\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{graphicx}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 256, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{7mm}}

\begin{document}
\title[\hfilneg EJDE-2016/256\hfil Regularization and error estimates]
{Regularization and error estimates for asymmetric backward nonhomogeneous
heat equations in a ball}

\author[L. M. Triet, L. H. Phong \hfil EJDE-2016/256\hfilneg]
{Le Minh Triet,  Luu Hong Phong}

\address{Le Minh Triet \newline
Division of Computational Mathematics and Engineering,
Institute for Computational Science. \newline
Faculty of Mathematics and Statistics,
Ton Duc Thang University, Ho Chi Minh City, Vietnam}
\email{leminhtriet@tdt.edu.vn}

\address{Luu Hong Phong \newline
Faculty of Mathematics,
University of Science,
Vietnam National University,
Ho chi Minh city, Vietnam}
\email{luuhongphong2812@gmail.com}

\thanks{Submitted September 2, 2016. Published September 21, 2016.}
\subjclass[2010]{35R25, 35R30, 65M30}
\keywords{Backward heat problem; quasi-boundary value method;
\hfill\break\indent spherical coordinates; ill-posed problem}

\begin{abstract}
 The backward heat problem (BHP) has been researched by many
 authors in the last five decades; it consists in recovering the initial
 distribution from the final temperature data.
 There are some articles \cite{W.L1,W.L2,W.L3} related the axi-symmetric BHP in
 a disk but the study in spherical coordinates is rare. Therefore, we wish to
 study a backward problem for nonhomogenous heat equation associated with
 asymmetric final data in a ball. In this article, we modify the
 quasi-boundary value method to construct a stable approximate solution
 for this problem. As a result, we obtain regularized solution and a sharp
 estimates for its error. At the end, a numerical
 experiment is provided to illustrate our method.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Inverse problems for partial differential equations play a vital role
in many physical areas. A typical example of these problems is the backward
heat problem (BHP) which is also known as the final value problem. The
purpose of the BHP is to retrieve the temperature distribution at a
particular time $t<T$ from the final temperature data. As we known, the
BHP is severely ill-posed in Hadamard's sense, i.e., the solution does not
always exist. Even if the solution exists, it may not depend continuously on
the given data. Therefore, an appropriate regularization is required so as
to get a stable solution.

There have been a lot of research related to the BHP in
different kinds of domains. For instance, the BHP has been investigated in
rectangular coordinates by many authors \cite{Fu1,Q1,Tr&H.T,Tr&H.T2,H.T&Tr},
to list just a few of them.
Recently, some works have considered  polar coordinates and
cylindrical coordinates. In particular, Cheng and Fu
\cite{W.L1,W.L2,W.L3} studied the axisymmetric backward heat
problem in a disk.
Cheng  and Fu \cite{W.L1,W.L3}  used the modified
Tikhonov method for regularizing the  problem
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}=\frac{\partial ^{2}u}{\partial r^{2}}+\frac{
1}{r}\frac{\partial u}{\partial r}, \quad 0<r\leq r_0, \; 0<t<T, \\
u(r,T)=\varphi (r), \quad 0\leq r\leq r_0,  \\
u(r_0,t)=0, \quad 0\leq t\leq T,  \\
| u(0,t)| <\infty , \quad  0\leq t\leq T,
\end{gathered}  \label{sec:15}
\end{equation}
where the function $\varphi (\cdot )$ in the problem \eqref{sec:15} is
radially symmetric or axisymmetric, i.e. it depends only on the radius $r$
and not on $\theta $.

 Cheng W. et al.\ \cite{W.L2} considered a problem which is similar to
\eqref{sec:15}. However, there are some differences in initial
condition which is expressed as follows
\begin{equation}
\begin{gathered}
\frac{\partial u}{\partial t}=\frac{\partial ^{2}u}{\partial r^{2}}+\frac{
1}{r}\frac{\partial u}{\partial r}, \quad 0<r\leq R, \; 0<t, \\
u(r,0)=0, \quad 0\leq r\leq R,   \\
u(r_1,t)=g(t) , \quad 0\leq t,  \\
| u(0,t)| <\infty , \quad 0\leq t,
\end{gathered}  \label{sec:16}
\end{equation}
in which $r$ is the radius coordinate and $g(\cdot )$ is the temperature
distribution at one fixed radius $r_1\leq R$ of a cylinder. By applying
the Fourier transform, the authors found the exact solution of the problem (
\ref{sec:16}) and used the modified Tikhonov method to construct the
regularized solutions. In the above papers \cite{W.L1,W.L2,W.L3}, although
the authors suggested some methods  to regularize  \eqref{sec:15} and
\eqref{sec:16}, they still did not give any numerical test
to prove the effectiveness of their regularization.

From the above problems, we see that BHP was considered in a rectangular
domain or a disk. In our knowledge, the works for BHP in a ball are rarely
studied and even we have not ever seen any results dealt with the asymmetric
case. Motivated by this reason, we focus on the problem of
determining the temperature distribution $u(r,\theta ,\phi ,t)$, for
 $(r,\theta ,\phi ,t)\in (0,a) \times (0,\pi ) \times
(0,2\pi ) \times (0,T) $, satisfying
\begin{gather}
u_{t} =c^{2}\Big\{ \frac{\partial ^{2}u}{\partial r^{2}}+\frac{2}{r}
\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\Big(\frac{\partial ^{2}u}{
\partial \theta ^{2}}+\cot \theta \frac{\partial u}{\partial \theta }+\csc
^{2}\theta \frac{\partial ^{2}u}{\partial \phi ^{2}}\Big) \Big\}
+q(r,\theta ,\phi ),  \label{pt1} \\
u(a,\theta ,\phi ,t) =0,  \label{pt2} \\
u(r,\theta ,\phi ,T) =f(r,\theta ,\phi ),  \label{pt3} \\
| u(0,\theta ,\phi, t)|  <\infty ,  \label{pt4}
\end{gather}
where $a$ is the radius coordinate and $f(\cdot ,\theta ,\phi )\in L^{2}
[[0;a] ;r] $ is the final temperature. In practice,
we cannot always obtain radially symmetric or axisymmetric form of the data
function $f$. Additionally, in physical applications, not only does the
initial temperature depend on the final data but it also depends on the heat
source. Hence, the heat source $q$ is not often homogeneous. Thus,
problem \eqref{pt1}-\eqref{pt4} is more general than problem
\eqref{sec:15} and \eqref{sec:16}. From that,  problem
\eqref{pt1}-\eqref{pt4} is more practical and applicable than
\eqref{sec:15} and \eqref{sec:16}. In this paper, we apply the
 modified quasi-boundary value method (MQBV) to formulate the approximate
solution for  \eqref{pt1}-\eqref{pt4}. As we known, the quasi-boundary
value (QBV) method which was given by  Showalter in 1983 is one of
effective regularization methods.
In \cite{Show}, the main idea of the QBV method is to add an
appropriate ``corrector term'' into the boundary condition. Based on this
idea, in \cite{Q1} we have modified the ``corrector term" to get a stable
error estimations so we called it the modified quasi-boundary value method.
By using the MQBV method, we can obtain the H\"{o}lder type estimate for 
the error between the regularized solution and the exact solution.
Furthermore, one advantage of the MQBV method is easier to make numerical
experiment for testing the feasibility of the method. Thus, we can make an
example to illustrate our results in this paper and it is a better point of
our paper when we compare with some previous papers \cite{W.L1,W.L2,W.L3}.

The rest of this article is organized as follows.
 In Section 2, some definitions and propositions are given.
In Section 3, we propose the regularized solutions for  problem
\eqref{pt1}-\eqref{pt4} and estimate the error between the regularized
solutions and the exact solution. Then, the proof of our results is given in
Section 4. Finally, we present a numerical experiment to illustrate the main
results in Section 5.

\section{Some definitions and propositions}

\begin{definition} \label{def2.1} \rm
Let $a>0$ and $L^{2}[[0;a] ;r] =\{ f:[0;a] \to \mathbb{R}:
f$  is Lebesgue measurable with weigh $r$ on $[0;a]\}$.
The above space is equipped with norm
\[
\| f\| _2=\Big(\int_0^a r|f(r)| ^{2}dr\Big) ^{1/2}.
\]
\end{definition}

Next some definitions and propositions, presented in
\cite{F.Bowman,Nakhle,Watson}, are restated.

\begin{proposition} \label{prop}
Let $n$ be a non-negative integer. Then, the spherical Bessel functions
of the $1^{st}$ kind of order $n$ are defined as
\begin{equation*}
j_{n}(x)=(\frac{\pi }{2x}) ^{1/2}{J_{n+\frac{1}{2}}}(x),
\end{equation*}
where ${J_{n+\frac{1}{2}}}$ is the Bessel function of the $1^{st}$-kind of
order $n+\frac{1}{2}$.
\end{proposition}

\begin{proposition} \label{prop2.3}
Let $n$ be a non-negative integer and the spherical Bessel's equation of
order $n$ be defined by
\begin{equation}
x^{2}y''+2xy'+(\lambda ^{2}x^{2}-n(n+1))y=0,\quad 0<x<a,\; y(a) =0.  \label{SBessel}
\end{equation}
Then, we obtain the following solutions for equation \eqref{SBessel},
\begin{equation*}
y_{n,j}(x) =j_{n}(\lambda _{n,j}x) ,\quad n=0,1,2,\dots ,j=1,2,\dots ,
\end{equation*}
where $\lambda =\lambda _{n,j}=\frac{\alpha _{n+1/2,j}}{a}$, for
$\alpha_{n+1/2,j}$ denotes the $j th$ positive zero of
$J_{n+\frac{1}{2}}$.
\end{proposition}

\begin{proposition} \label{prop2.4}
Let $n$ be a non-negative integer. Then, we have the Legendre polynomial of
the $1^{st}$ kind of degree $n$,
\begin{equation}
P_{n}(x) =\frac{1}{2^{n}}\sum_{m=0}^{M}(-1)^{m}\frac{(
2n-2m) !}{m!(n-m) !(n-2m) !}x^{n-2m},
\label{Legendre1}
\end{equation}
in which $M=n/2$ if $n$ is even or $M=(n-1)/2$ if $n$ is odd. Moreover, we have
the Legendre function of the $2^{nd}$ kind of degree $n$,
\begin{equation}
Q_{n}(x) =P_{n}(x) \int \frac{1}{[P_{n}(x) ] ^{2}(1-x^{2}) }dx,\quad 
(n=0,1,2,\dots ) .  \label{Legendre2}
\end{equation}
\end{proposition}

\begin{proposition} \label{prop2.5}
For $n=0,1,2,\dots $, Legendre's equation of degree $n$,
\begin{equation}
(1-x^{2}) y''-2xy'+n(n+1)y=0,\quad -1<x<1.  \label{Legendre}
\end{equation}
From which, the general solution of  \eqref{Legendre} is
\begin{equation*}
y(x)=c_1P_{n}(x) +c_2Q_{n}(x) ,
\end{equation*}
where $P_{n}(x) ,Q_{n}(x) $ are defined by \eqref{Legendre1} and \eqref{Legendre2}, 
respectively, and $c_1$, $c_2$ are arbitrary constants.
\end{proposition}

\begin{remark} \label{rmk2.6} \rm 
(i)
 For $n=0,1,2,\dots$ and $m=0,1,2,\dots $, the associated Legendre
function $P_{n}^{m}(x) $ is defined in terms of the $m-th$
derivative of the Legendre polynomial of degree $n$ by
\begin{equation}
P_{n}^{m}(x) =(-1) ^{m}(1-x^{2})^{m/2}\frac{
d^{m}P_{n}(x)}{dx^{m}}.  \label{Legendre3}
\end{equation}
Since $P_{n}$ is a polynomial of degree $n$, for $P_{n}^{m}$ to be nonzero,
we must take $0\leq m\leq n$. Moreover, if $m$ is negative integer, we
defined $P_{n}^{m}$ by
\begin{equation*}
P_{n}^{m}(x) =(-1) ^{m}\frac{(n+m)!}{(n-m) !}P_{n}^{-m}(x) .
\end{equation*}
This extends the definition of the associated Legendre function for 
$n=0,1,2,\dots$ and $m=-n,-(n-1) ,\dots ,n-1,n$.

(ii) After that, we define the spherical harmonics 
$Y_{n,m}( \theta ,\phi ) $ by
\begin{equation}
Y_{n,m}(\theta ,\phi ) =\sqrt{\frac{2n+1}{4\pi }\frac{(n-m)!}{
(n+m)!}}P_{n}^{m}(\cos \theta ) e^{im\phi },  \label{harmonic}
\end{equation}
where $n=0,1,2,\dots$ and $m=-n,-(n-1) ,\dots ,n-1,n$.
\end{remark}

\begin{proposition} \label{prop2.7}
Let $n$ be a non-negative integer and the differential equation for the
spherical harmonics be defined by
\begin{equation*}
\frac{\partial ^{2}Y}{\partial \theta ^{2}}+\cot \theta \frac{\partial Y}{
\partial \theta }+\csc ^{2}\theta \frac{\partial ^{2}Y}{\partial \phi ^{2}}
+n(n+1)Y=0,
\end{equation*}
where $0<\theta <\pi$, $0<\phi <2\pi $. Then, we have $2n+1$ nontrivial
solutions given by the spherical harmonics
\begin{equation*}
Y(\theta ,\phi ) =Y_{n,m}(\theta ,\phi ) ,\quad | m| \leq n,
\end{equation*}
where $Y_{n,m}(\theta ,\phi ) $ is defined by \eqref{harmonic}.
\end{proposition}

\begin{proposition} \label{prop2.8}
Let $f(r,\theta ,\phi ) $ be a square integrable function,
defined for $0<r<a$, $0<\theta <\pi $, $0<\phi <2\pi $, and $2\pi $-periodic
in $\phi $. Then, we have
\begin{equation*}
f(r,\theta ,\phi ) 
=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty }\sum_{m=-n}^{n}A_{jnm}j_{n}(
\lambda _{n,j}r) Y_{n,m}(\theta ,\phi ) ,
\end{equation*}
where
\begin{equation*}
A_{jnm}=\frac{2}{a^{3}j_{n+1}^{2}(\alpha _{n+\frac{1}{2},j}) }
 \int_0^a \int_0^{2\pi }\int_0^{\pi
}f(r,\theta ,\phi ) j_{n}(\lambda _{n,j}r) \overline{
Y}_{n,m}(\theta ,\phi ) r^{2}\sin \theta \,d\theta\,d\phi\,dr,
\end{equation*}
and $\overline{Y}_{n,m}$ is the complex conjugate of $Y_{n,m}$.
\end{proposition}

\section{Regularization and main results}

By employing the method of separation of variables, the exact
solution $u$ of the problem \eqref{pt1}-\eqref{pt3} corresponding to the
exact data $f$ can be found out as follows
\begin{equation}
u(r,\theta ,\phi ,t)=\sum_{j=1}^{\infty }\sum_{n=0}^{\infty
}\sum_{m=-n}^{n}A_{jnm}(t) j_{n}(\lambda
_{n,j}r) Y_{n,m}(\theta ,\phi ) ,  \label{ucx}
\end{equation}
where
\begin{gather*} %  \label{u1cx}
A_{jnm}(t) =\exp \{c^{2}\lambda _{n,j}^{2}(T-t)
\}\Big(f_{jnm}-\frac{q_{jnm}}{c^{2}\lambda _{n,j}^{2}}\Big) 
+\frac{q_{jnm}}{c^{2}\lambda _{n,j}^{2}},   
\\
f_{jnm} = \frac{2}{a^{3}j_{n+1}^{2}(\alpha _{n+1/2,j}) }
\int_0^a \int_0^{2\pi }\int_0^{\pi
}f(r,\theta ,\phi )j_{n}(\lambda _{n,j}r) \overline{Y}
_{n,m}(\theta ,\phi ) r^{2}\sin \theta \,d\theta\,d\phi\,dr,
\\
q_{jnm} = \frac{2}{a^{3}j_{n+1}^{2}(\alpha _{n+1/2,j}) }
\int_0^a \int_0^{2\pi }\int_0^{\pi
}q(r,\theta ,\phi )j_{n}(\lambda _{n,j}r) \overline{Y}
_{n,m}(\theta ,\phi ) r^{2}\sin \theta \,d\theta\,d\phi\,dr.
\end{gather*}
From  \eqref{ucx}, we can see that the term $\exp \{c^{2}\lambda
_{n,j}^{2}(T-t) \}$ becomes large as $n$ tends to infinity. This
term causes the instability of  problem \eqref{pt1}-\eqref{pt3} so that
we replace this term by a better term. In fact, if we use the QBV method;
the regularized problem shall be as follows
\begin{gather}
\omega _{t}^{\varepsilon } =c^{2}\nabla ^{2}\omega ^{\varepsilon
}+q(r,\theta ,\phi ),  \label{ptQBV1} \\
\omega ^{\varepsilon }(a,\theta ,\phi ,t) =0,  \label{ptQBV2} \\
\omega ^{\varepsilon }(r,\theta ,\phi ,T)+\varepsilon \omega ^{\varepsilon
}(r,\theta ,\phi ,0) =f^{\varepsilon }(r,\theta ,\phi ),  \label{ptQBV3} \\
| \omega ^{\varepsilon }(0,\theta ,\phi,t)|  <\infty ,  \label{ptQBV4}
\end{gather}
where $\nabla ^{2}$ is the spherical form of the Laplacian, i.e, 
\[
\nabla ^{2}\omega ^{\varepsilon }=\frac{\partial ^{2}\omega ^{\varepsilon }}{
\partial r^{2}}+\frac{2}{r}\frac{\partial \omega ^{\varepsilon }}{\partial
r}+\frac{1}{r^{2}}(\frac{\partial ^{2}\omega ^{\varepsilon }}{
\partial \theta ^{2}}+\cot \theta \frac{\partial \omega ^{\varepsilon }}{
\partial \theta }+\csc ^{2}\theta \frac{\partial ^{2\omega \varepsilon }}{
\partial \phi ^{2}}) .
\]
Then, we have the following regularized
solution of  \eqref{ptQBV1}-\eqref{ptQBV4},
\begin{equation*}
\omega ^{\varepsilon }(r,\theta ,\phi ,t)=\sum_{j=1}^{\infty
}\sum_{n=0}^{\infty }\sum_{m=-n}^{n}A_{jnm}^{\varepsilon
}(t)j_{n}(\lambda _{n,j}r) Y_{n,m}(\theta ,\phi ) ,
\end{equation*}
in which
\begin{gather*}
A_{jnm}^{\varepsilon }(t) 
=\frac{\exp \{ -c^{2}\lambda
_{n,j}^{2}t\} }{\varepsilon +\exp \{ -c^{2}\lambda
_{n,j}^{2}T\} }\Big(f_{jnm}^{\varepsilon }-\frac{q_{jnm}}{
c^{2}\lambda _{n,j}^{2}}\Big) +\frac{q_{jnm}}{c^{2}\lambda _{n,j}^{2}}, 
\\
f_{jnm}^{\varepsilon } 
=\frac{2}{a^{3}j_{n+1}^{2}(\alpha_{n+1/2,j}) }
\!\int_0^a \!\int_0^{2\pi}\!\int_0^{\pi }f^{\varepsilon }(r,\theta ,\phi )j_{n}(\lambda
_{n,j}r) \overline{Y}_{n,m}(\theta ,\phi ) r^{2}\sin
\theta \,d\theta\,d\phi\,dr.
\end{gather*}

In this article, we modify the regularized parameter of
 $\omega^{\varepsilon }$ by a different one to get a H\"{o}lder type estimate
for the error between the regularized solution and the exact solution. So we call
this method the modified quasi-boundary value method. In particular, we
construct the regularized solutions $u^{\varepsilon },v^{\varepsilon }$
corresponding to the measured data $f^{\varepsilon }$ and the exact data $f$, 
respectively
\begin{equation}
u^{\varepsilon }(r,\theta ,\phi ,t)=\sum_{j=1}^{\infty
}\sum_{n=0}^{\infty }\sum_{m=-n}^{n}B_{jnm}^{\varepsilon
}(t)j_{n}(\lambda _{n,j}r) Y_{n,m}(\theta ,\phi ) ,
\label{uch1}
\end{equation}
where
\begin{equation*}
B_{jnm}^{\varepsilon }(t)=\frac{\exp \{ -c^{2}\lambda
_{n,j}^{2}t\} }{\alpha (\varepsilon ) c^{2}\lambda
_{n,j}^{2}+\exp \{ -c^{2}\lambda _{n,j}^{2}T\} }\Big(
f_{jnm}^{\varepsilon }-\frac{q_{jnm}}{c^{2}\lambda _{n,j}^{2}}\Big) 
+\frac{q_{jnm}}{c^{2}\lambda _{n,j}^{2}},
\end{equation*}
and
\begin{equation}
v^{\varepsilon }(r,\theta ,\phi ,t)=\sum_{j=1}^{\infty
}\sum_{n=0}^{\infty }\sum_{m=-n}^{n}B_{jnm}(t)j_{n}(
\lambda _{n,j}r) Y_{n,m}(\theta ,\phi ) ,  \label{uch2}
\end{equation}
in which
\begin{equation*}
B_{jnm}(t)=\frac{\exp \{ -c^{2}\lambda _{n,j}^{2}t\} }{\alpha
(\varepsilon ) c^{2}\lambda _{n,j}^{2}+\exp \{
-c^{2}\lambda _{n,j}^{2}T\} }\Big(f_{jnm}-\frac{q_{jnm}}{
c^{2}\lambda _{n,j}^{2}}\Big) +\frac{q_{jnm}}{c^{2}\lambda _{n,j}^{2}}.
\end{equation*}
and $\alpha (\varepsilon ) $ is regularization parameter such
that $\alpha (\varepsilon ) \to 0$ when $\varepsilon
\to 0$. For short notation, we denote $\alpha =\alpha (\varepsilon)$.

\begin{lemma}
\label{bo de 1}For $0<\alpha <T$, $a>0$, we have the following inequality
\begin{equation*}
\frac{1}{\alpha a+\exp \{ -aT\} }\leq \frac{T}{\alpha }(\ln
(\frac{T}{\alpha }) ) ^{-1}.
\end{equation*}
\end{lemma}

\begin{lemma}\label{bo de 2}
For $0\leq t\leq s\leq T$, $0<\alpha <T$, $a>0$ and denote
 $\widetilde{T}=\max \{ 1,T\} $, we get the following inequalities
\begin{itemize}
\item[(i)]
\begin{equation*}
\frac{\exp \{ (s-t-T) a\} }{\alpha a+\exp \{-aT\} }
\leq \widetilde{T}\Big(\alpha \ln (\frac{T}{\alpha }) \Big) ^{\frac{t-s}{T}}.
\end{equation*}

\item[ii)] For $s=T$, we obtain
\begin{equation*}
\frac{\exp \{ -ta\} }{\alpha a+\exp \{ -aT\} }\leq
\widetilde{T}\Big(\alpha \ln (\frac{T}{\alpha }) \Big) ^{
\frac{t}{T}-1}.
\end{equation*}
\end{itemize}
\end{lemma}

In this article, we require some assumptions on the exact data $f$ and the
measured data $f^{\varepsilon }$ as follows
\begin{itemize}
\item[(H1)] Let $f(\cdot ,\theta ,\phi ) $, $f^{\varepsilon }(
\cdot ,\theta ,\phi ) \in L^{2}[[0;a] ;r] $ be
the exact data and the measured data such that
\begin{equation*}
\| f^{\varepsilon }(\cdot ,\theta ,\phi ) -f(\cdot
,\theta ,\phi ) \| _2\leq \varepsilon ,
\end{equation*}
for $(\theta ,\phi ) \in (0,\pi ) \times (0,2\pi ) $.

\item[(H2)] There exists a non-negative number $A$ such that
\begin{equation*}
\sup_{(\theta ,\phi ) \in [0;\pi ] \times
[0;2\pi ] }\| \frac{\partial u}{\partial t}(\cdot,\theta ,\phi ,0) \| _2\leq A.
\end{equation*}
\end{itemize}
In the following theorem, we give the stability of the modified method for
 problem \eqref{uch1}.

\begin{theorem}\label{dinh ly 1}
Let $\alpha \in (0;1) $, $f^{\varepsilon}(\cdot ,\theta ,\phi ) $,
 $f(\cdot ,\theta ,\phi ) $ satisfy {\rm (H1)} for all 
$(\theta ,\phi ) \in (0,\pi ) \times (0,2\pi ) $. Assume that
 $u^{\varepsilon }$ and $ v^{\varepsilon }$ are defined by \eqref{uch1} 
and \eqref{uch2} corresponding to the final data 
$f^{\varepsilon }(\cdot ,\theta ,\phi ) $ and 
$f(\cdot ,\theta ,\phi ) $, respectively. Then, we obtain
\begin{equation*}
\| u^{\varepsilon }(\cdot ,\theta ,\phi ,t)-v^{\varepsilon }(\cdot
,\theta ,\phi ,t)\| _2
\leq \widetilde{T}\Big(\alpha \ln (
\frac{T}{\alpha }) \Big) ^{\frac{t}{T}-1}\varepsilon ,
\end{equation*}
for $(\theta ,\phi ,t) \in (0,\pi ) \times (0,2\pi ) \times (0,T) $.
\end{theorem}

Finally, we estimate the error between the regularized solution
corresponding to the measured data $f^{\varepsilon }$ and the exact solution
of problem \eqref{pt1}-\eqref{pt3}.

\begin{theorem}\label{dinhly2}
Let $f$, $f^{\varepsilon }$ be as in Theorem \ref{dinh ly 1} and
$0<\alpha <\min \{1;T\}$. Suppose that $u^{\varepsilon }$ is defined by 
\eqref{uch1} corresponding to the perturbed datum $f^{\varepsilon }$ and $u$ be
the exact solution of  \eqref{pt1}-\eqref{pt3} satisfying {\rm (H2)}.
Then, we have
\begin{equation}
\| u^{\varepsilon }(\cdot ,\theta ,\phi ,t)-u(\cdot ,\theta ,\phi,t)\| _2
\leq \widetilde{T}\varepsilon ^{\frac{t}{T}}\Big(\ln
(\frac{T}{\varepsilon }) \Big) ^{\frac{t}{T}-1}(A+1) .  \label{kqc}
\end{equation}
for $(\theta ,\phi ,t) \in (0,\pi ) \times (0,2\pi ) \times (0,T) $.
\end{theorem}

\section{Proofs of main results}

\begin{proof}[Proof of Lemma \protect\ref{bo de 1}]
Let $0<\alpha <T$ and $\psi (a) =\frac{1}{\alpha a+\exp \{-aT\} }$.
 By simple calculations, we have
\begin{equation*}
\psi (a) \leq \frac{T}{\alpha (1+\ln (T/\alpha )) }
\leq \frac{T}{\alpha \ln (T/\alpha )},
\end{equation*}
for $a>0$.
This completes the proof.
\end{proof}

\begin{proof}[Proof of Lemma \protect\ref{bo de 2}]
(i) From Lemma \ref{bo de 1}, we have
\begin{align*}
\frac{\exp \{ (s-t-T) a\} }{\alpha a+\exp \{-aT\} } 
&\leq \frac{\exp \{ (s-t-T) a\} }{
(\alpha a+\exp \{ -aT\} ) ^{\frac{s-t}{T}}(
\alpha a+\exp \{ -aT\} ) ^{\frac{T+t-s}{T}}} \\
&\leq \frac{\exp \{ (s-t-T) a\} }{(\alpha
a+\exp \{ -aT\} ) ^{\frac{s-t}{T}}(\exp \{
-aT\} ) ^{\frac{T+t-s}{T}}} \\
&\leq \Big(\frac{T}{\alpha \ln (T/\alpha )}\Big) ^{\frac{s-t}{T}} \\
&\leq \widetilde{T}[\alpha \ln (T/\alpha )] ^{\frac{t-s}{T}},
\end{align*}
where $\widetilde{T}=\max\{ 1,T\}$.

(ii) Let $s=T$, we obtain 
\[
\frac{\exp \{ -ta\} }{\alpha a+\exp \{ -aT\} }
\leq \widetilde{T}[\alpha \ln (T/\alpha )] ^{\frac{t-T}{T}}.
\]
This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \protect\ref{dinh ly 1}]
From \eqref{uch1}, \eqref{uch2} and Lemma \ref{bo de 2}, we have the
 estimate
\begin{align}
&\| u^{\varepsilon }(\cdot ,\theta ,\phi ,t)-v^{\varepsilon }(\cdot
,\theta ,\phi ,t)\| _2  \nonumber\\
&= \| \sum_{j=1}^{\infty }\sum_{n=0}^{\infty
}\sum_{m=-n}^{n}\frac{\exp \{ -c^{2}\lambda _{n,j}^{2}t\}
}{\alpha c^{2}\lambda _{n,j}^{2}+\exp \{ -c^{2}\lambda
_{n,j}^{2}T\} }(f_{jnm}^{\varepsilon }-f_{jnm})
j_{n}(\lambda _{n,j}\cdot ) Y_{n,m}(\theta ,\phi )\| _2
 \nonumber \\
&\leq \widetilde{T}\Big(\alpha \ln (\frac{T}{\alpha })
\Big) ^{\frac{t}{T}-1}\| \sum_{j=1}^{\infty
}\sum_{n=0}^{\infty }\sum_{m=-n}^{n}(
f_{jnm}^{\varepsilon }-f_{jnm}) j_{n}(\lambda _{n,j}\cdot
) Y_{n,m}(\theta ,\phi ) \| _2    \label{Bdtp1} \\ 
&=\widetilde{T}\Big(\alpha \ln (\frac{T}{\alpha }) 
 \Big) ^{\frac{t}{T}-1}\| f^{\varepsilon }(\cdot ,\theta ,\phi )
-f(\cdot ,\theta ,\phi ) \| _2  \nonumber \\
&\leq \widetilde{T}\big(\alpha \ln (\frac{T}{\alpha })
\Big) ^{\frac{t}{T}-1}\varepsilon .  \nonumber
\end{align}
This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \protect\ref{dinhly2}]
Using the triangle inequality, 
\begin{equation}
\begin{aligned}
&\| u^{\varepsilon }(\cdot ,\theta ,\phi ,t)-u(\cdot ,\theta ,\phi,t)\| _2\\
&\leq \| u^{\varepsilon }(\cdot ,\theta ,\phi,t)
 -v^{\varepsilon }(\cdot ,\theta ,\phi ,t)\| _2
 +\|v^{\varepsilon }(\cdot ,\theta ,\phi ,t)-u(\cdot ,\theta ,\phi,t)\| _2. 
\end{aligned} \label{bdttg}
\end{equation}
From \eqref{ucx} and \eqref{uch2}, we obtain
\begin{align}
&\| v^{\varepsilon }(\cdot ,\theta ,\phi ,t)-u(\cdot ,\theta ,\phi
,t)\| _2   \nonumber \\
&=\| \sum_{j=1}^{\infty }\sum_{n=0}^{\infty}\sum_{m=-n}^{n}
\Big(\frac{\exp \{ -c^{2}\lambda
_{n,j}^{2}t\} }{\alpha c^{2}\lambda _{n,j}^{2}+\exp \{
-c^{2}\lambda _{n,j}^{2}T\} }-\exp \{c^{2}\lambda _{n,j}^{2}(
T-t) \}\Big)  \nonumber  \\
&\quad \times \big(f_{jnm}-\frac{q_{jnm}}{c^{2}\lambda _{n,j}^{2}}
\big) j_{n}(\lambda _{n,j}\cdot ) Y_{n,m}(\theta ,\phi
) \| _2  \nonumber  \\
&\leq \alpha \widetilde{T}\big(\alpha \ln (\frac{T}{\alpha }) 
 \Big) ^{\frac{t}{T}-1}   
 \| \sum_{j=1}^{\infty }\sum_{n=0}^{\infty
}\sum_{m=-n}^{n}c^{2}\lambda _{n,j}^{2}\exp \{ c^{2}\lambda
_{n,j}^{2}T\}  \nonumber \\
&\quad\times \Big(f_{jnm}-\frac{q_{jnm}}{c^{2}\lambda _{n,j}^{2}}
\Big) j_{n}(\lambda _{n,j}\cdot ) Y_{n,m}(\theta ,\phi
) \| _2  \nonumber  \\
&=\alpha \widetilde{T}\Big(\alpha \ln (\frac{T}{\alpha })
\Big) ^{\frac{t}{T}-1}\| \frac{\partial u}{\partial t}(
\cdot ,\theta ,\phi ,0) \| _2  \nonumber \\
&\leq \alpha \widetilde{T}(\alpha \ln (\frac{T}{\alpha }
) ) ^{\frac{t}{T}-1}A.  \label{Bdtp2}
\end{align}
Combining Theorem \ref{dinh ly 1} and \eqref{Bdtp2}, choosing $\alpha
=\varepsilon $, we have the  estimate
\begin{equation*}
\| u^{\varepsilon }(\cdot ,\theta ,\phi ,t)-u(\cdot ,\theta ,\phi
,t)\| _2
\leq \widetilde{T}\varepsilon ^{\frac{t}{T}}
\Big(\ln (\frac{T}{\varepsilon }) \Big) ^{\frac{t}{T}-1}(A+1) .
\end{equation*}
This completes the proof.
\end{proof}

\section{Numerical experiments}


In this section, we consider the backward nonhomogeneous heat
equation in a ball,
\begin{gather}
u_{t} =c^{2}\Big\{ \frac{\partial ^{2}u}{\partial r^{2}}+\frac{2}{r}
\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\Big(\frac{\partial ^{2}u}{
\partial \theta ^{2}}+\cot \theta \frac{\partial u}{\partial \theta }+\csc
^{2}\theta \frac{\partial ^{2}u}{\partial \phi ^{2}}\Big) \Big\}
+q(r,\theta ,\phi ),  \label{ptvd1} \\
u(a,\theta ,\phi ,t) =0,  \label{ptvd2} \\
u(r,\theta ,\phi ,T) =f(r,\theta ,\phi ),  \label{ptvd3}
\end{gather}
where $(r,\theta ,\phi ,t)\in (0,1) \times (0,\pi )
\times (0,2\pi ) \times (0,1)$, $c=0.05$ and $q,f$ are
defined as follows
\begin{gather}
f(r,\theta ,\phi ) = 100,  \label{dlieucx} \\
q(r,\theta ,\phi ) = j_{12}(\alpha _{25/2,1}r)[Y_{12,-12}(
\theta ,\phi ) +Y_{12,12}(\theta ,\phi ) ] .
\label{nguon}
\end{gather}
By simple calculations, we have
\begin{gather*}
f_{jnm}=0 \quad \text{for all $j,n\neq 0$  or $m\in [-n,n]/backslash \{ 0\}$}, \\
f_{j00}=\frac{400\sqrt{2}}{\sqrt{\alpha _{1/2,j}}J_{3/2}(\alpha _{1/2,j})}
\quad \text{for all }j, \\
q_{jnm}=0, \quad \text{for all $(j,n,m) \neq (1,12,-12) $ and $(1,12,12)$}, \\
q_{jnm}=1, \quad \text{for $(j,n,m) =(1,12,-12)$ or $(1,12,12) $}.
\end{gather*}
We also obtain
\begin{gather*}
Y_{12,12}(\theta ,\phi ) = \sqrt{\frac{25}{24!.4\pi }}
P_{12}^{12}(\cos \theta )e^{i12\theta }, \\
P_{12}^{12}(x) = (-1)^{12}(1-x^{2})^{6}\frac{d^{12}P_{12}(x)}{dx^{12}}, \\
P_{12}(x)  = \frac{1}{2^{12}}\sum_{m=1}^{6}(-1) ^{m}
\frac{(24-2m) !}{m!(12-m) !(12-2m) !}x^{12-2m}, \\
Y_{12,-12}(\theta ,\phi ) = (-1) ^{12}\overline{Y}_{12,12}(\theta ,\phi ) .
\end{gather*}
From which, we get the exact solution $u$ corresponding to $f$, $q$ which are
defined by \eqref{dlieucx} and \eqref{nguon}, respectively.
\begin{equation}
\begin{aligned}
&u(r,\theta ,\phi ,t) \\
&=\sum_{j=1}^{\infty }\exp (\alpha _{1/2,j}^{2}c^{2}(1-t))\frac{400
\sqrt{2}}{\sqrt{\alpha _{1/2,j}}J_{3/2}(\alpha _{1/2,j})}j_0(\alpha
_{1/2,j}r)Y_{0,0}(\theta ,\phi ) \\
&\quad+\big(1-\exp (\alpha _{25/2,1}^{2}c^{2}(1-t))\big) 
\frac{1}{c^{2}\alpha _{25/2,1}^{2}}j_{12}(\alpha _{25/2,1}r) \\
&\quad \times (Y_{12,-12}(\theta ,\phi ) +Y_{12,12}(\theta ,\phi ) )
\\
&=\sum_{j=1}^{\infty }\exp (\alpha _{1/2,j}^{2}c^{2}(1-t))\frac{200
\sqrt{2}}{\sqrt{\alpha _{1/2,j}}J_{3/2}(\alpha _{1/2,j})}(\frac{1}{
2\alpha _{1/2,j}r}) ^{1/2}J_{1/2}(\alpha _{1/2,j}r)   \\
&\quad +2(1-\exp (\alpha _{25/2,1}^{2}c^{2}(1-t))) \frac{1}{
c^{2}\alpha _{25/2,1}^{2}}\big(\frac{\pi }{2\alpha _{25/2,1}r}\big)
^{1/2} \\
&\quad\times J_{25/2}(\alpha _{25/2,1}r)P_{12}^{12}(\cos \theta )\cos 12\phi .
 \end{aligned} \label{ncxvd}
\end{equation}


\begin{figure}[htb]
\begin{center}
 \includegraphics[width=0.7\textwidth]{fig1} % Fig1
\end{center}
\caption{Exact and  regularized solutions corresponding to 
$\varepsilon _i$, $i=1,2,3$ when $r=0.5,\theta =\frac{\protect\pi }{6}$.}
\label{fig:matcat}
\end{figure}

\begin{figure}[htb]
\begin{center}
 \includegraphics[width=0.7\textwidth]{fig2} % Fig2
\end{center}
\caption{Exact and  regularized solution corresponding to 
$\varepsilon_1$}
\label{fig:ncx+1+2}
\end{figure}

\begin{figure}[htb]
\begin{center}
 \includegraphics[width=0.7\textwidth]{fig3}
\end{center}
\caption{Regularized solutions corresponding to $\varepsilon_i$, $i=2,3$.}
\label{fig:3+4chinh}
\end{figure}


Then, we consider the measured data
\begin{equation}
f^{\varepsilon }(r,\theta ,\phi )=100+\varepsilon .  \label{dlieudo}
\end{equation}
From \eqref{dlieucx} and \eqref{dlieudo}, we have
\begin{equation*}
\| f^{\varepsilon }(\cdot ,\theta ,\phi ) -f(\cdot,\theta ,\phi ) \| _2
=\Big(\int_0^{1}r\varepsilon ^{2}dr\Big) ^{1/2}
\leq \varepsilon .
\end{equation*}
From \eqref{uch1} and \eqref{dlieudo}, we have the regularized solution 
$u^{\varepsilon }$ as follows
\begin{align}
&u^{\varepsilon }(r,\theta ,\phi ,t)  \nonumber \\
&=\sum_{j=1}^{\infty }\frac{\exp (-\alpha _{1/2,j}^{2}c^{2}t)}{
\varepsilon \alpha _{1/2,j}^{2}c^{2}+\exp (-\alpha _{1/2,j}^{2}c^{2})}
\frac{4(100+\varepsilon ) \sqrt{2}}{\sqrt{\alpha _{1/2,j}}
J_{3/2}(\alpha _{1/2,j})}j_0(\alpha _{1/2,j}r)Y_{0,0}(\theta ,\phi)  \nonumber \\
&\quad+\Big(1-\frac{\exp (-\alpha _{25/2,1}^{2}c^{2}t)}{\varepsilon \alpha
_{25/2,1}^{2}c^{2}+\exp (-\alpha _{25/2,1}^{2}c^{2})}\Big) 
\frac{1}{c^{2}\alpha _{25/2,1}^{2}}j_{12}(\alpha _{25/2,1}r) \nonumber \\
&\quad\times (Y_{12,-12}(\theta ,\phi ) +Y_{12,12}(\theta ,\phi ) ) \nonumber  \\
&= \sum_{j=1}^{\infty }\frac{\exp (-\alpha _{1/2,j}^{2}c^{2}t)}{
\varepsilon \alpha _{1/2,j}^{2}c^{2}+\exp (-\alpha _{1/2,j}^{2}c^{2})}
\frac{2(100+\varepsilon ) \sqrt{2}}{\sqrt{\alpha _{1/2,j}}
J_{3/2}(\alpha _{1/2,j})}\big(\frac{1}{2\alpha _{1/2,j}r}\big)^{1/2} \label{nchvd} \\
&\quad\times J_{1/2}(\alpha _{1/2,j}r)  \nonumber \\
&\quad +2\Big(1-\frac{\exp (-\alpha _{25/2,1}^{2}c^{2}t)}{\varepsilon \alpha
_{25/2,1}^{2}c^{2}+\exp (-\alpha _{25/2,1}^{2}c^{2})}\Big) \frac{1}{
c^{2}\alpha _{25/2,1}^{2}}\big(\frac{\pi }{2\alpha _{5/2,1}r}\big) ^{1/2} \nonumber\\
&\quad \times J_{25/2}(\alpha _{25/2,1}r)P_{12}^{12}(\cos \theta )\cos 12\phi .
\nonumber
\end{align}

Next, we calculate the first seven coefficients of \eqref{ncxvd} and 
\eqref{nchvd} at various values of $t$. Let $\varepsilon $ be 
$\varepsilon_1=10^{-3}$, $\varepsilon _2=10^{-4}$, 
$\varepsilon _{3}=10^{-5}$, respectively and $t\in \{ 0;0.5\} $. 
The following table shows estimates for the error  between the exact 
solution \eqref{ncxvd} and the regularized solutions \eqref{nchvd}.

\begin{table}[htb]
\caption{Error between exact and regularized solutions when
$(\theta ,\phi ) =(\frac{\pi }{6},\frac{\pi }{6})$.}
\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
& \multicolumn{3}{|c|}{$\| u^{\varepsilon }(\cdot ,\frac{\pi }{6},
\frac{\pi }{6},t)-u(\cdot ,\frac{\pi }{6},\frac{\pi }{6},t)\|
_2$} \\ \hline
$t$ & $\varepsilon _1=10^{-3}$ & $\varepsilon _2=10^{-4}$ & $\varepsilon
_{3}=10^{-5}$ \\ \hline
$0$ & $1.2431\times 10^{-1}$ & $1.2475\times 10^{-2}$ & $1.2479\times
10^{-3} $ \\ \hline
$0.5$ & $6.9674\times 10^{-2}$ & $6.9906\times 10^{-3}$ & $6.9929\times
10^{-4}$ \\ \hline
\end{tabular}
\end{center}
\end{table}

 Figure \ref{fig:matcat} shows the exact 
and regularized solutions $u^{\varepsilon _i}$, $i=1,2,3$ at the time
$t=0.5$ when $r=0.5$ and $\theta =\frac{\pi }{6}$. Finally, we
plot the graphs of the exact and  regularized solutions 
$u^{\varepsilon _i}$, $i=1,2,3$ at the time $t=0.5$ corresponding to
$\theta =\frac{\pi }{6}$ in Figures 
\ref{fig:ncx+1+2}--\ref{fig:3+4chinh}.


\subsection*{Acknowledgements} 
The authors were supported by the National
Foundation for Science and Technology Development (NAFOSTED), Project
101.02-2015.23.

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\end{document}