\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 21, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/21\hfil Identification of the density dependent coefficient] {Identification of the density dependent coefficient in an inverse reaction-diffusion problem from a single boundary data} \author[R. Tinaztepe, S. Tatar , S. Ulusoy \hfil EJDE-2014/21\hfilneg] {Ramazan Tinaztepe, Salih Tatar, S\"uleyman Ulusoy} % in alphabetical order \address{Ramazan Tinaztepe \newline Department of Mathematics, Faculty of Education, Zirve University, Sahinbey \newline Gaziantep 27270, Turkey} \email{ramazan.tinaztepe@zirve.edu.tr} \urladdr{http://person.zirve.edu.tr/tinaztepe/} \address{Salih Tatar \newline Department of Mathematics, Faculty of Education, Zirve University, Sahinbey \newline Gaziantep 27270, Turkey} \email{salih.tatar@zirve.edu.tr} \urladdr{http://person.zirve.edu.tr/statar/} \address{S\"uleyman Ulusoy \newline Department of Mathematics, Faculty of Education, Zirve University, Sahinbey \newline Gaziantep 27270, Turkey} \email{suleyman.ulusoy@zirve.edu.tr} \urladdr{http://person.zirve.edu.tr/ulusoy/} \thanks{Submitted November 8, 2013. Published January 10, 2014.} \subjclass[2000]{45K05, 35R30, 65M32} \keywords{Fractional derivative; fractional Laplacian; weak solution; \hfill\break\indent inverse problem; Mittag-Leffler function; Cauchy problem} \begin{abstract} This study is devoted to the numerical solution of an inverse coefficient problem for a density dependent nonlinear reaction-diffusion equation. The method is based on approximating the unknown coefficient by polynomials. An optimal idea for solving the inverse problem is to minimize an error functional between the output data and the additional data. For this purpose, we find a polynomial of degree $n$ that minimizes the error functional; i.e, $n^{th}$ degree polynomial approximation of the unknown coefficient for the desired $n$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} \label{sec:intro} Problems involving the determination of unknown coefficients in ordinary and partial differential equations by some additional conditions are well known in the mathematical literature as the inverse coefficient problems. These additional conditions may be given on the whole domain, on the boundary of the domain, or at the final time. As it is known, a direct problem aims to find a solution that satisfies given ordinary or partial differential equation with initial and boundary conditions. In some problems the main ordinary or partial differential equation and the initial and boundary conditions are not sufficient to obtain the solution, but, instead some additional conditions are required. Such problems are called the inverse problems. A problem is said to be well-posed or properly posed in the sense of Hadamard \cite{J} if it satisfies the following three conditions: there exists a solution of the problem (existence), there is at most one solution of the problem (uniqueness), and the solution depends continuously on the data (stability). If at least one of these properties does not hold, then the equation is called ill-posed. In this context, another definition of the inverse problems can be given as follows: If one of two problems which are inverse to each other is ill-posed, we call it the inverse problem and the other one the direct problem. It is well-known that inverse problems are often ill-posed. In this paper, we consider the one dimensional nonlinear inverse reaction-diffusion problem \begin{equation}\label{inverseprob} \begin{gathered} u_t=(a(u) u_x)_x+| u | ^p, \quad (x,t)\in\Omega_T, p \geq 2, \\ u(x,0)=0, \quad x \in \overline \Omega,\\\ -a(u(0,t))u_x(0,t)=g(t), \quad t \in [0,T],\\ u_x(1,t)=0, \quad t \in [0,T],\\ u(0,t)=f(t), \quad t \in [0,T],\\ \end{gathered} \end{equation} where $\Omega$ is an open interval in $\mathbb{R}$, $\Omega_T:=\Omega\times (0,T)$ a domain in $\mathbb{R}\times \mathbb{R^{+}}$. The inverse problem here consists of determining the unknown coefficient $a(u)$ in the inverse problem \eqref{inverseprob}. The last condition, i.e., $u(0,t)=f(t)$ is taken to be an additional condition. In this context, for given inputs $a(u)$, $g(t)$ and $p$ the nonlinear problem \eqref{inverseprob}, without the additional condition, is defined as a direct (forward) problem. Henceforth, the expression {\it direct problem} will mean the studied problem without the additional condition. In the problem \eqref{inverseprob}, the compatibility condition $f(0)=0$ is satisfied. The density dependent nonlinear reaction-diffusion equation $u_t=(a(u) u_x)_x+R(u)$ models many transport phenomena where $a(u)$ and $R(u)$ are called the diffusion coefficient and the reaction term respectively. The applications of this equations have had wide variety involving transport in porous medium, population dynamics, plasma physics and combustion theory. The density dependent nonlinear reaction-diffusion equation becomes $u_t=(a(u) u_x)_x+| u | ^p$ for the reaction term $R(u)=| u | ^p$ and has been used to model many different applications. For instance, it is used to model the flow of groundwater in a homogeneous, isotropic, rigid and unsaturated porous medium \cite{B}. If we choose the coordinate $x$ to measure the vertical height from ground level and pointing upward, the soil is represented by the vertical column $(-L, 0)$ \cite{BA}, however as noted in \cite{BA}, they assumed that absorption and chemical osmotic and thermal effects are negligible and there is no source inside the material. But in our work, we consider the effect of the nonlinear source term as well. This equation can be obtained easily by combining Darcy's law and the continuity equation. First initial condition and third boundary conditions in \eqref{inverseprob} represent the initial moisture content and moisture content at $x = 1$ respectively. The flux of moisture and moisture content at $x = 0 $ are identified by the second and last conditions in problem \eqref{inverseprob} respectively. In this context, source term can be interpreted as material source. Some numerical methods have been introduced for nonlinear diffusion equations. In \cite{C1, D1} for example, the authors considered an inverse problem for the nonlinear diffusion equation \[ u_t=(a(u)u_x)_x, \quad (x,t)\in\Omega_T. \] The inverse problem is reformulated as an auxiliary inverse problem. Also it is proved that this auxiliary inverse problem has at least one solution in a specific admissible class. Finally, the auxiliary inverse problem is approximated by an associated identification problem. In addition to these, the authors presented a numerical method to solve the inverse problem for a special class of admissible coefficients. In the method, the partial differential equation is solved directly employing finite difference approach and the optimization part is solved using the program ZXMIN of the IMSL package. Furthermore, the intersecting graph technique is defined as a second numerical method \cite{C1,D1}. We note that there are some other numerical methods introduced for numerical solution of the nonlinear inverse diffusion problem. A numerical algorithm based on the finite difference method and the least-squares scheme was given in \cite{N1}. According to this algorithm, Taylor's expansion is employed to linearize nonlinear terms and then finite difference method is applied to discretize the problem domain. Also this approach rearranges the matrix form of the governing differential equations and estimate the unknown coefficient. In \cite{N2}, the given algorithm is based on linearizing nonlinear terms by Taylor's series expansion and removing the time-dependent terms by Laplace transform. Finite difference technique is used to discretize the problem. In this paper, we develop a numerical algorithm to solve the inverse coefficient problem \eqref{inverseprob}. The algorithm is based on the optimization of an error functional between the output data and the additional data. The algorithm attempts to minimize the error functional by using polynomials of a predetermined degree $n$. In doing so, it is assumed that the error functional is differentiable with respect to the coefficients of the polynomial which enables us to use the gradient descent method. The numerical experiments show that the algorithm is effective in practical use. A detailed analysis of the factors affecting the algorithm is also given. The remainder of this paper comprises of five sections: In the next section, some theoretical background is recalled for the inverse problem including the existence and uniqueness of the solution. Our numerical method is given in section 3. Some numerical examples are presented to show the efficiency of the method in section 4. In section 5, analysis of the results are given. The final section of the paper contains discussions and comments on planned studies. \section{Existence and uniqueness for the inverse coefficient problem} The theoretical aspect of an inverse problem, similar to \eqref{inverseprob} is studied in \cite{FSS1}. The inverse problem \eqref{inverseprob} differs from the inverse problem in \cite{FSS1} in that it imposes the condition $-a(u(0,t))u_x(0,t)=g(t)$, (see \cite{C1}); whereas in \cite{FSS1}, the condition $-a(u(0,t))u_t(0,t)=g(t)$, (see \cite{APS}) is used instead. In this paper, the authors have proved that the inverse problem has a unique solution under certain conditions. But the existence and uniqueness theorem also holds for the inverse problem \eqref{inverseprob}. In this section, we present the existence and uniqueness theorem for the self-containment of the paper, but first we give some preliminaries. We define the following norms and function spaces: \begin{gather*} | u |_D=\sup \{u(s), s \in D \},\\ H_{\alpha}(u)=\sup \big\{\frac{u(p)-u(q)}{d(p,g)^{\alpha}} : p,q \in D, p \ne q \big\},\\ | u |_\alpha=| u |_D+H_{\alpha}(u),\quad | u |_{1+\alpha}=| u |_\alpha+\big| \frac{\partial u}{\partial x} \big|_\alpha,\\ | u |_{2+\alpha}=| u |_\alpha+\big| \frac{\partial u}{\partial x} \big|_\alpha +\big| \frac{\partial^2 u}{\partial x^2 } \big|_\alpha +\big| \frac{\partial u}{\partial t} \big|_\alpha, \end{gather*} where $D=\Omega_T$, $d(p,q)$ is the usual Euclidean metric for the points $p$ and $q$ in $D$ and $\alpha>0$ is a constant. The space of all functions $u$ for which $| u |_{2+\alpha}<\alpha$ is denoted by $C_{2+\alpha}(D)$. In \cite{FRD}, it is proved that the space $C_{2+\alpha}(D)$ is a Banach space with the corresponding norm. \begin{definition} \label{def1}\rm A set $\mathbb{A}$ satisfying the following conditions is called the {\it class of admissible coefficients} for the inverse coefficient problem \eqref{inverseprob} \begin{itemize} \item[(C1)] $a \in C_{2+\alpha}(I)$ with $| a |_{2+\alpha} \leq c_1$, \item[(C2)] $\nu \leq a \leq \mu$ and $a'(s)>0$, for $s \in I$, \item[(C3)] $| a' | \leq \delta$ and $| a'' | \leq \delta$ for $s \in I$, \end{itemize} where $\alpha \in (0,1)$, I is a closed interval, $a:I \to \mathbb{R}$ and $c_1, \nu, \mu, \delta$ are positive constants. \end{definition} Inspired by \cite{C1,D1}, the authors in \cite{FSS1} use the transformation $v(x,t)=T_a(u(x,t))=\int_0^{u(x,t)}a(s)ds$ to transform the inverse problem \eqref{inverseprob} into the problem \begin{equation} \label{newinverseprob} \begin{gathered} v_t=a(T_a^{-1}(v))v_{xx}+a(T_a^{-1}(v))| (T_a^{-1}(v)) |^p,\quad (x,t)\in\Omega_T,~p\geq 2, \\ v(x,0)=0,\quad x \in \overline \Omega,\\ -v_x(0,t)=g(t),\quad t \in [0,T],\\ v_x(1,t)=0,\quad t \in [0,T],\\ v(0,t)=F(t),\quad t \in [0,T],\\ \end{gathered} \end{equation} where $F(t)= \int_0^{f(t)}a(s)ds$. In problem \eqref{newinverseprob}, the compatibility condition $F(0)=0$ also holds. We note that $\frac {d}{du}T_a(u)\geq\nu>0$ implies that $T_a(u(x,t))$ is invertible. So the term $a(T_a^{-1}(v))$ in \eqref{newinverseprob} makes sense. It is clear that the unknown coefficient is not in divergence form in \eqref{newinverseprob}, that is why the inverse problem \eqref{inverseprob} is needed to be transformed into a new one. Moreover, determination of the unknown coefficient $a(u)$ in the problem \eqref{newinverseprob} is equivalent to determination of the unknown coefficient $A(v):=a(T_a^{-1}(v))$ in the problem \eqref{newinverseprob}. Therefore the authors study the inverse problem \eqref{newinverseprob} instead of \eqref{inverseprob}. Before we state the existence and uniqueness theorem, we need the following lemmas for the functions that belong the class of admissible coefficients $\mathbb{A}$. \begin{lemma}[\cite{C1}] \label{lem1} For each $a \in \mathbb{A}$, there exists a unique function $p_a(u,v)$ defined on $I \times I$ such that $p_a(u,v)$ is a number between $u$ and $v$. Moreover, the following equality holds \[ a(u)-a(v)=a'(p_a(u,v))(u-v). \] \end{lemma} It is important to emphasize that the above lemma can be applied to $T_a^{-1}(v)$. Because the following equalities imply that the inverse function $T_a^{-1}(v)$ also belongs to the set $\mathbb{A}$, \begin{gather*} \frac{\partial}{\partial v} (T_a^{-1}(v))=\frac{1}{T_a'(v)}=\frac{1}{a(v)},\\ \frac{\partial^2}{\partial v^2} (T_a^{-1}(v)) = \frac{\partial}{\partial v} \frac{1}{a(v)} =-\frac{a'(v)}{a(v)^2}. \end{gather*} Hence we have the following lemma. \begin{lemma}[\cite{C1}] \label{lem2} There exists a unique function $q_a(\cdot,\cdot)$ such that $q_a(u,v)$ is a number between $u$ and $v$. Moreover, the following equality holds, \[ T_a^{-1}(u)-T_a^{-1}(v)= (T_a^{-1})'(q_a(u,v))(u-v). \] \end{lemma} \begin{lemma}[\cite{APS}] \label{lem3} Suppose that $\{w_n\}$ is a bounded and monotone increasing sequence of functions in $C_{2+\alpha}(\Omega_T)$. Then, there exists a function $w \in C_{2+\alpha}(\Omega_T) $ such that $D^{\beta}D_t^j w_n \to D^{\beta}D_t^j w, |\beta| \leq 2$, $0 \leq j \leq 1$ uniformly (on compact subsets of D), where $D^{\beta} u(x,t)=\frac{\partial^ {| \beta |} }{\partial x_1^{m_1} \dots\partial x_n^{m_n}}u(x,t), \beta=(m_1 ,m_2,\dots, m_n), | \beta | =m_1+m_2+\dots+m_n $ and $D_t^{m} u(x,t)=\frac{\partial ^m}{\partial t^m}u(x,t)$. \end{lemma} Using the above lemmas, following \cite{FSS1} closely, it can be proved that the inverse coefficient problem \eqref{newinverseprob} has a unique solution under certain conditions. This is stated in the following theorem. For the sake of completeness we provide a sketch of the proof. \begin{theorem} \label{thm1} Assume that $\frac{dF}{dt}$ and $g(t)$ are positive continuous functions on $[0,T]$ and $C^1([0,T])$ respectively. Then the inverse problem \eqref{newinverseprob} (equivalently, \eqref{inverseprob} ) has a unique solution. \end{theorem} \begin{proof}[Sketch of the proof] {\it Step 1 (Existence).} Let ${\hat{v}_0}=0$ and ${\hat{v}_n}$, $n=1,2,\dots , $ be solution of the problem \begin{gather*} (\hat {v}_n)_t=a(T_a^{-1}((\hat {v}_{n-1})))v_{xx} +a(T_a^{-1}((\hat {v}_{n-1})))| (T_a^{-1}((\hat {v}_{n-1}))) |^p, \; (x,t)\in\Omega_T,\, p\geq 2, \\ \hat {v}_n(x,0)=0,\quad x \in \overline \Omega,\\ -(\hat {v}_n)_x(0,t)=g(t),\quad t \in [0,T],\\ (\hat {v}_n)_x(1,t)=0,\quad t \in [0,T],\\ \hat {v}_n(0,t)=F(t),\quad t \in [0,T]. \end{gather*} First it is not difficult to show that the sequence $\{\hat {v}_n\}$ is monotone increasing. Also, by applying Lemma \ref{lem3} for $\beta=1$ and $0 \leq j \leq 1$, we deduce \[ D^{\beta}D_t^j {\hat{v}_{n}} \to D^{\beta}D_t^j{\hat{v}}. \] Since ${\hat{v}_{n}} $ is the a solution of the problem in Step 1, we have \begin{equation} (\hat {v}_n)_t=a(T_a^{-1}((\hat {v}_{n-1})))v_{xx}+a(T_a^{-1} ((\hat {v}_{n-1})))| (T_a^{-1}((\hat {v}_{n-1}))) |^p. \end{equation} \label{e2.2} Letting $n \to \infty$, we deduce that $\hat {v}$ is a solution of \eqref{newinverseprob}. {\it Step 2 (Uniqueness).} Suppose $v(x,t)$ and $u(x,t)$ are two solutions of \eqref{newinverseprob}. Let $z(x,t)=v(x,t)-u(x,t)$. Then $z(x,t)$ must satisfy the problem \begin{equation} \label{e2.3} \begin{gathered} z_t=a(T_a^{-1}(v))z_{xx}+C_*(x,t)z,\\ z(x,0)=0,\quad x \in \overline \Omega,\\ z_x(0,t)=z_x(1,t)=0,\quad t \in [0,T],\\ z(0,t)=0,\quad t \in [0,T], \end{gathered} \end{equation} where \begin{gather*} C_*(x,t)=C(x,t)+ \frac{h'(T_a^{-1}(\bar u))}{a(q_a(v(x,t),u(x,t)))}, \\ C(x,t)= \frac{a'\Big(p_a\Big(T_a^{-1}(v(x,t)),T_a^{-1}(u(x,t))\Big)\Big)} {a\Big(q_a\Big(v(x,t),u(x,t)\Big)\Big)}. \end{gather*} By using the maximum principle we conclude that $z(x,t) \equiv 0$. Therefore the solution of problem \eqref{newinverseprob} must be unique. \end{proof} \section{Overview of the method} \label{sec:over} In this section, we present our numerical method. The essence of the method is to approximate the unknown diffusion coefficient $a(u)$ by polynomials. Since the unknown diffusion coefficient $a(u)$ is continuous on a compact domain $\Omega_{T}$ in the problem (3), there exists a sequence of polynomials converging to $a(u)$. However, finding such a sequence which guarantees the solution of the inverse problem is difficult. It is known that the direct problem has a unique solution if $a(u)$ satisfies certain conditions \cite{FRD}. Our starting point is that the correct $a(u)$ will yield the solution satisfying the condition $u(0,t)=f(t)$, hence $a(u)$ will minimize the functional \[ F(c)=\| u(c,0,t)-f(t)\| _2^2, \] where $u(c,x,t)$ is the solution of the direct problem with the diffusion coefficient $c(u)$ and $\| \cdot\| _2$ is the $L^2$ norm on $\Omega$. Hence, our strategy is to find a polynomial of degree $n$ that minimizes $F(c)$, i.e, $n^{th}$ degree polynomial approximation of $a(u)$ for the desired $n$. From now on we take $c(u)=c_{0}+c_{1}u+\dots+c_{n}u^{n}$ as $c=(c_{0},\dots,c_{n}$) hence $F(c)$ is a function of $n$ variables. To overcome the ill-posedness of the inverse problem, Tikhonov regularization is applied. A regularization term with a regularization parameter $\lambda$ is added to $F(c)$ \[ G(c)=\| u(c,0,t)-f(t)\| _2^2+\lambda\| c\| ^2, \] where $\| c\| $ denotes the Euclidean norm of $c$. From now on, we fix $n$ and $\lambda$ and we leave the discussions about the regularization parameter to the next section. The method for minimizing $G(c)$ depends on the properties of $F(c)$, e.g., convexity, differentiability etc. In our case, the convexity or differentiability of $F(c)$ is not clear due to the term $u(c,x,t)$. However, we do not envision a major drawback in assuming the differentiability of $F(c)$ in numerical implementations. For this reason, we proceed the minimization of $G(c)$ by the steepest descent method which will utilize the gradient of $F$. In this method, the algorithm starts with an initial point $b_{0}$, then the point providing the minimum is approximated by the points \[ b_{i+1}=b_{i}+\Delta b_{i}, \] where $\Delta b_{i}$ is the feasible direction which minimizes \[ E(\Delta b)=G(b_{i}+\Delta b). \] This procedure is repeated until a stop criterion is satisfied; i.e, $\| \Delta b_{i}\| <\epsilon$ or $|G(b_{i+1})-G(b_{i})|<\epsilon$ or a certain number of iterations. In the minimization of $E(\Delta b)$, we use the following estimate on $u(b_{i}+\Delta b,0,t)$; \[ u(b_{i}+\Delta b,0,t)\simeq u(b_{i},0,t)+\nabla u(b_{i},0,t)\cdot\Delta b, \] where $\nabla$denotes the gradient of $u(b,0,t)$ with respect to $b$. Hence $E(\Delta b)$ turns out to be \[ E(\Delta b)=\| \nabla u(b_{i},0,t)\cdot\Delta b+u(b_{i},0,t)-f(t)\| _2^2 +\lambda\| \Delta b\| _2^2. \] In numerical calculations, we note that $\| \cdot\| _2$ can be discretized by using a finite number of points in $[0,T]$, i.e., for $t_{1}=0