g(Y ) = ss e–s Y ≥ Since (), (), we have  +Y – ln( ) T – ln( Y ) s ≤ ss e–s () Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Page of 15 From (), we get ( + X)s ( + e–TX ) s T ln(/ ) ≤ ss e–s ≤ ss e–s  + T –s T ln(/ ) s Lemma  Let a : [, T] → R be a continuous function on [, T] Let p = inf≤t≤T a(t), q = sup≤t≤T a(t) Then we have T – t exp n (i)    (ii)  , T ≤ a(s) ds dt ( + n )k (α( ) + e–n A(T) ) ≤ () B(q, k, T) qT k , α( ) ln( α( ) ) () where B(q, k, T) = k k e–k  + (qT)–k Proof (i) Since a(t) ≥ p, we have T – t exp n  a(s) ds dt =  ≤ =  A(T) (ii) Since a(t) ≤ q, we get e–n  ( + n )k (α( ) + e–n A(T) ) ≤ ≤  T  exp(n t  a(s) ds) dt t  p ds) dt = pn  ≤ T pn T e –  T  exp(n  T pn t  e  qT ≥ e–n dt () Then using Lemma , we get  ( + n )k (α( ) + e–n qT ) B(q, k, T) qT k α( ) ln( α( ) ) () 2.1 Regularization by a quasi-boundary value method Denote by · k the norm in Sobolev space H k (, π) defined by ∞ f k =  k +n   |fn |  , n= where fn = π f (x), sin nx We modify the problem ()-() by perturbing the Fourier expansion of final value g as follows: ⎧ ∂u ∂ – ∂x (a(t) ∂u ) = ϕ (t)f (x), x ∈ (, π),  < t < T, ⎪ ⎪ ∂t ∂x ⎪ ⎪ ⎨ u (x, ) = , x ∈ (, π), u (, t) = u (π, t) = , t ∈ (, T), ⎪ ⎪ ⎪  ⎪ e–A(T)n ⎩ u (x, T) = ∞ g sin nx, x ∈ (, π), n= α( )+e–A(T)n n () Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Page of 15 where gn = π g (x), sin nx and α( ) is a regularization parameter such that lim → α( ) =  This problem is based on the quasi-boundary regularization method which is given in [] This method has been studied for solving various types of inverse problem [, ] The solution of this problem is given by ∞ T  α( ) + e–n A(T) f (x) = n=  A(t) en – ϕ (t) dt gn sin nx  () Now we will give an error estimate between the regularization solution and the exact solution by the following theorem Theorem  Suppose that f , g ∈ L (, π) such that f k < ∞ and g k+  < ∞ for some  k ≥  Let g ∈ L (, π) be measured data at t = T satisfying () Let f be the regularized solution given by () If we select α( ) such that lim → then lim α( ) → = , f – f =  and we have following estimate: f –f ≤ C Tα( ) + C(p, q, k, T) + B(q, k, T) qT ln( α( ) ) k  α( ) g ln( α( ) ) k+  k f k () Proof We define ∞ h (x) = n= T  α( ) + e–n A(T) –  A(t) ϕ (t) dt  A(t) ϕ(t) dt en gn sin nx () gn sin nx ()  and ∞ p (x) = n= T  α( ) + e–n A(T) en –  We divide the proof into three steps Step  Estimate f – h From () and (), we have ∞ f –h  = n=  (α( ) + e–n A(T) ) ∞ ≤ n= ≤ ≤ ( T  C  T  C dt)  A(t) en – ϕ (t) dt     ln ( qT )  |α( )| ( T dt) g –g gn – gn gn – gn     C T  |α( )| () Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Page of 15 Step  Estimate h – p From (), (), and (), we have  h –p ∞ n= ∞  A(t) en n= B(q, k, T) ≤ α( )  B(q, k, T) α( )  ∞ [ × n= qT ln( α( ) ) qT ln( α( ) ) – ϕ (t) dt T  A(t) en –  –  ϕ(t) dt gn  T n A(t) (ϕ(t) – ϕ (t)) dt)  e T n A(t) T ϕ (t) dt) (  en A(t) ϕ(t) dt)  e (  ( + n )k (α( ) + e–n A(T) ) ( = ≤ T  (α( ) + e–n A(T) ) = k ∞ [ ( n= T n A(t) T dt][  |ϕ (t) – ϕ(t)| dt]  e T n A(t) T ϕ(t) dt) (  en A(t) ϕ (t) dt)  e  + n  + n k  gn k  gn k T n A(t) T dt][  |ϕ (t) – ϕ(t)| dt]  e T C T  (  en A(t) dt)  + n k  gn () On other hand, we have  A(T) en  A() – en  A(T) = en T  A(t) en –= (t) dt  T  A(t) n A (t)en = T  A(t) n a(t)en dt =  dt  Since p ≤ a(t) ≤ q, we get T  A(t) en p T dt ≤   A(t) T dt ≤ q a(t)en  A(t) en  dt  Hence  A(T) en – qn T ≤ n A(t) e dt ≤  A(T) en pn  – () It follows from () and () that h –p  B(q, k, T) ≤ α( )  qT ln( α( ) ) k ∞ n=  q (en A(T) – ) ϕ (t) – ϕ(t) pC T  (en A(T) – ) Since   en A(T) –   – e–n A(T) = ≤  A(T) n  (e – ) ( – e–n A(T) ) ≤   – e–A(T)  ≤    – e–n A(T)   – e–pT    + n k+  gn () Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 and ϕ (t) – ϕ(t)  h –p  ≤ ≤  Page of 15 , we obtain q pC T  ( – e–pT ) = C(p, q, k, T) B(q, k, T) α( )   qT ln( α( ) ) k ∞  + n k+  gn n= k  α( )   g ln( α( ) ) k+  Here q C(p, q, k, T) = √   B(q, k, T)(qT)k  pC T ( – e–pT ) Hence ≤ C(p, q, k, T) h –p k  α( ) g ln( α( ) ) k+  () Step  Estimate p – f In fact, using the Fourier expansion of f , we have ∞ p –f     – en A(T) α( ) + e–n A(T) = n= ∞ = n= ∞ = n= α( ) α( ) + e–n A(T)  α( ) α( ) + e–n A(T)  T  A(t) en – ϕ(t) dt   A(T) gn  en gn T n A(t) ϕ(t) dt  e fn Using Lemma , we obtain ∞ p –f  = n= |α( )|  + n ) + e–n A(T) ) ( + n )k (α(  ≤ B(q, k, T) qT ln( α( ) ) k  fn k  k f This implies that p – f ≤ B(q, k, T) qT ln( α( ) ) k f k () Combining Steps , , and  and using the triangle inequality, we get f –f ≤ f –h ≤ C Tα( ) + h –p + p –f + C(p, q, k, T) + B(q, k, T) qT ln( α( ) )  α( ) ln( α( ) ) k g k+  k f k () Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Remark  If we choose α( ) = m Page of 15 ,  < m < , then () holds Remark  In this theorem, with the assumption f ∈ H k (, π), we have an error f – f of logarithmic order In the next section, we introduce a truncation method which improves the order of the error We present the error of Hölder estimates (the order is α ,  < α < ) with a weaker assumption of f , i.e., f ∈ H  (, π) 2.2 Regularization by a truncation method Theorem  Suppose that f ∈ H  (, π) Let g ∈ L (, π) be measured data at t = T satisfying () Put N T  A(T) en f (x) = k–  – ϕ (t) dt gn sin nx,  n= where N = [  A(t) en () ] + , k ∈ (, ) Then the following estimate holds: f –f ≤Q –k  + P k , () where P= q g   + ,   C pC ( – e–pT ) √ Q=  π + π  f H  (,π ) Proof From () and (), we have f (x) – f (x) ∞ N  A(T) en = T n A(t) ϕ(t) dt  e n= ∞  A(T) en gn sin nx – n= N  A(T) en gn sin nx + T n A(t) ϕ(t) dt n=  e = n=N+ N – n= T n A(t) ϕ  e (t) dt gn sin nx  A(T) en gn sin nx T n A(t) ϕ(t) dt  e  A(T) en T n A(t) ϕ  e (t) dt gn sin nx = I  + I , () where ∞ I = n=N+  A(T) en gn sin nx T n A(t) ϕ(t) dt  e () and N I = n= N  A(T) en T n A(t) ϕ(t) dt  e gn sin nx – n=  A(T) en T n A(t) ϕ  e (t) dt gn sin nx () Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Page 10 of 15 Step  We estimate I In fact, since (), we get ∞ I  ∞  A(T) en = n=N+ T n A(t) ϕ(t) dt)  e ( gn fn = () n=N+ Using integration by parts, we have π fn = f (x) sin nx dx = –  cos nx f (x) n x=π + x= π  n f (x) cos nx dx  (–)n   f () – f (π) + n n n = π f (x) cos nx dx ()  Hence |fn | ≤ |f ()| + |f (π)| + n π  f (x) n () On the other hand, since H  (, π) is embedded continuously in C[, π] we can assume π that u ∈ C[, π] So, there exists an m ∈ [, π] such that f (m) = π  f (x) dx We have π f (π) = f (m) + f (x) dx, m m f () = f (m) – () f (x) dx  It follows that f (π) ≤ f (m) + π f (x) dx ≤ m π π ≤  f (x) + f (x) π  π  π f (x) dx f (x) dx +  dx =  √ π f H  (,π ) ()  In a similar way, we also obtain |f ()| ≤ This implies that √ ( π + n n=N+ ∞ I  ≤ π  )  √ ≤  π+ π   √ ≤  π+ π   f √ π f H  (,π ) Hence |fn | ≤ √ √  π + π n f  H  (,π ) ∞ f  H  (,π )  –n n n=N+ f  H  (,π )  N () Step  We estimate I The term () can be rewritten as follows: N I =  A(T) en ( n= N = n=  A(T) en  T n A(t) T ϕ (t) dt – gn  en A(t) ϕ(t) dt]  e T n A(t) T ϕ(t) dt)(  en A(t) ϕ (t) dt)  e [gn ( sin nx  T n A(t) T ϕ (t) dt + gn  en A(t) (ϕ  e T n A(t) T ϕ(t) dt)(  en A(t) ϕ (t) dt)  e [(gn – gn ) H  (,π ) (t) – ϕ(t)) dt] sin nx Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Page 11 of 15 Then N I Using  en A(T) – qn N (gn – gn ) T n A(t) ϕ(t) dt)  e n= ( ≤ T n A(t) dt,  e  A(T) en n=  A(T) en ≤  ( gn [ () we have ≤ T n A(t) ϕ(t) dt)  e ( T n A(t) (ϕ (t) – ϕ(t)) dt]  e T n A(t) T ϕ(t) dt) (  en A(t) ϕ (t) dt)  e en n= N (gn – gn )  A(T)  N + n= N ≤ n= N ≤ n=  A(T) en C ( gn – gn   n q en A(T) gn – gn C (en A(T) – )  n q gn – gn C ( – e–n A(T) ) N  q C ≤ T n A(t) dt)  e   () In a similar way and using (), we also obtain  A(T)  N T n A(t) (ϕ (t) – ϕ(t)) dt]  e T n A(t) T ϕ(t) dt) (  en t ϕ (t) dt)  e en ( n= gn [  A(T)  N en ≤ ( n= T n A(t) T dt][  |ϕ (t) – ϕ(t)| dt]  e T n A(t) T ϕ(t) dt) (  en A(t) ϕ (t) dt)  e gn [  A(T)  N en ≤ gn q[ n= N T n A(t) T dt][  |ϕ  e T C (  en A(t) dt)  ≤ n= N  n q en A(T) (en A(T) – )gn pC (en A(T) – ) (t) – ϕ(t)| dt]   ≤ n= n q ( – e–n A(T) )gn  pC ( – e–n A(T) ) It is easy to see that   –e–n A(T) ≤  –e–pT () It implies that  A(T)  N T n A(t) (ϕ (t) – ϕ(t)) dt]  e T n A(t) T ϕ(t) dt) (  en A(t) ϕ (t) dt)  e en n= ( N ≤ n= ≤ ≤ gn [ N  gn  C ( – e–pT ) N  q  pC ( – e–pT )    ∞ gn n=  g N q pC ( – e–pT ) () Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Page 12 of 15 Therefore I  ≤ where P = N  C  C +  + N  q  g  ≤ N   P , pC ( – e–pT ) q g  pC (–e–pT ) Hence I ≤ N  P () Combining (), (), and (), we obtain f –f = I + I  ≤ I  + I  √ ≤  π+ Since N = [ f –f k–  π  f H  (,π )  + PN  N () ] + , we obtain ≤Q –k  √ where Q = ( π + + P k , π )  f () H  (,π ) Numerical results In this section, we consider some examples simulation for the theory in Section  In numerical experiments, we are interested in the error between exact source and source with approximation as RMSE: RMSE(f , f ) :=  N N f (xn ) – f (xn )  n= with f (xn ), f (xn ) a discretization of function f , f Now, we consider ⎧ ⎪ ⎨ ut – a(t)uxx = ϕ(t)f (x), x ∈ (, π), t ∈ (, ), u(, t) = u(π, t) = , t ∈ (, ), ⎪ ⎩ u(x, T) = g(x), x ∈ (, π), where a(t) = t + , ϕ(t) = t  + t + , g(x) = sin x We can see the exact source f (x) = sin(x) Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Page 13 of 15 Using FORTRAN , we have a generator for noise data from routine rand() which is a random variable with the uniform distribution on [, ] Therefore, we have measurement data with noise g (x) = sin x + ∗ rand(), ϕ (t) = t  + t +  + ∗ rand(), where = –r , with r = , , , , works as the amplitude of noise We can easily see g –g √ π, √ < π < φ–φ and we have convergence to zero From Figure , we can compare between exact data and measured data We consider the source approximation with the quasi-reversibility regularization ∞   A(t) en f (x) = n= – ϕ (t) dt  gn sin nx + e–n A() We have the table of errors with = – , – , – and – (see Table ) and Figure  On the other hand, we have the source approximation with the truncation Fourier regularization N   A(t) en f (x) = n= – ϕ (t) dt  A() en  gn sin nx We have the table of errors = – , – , – and – (see Table ) and Figure  is as in Table  Figure Data for the problem Table The error estimation between exact solution and regularized solution by quasi-reversibility method RMSE(f , f ) 10–1 10–2 10–3 10–4 3.33236 × 10–1 4.82402 × 10–2 9.20728 × 10–4 1.11864 × 10–5 Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Page 14 of 15 Figure The approximation source Red is for the exact solution and green is for the approximation from the quasi-reversibility regularization Table The error estimation between exact solution and regularized solution by truncation method RMSE(f , f ) 10–1 10–2 10–3 10–4 1.74326 × 10–2 4.3520810–4 1.38719–5 5.67552–6 Figure The approximation source Red is for the exact solution and green is for the approximation from the truncation Fourier regularization Competing interests The authors declare that they have no competing interests Tuan et al Journal of Inequalities and Applications 2014, 2014:161 http://www.journalofinequalitiesandapplications.com/content/2014/1/161 Authors’ contributions All authors contributed equally to the writing of this paper All authors read and approved the final manuscript Author details Saigon Institute for Computational Science and Technology, Ho Chi Minh City, Vietnam Department of Mathematics, University of Natural Science, Vietnam National University, 227 Nguyen Van Cu, Distric 5, Ho Chi Minh City, Vietnam High School for the Gifted, Vietnam National University, Ho Chi Minh City, Vietnam Acknowledgements This research is funded by the Institute for Computational Science and Technology at Ho Chi Minh City (ICST HCMC) under the project name ‘Inverse parabolic equation and application to groundwater pollution source’ Received: 30 January 2014 Accepted: 25 March 2014 Published: 06 May 2014 References Atmadja, J, Bagtzoglou, AC: Marching-jury backward beam equation and quasi-reversibility methods for hydrologic inversion: application to 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2069-2080 (2012) 15 Denche, M, Bessila, K: A modified quasi-boundary value method for ill-posed problems J Math Anal Appl 301, 419-426 (2005) 10.1186/1029-242X-2014-161 Cite this article as: Tuan et al.: Identification of the pollution source of a parabolic equation with the time-dependent heat conduction Journal of Inequalities and Applications 2014, 2014:161 Page 15 of 15 ... this article as: Tuan et al.: Identification of the pollution source of a parabolic equation with the time-dependent heat conduction Journal of Inequalities and Applications 2014, 2014:161 Page... identifying an unknown source in the heat equation J Math Chem 49(3), 765-775 (2011) 14 Hasanov, A: Identification of spacewise and time dependent source terms in 1D heat conduction equation from temperature... Note that en A( T) increases rather quickly when n becomes large Thus the exact data function g(x) must satisfy the property that g(x), sin nx decays rapidly But in applications, the input data g(x)