General ⇒ Humble request

[kchari]

can you find out the error why it is not running in the texmaker and texstudio?
\documentclass{beamer}
\mode<presentation> {
\usetheme{AnnArbor}
\usepackage{graphicx}
\usepackage{booktabs} % Allows the use of \toprule, \midrule and \bottomrule in tables

%----------------------------------------------------------------------------------------
% TITLE PAGE
%----------------------------------------------------------------------------------------

\title[Project]{STUDY OF POISSON AND LAPLACE EQUATIONS AND THEIR SOLUTION BY USING FORTRAN CODE} % The short title appears at the bottom of every slide, the full title is only on the title page

\author{Hari Bahadur K.C} % Your name
\institute[CDP] % Your institution as it will appear on the bottom of every slide, may be shorthand to save space
{Central Department of Physics \\ % Your institution for the title page
\medskip
\text{ Ph.D. Reg.No.:100/2074/2075}

%\textit{bhandari.indra @ gmail.com} % Your email address
}
\date{\today} % Date, can be changed to a custom date
%......................................................................
\begin{document}

\begin{frame}
\titlepage % Print the title page as the first slide
\end{frame}

\begin{frame}
\frametitle{Contents} % Table of contents slide, comment this block out to remove it
\tableofcontents % Throughout your presentation, if you choose to use \section{} and \subsection{} commands, these will automatically be printed on this slide as an overview of your presentation
\end{frame}

%----------------------------------------------------------------------------------------
% PRESENTATION SLIDES
%----------------------------------------------------------------------------------------

%------------------------------------------------
\begin{frame}
\frametitle{Indroduction}
\begin{itemize}
\item The transfer of heat from the region of larger temperature to the region of lower temperature is called heat diffusion. \vspace{8pt}
\item The diffusion of heat was first described through a partial differential equation by Jean Baptise Joseph Forier (1768-1830) called heat diffusion equation.
\end{itemize}


\end{frame}
\section{Introduction} % Sections can be created in order to organize your presentation into discrete blocks, all sections and subsections are automatically printed in the table of contents as an overview of the talk
%------------------------------------------------

%\subsection{Subsection Example} % A subsection can be created just before a set of slides with a common theme to further break down your presentation into chunks

\begin{frame}
\frametitle{History}
\large{About the nature of heat:}
\begin{itemize}
\item Those of mechanistic school, including Biot believed that heat was a permeating fluid.
\item Those of dynamic school believed that heat was essentially a motion consisting of rapid molecular vibration.
\item Heat can be easily observed but can be measured in terms of temperature.
\end{itemize}
\end{frame}
\section{History}
%------------------------------------------------

\begin{frame}
\frametitle{History}
\large{Measurement of heat:}
\begin{itemize}
\item German physicist, Gabriel Daniel Fahrenheit perfected the closed tube mercury thermometer in 1714.
\item By 1724 he had established the Fahrenheit scale with melting of ice at 32 degree and the boiling of water at 212 degree.
\item Around 1760, Joseph Black noticed that when ice melts, it takes in heat with out changing temperature.
\item He propose the term \textbf{"Latent Heat"} to denote this type of heat.
\item He also noticed that equal masses of different substances needed different amounts of heat to raise their temperatures by the same amount. He invented the the term \textbf{"specific heat"} to denote this type of heat.
\end{itemize}
\end{frame}

%------------------------------------------------

\begin{frame}
\frametitle{History}
\begin{itemize}
\item In 1783,Lavoiser and Laplace invented an ice calorimeter with which they measured for the first time, the latent heat of melting of ice and the specific heat of different materials.\\
\textbf{Development of partial differential Equation:}
\item Laplace (1789) formulated the partial differential equation in potential theory.
\item The seventeenth century there was very active developments in the theory of ordinary and partial differential equations through the contribution of Daniel Bernoulli (1700-1782), Jean le Rond d'Alembert (1717-1783), Leonhard Euler (1707-1783), John- Louis Lagrange (1736-1813) and others.
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{History}
\begin{itemize}
\item Baptiste Biot (1804)explained the heat conduction in a thin bar. He observed that heat not only was conducted along the length but also lost to the exterior atmosphere transverse to the direction of conduction.
\item Biot's approach did not involve a temperature gradient, so necessary to the formulation of differential equation.
\item Finally, the representation of dynamic problems in continuum with help of trigonometric series were also known.
\end{itemize}
\end{frame}

%/////////////////////////////////////////////////////////////////////

\begin{frame}
\large{About Fourier:}
\begin{itemize}
\item Fourier was born in 1768 in Auxerre in Burgundy in central France.
\item He tought mathematics for a few years at the Ecole Polytechnique in Paris.
\item Fourier started work on heat conduction some time between 1802 and 1804.
\item Fourier initially formulated heat conduction as an n-body problem, stemming from the Laplace philosophy of action at a distance.
\item He was aware of Biot's work, having received a copy of the paper from Biot himself.
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{History}
\begin{itemize}
\item Fourier moved away from discontinuous bodies and towards continuous bodies.
\item Fourier took an empirical, observational approach to idealize how matter behave microscopically.
\item He avoided discussion about nature of heat. Rather than assuming that the behavior of temperature at a point was influenced by all point's in it's vicinity.
\item Fourier assumed that the temperature in an infinitesimal lamina or element was dependent only on the conditions at the lamina or element immediately upstream and down stream of it. He thus formulated heat diffusion equation in a continuum.
\item The temperature, quantity of heat and transport of heat as well as the relation between quantity of heat and temperature are fundamental to Fourier heat conduction model.
\end{itemize}
\end{frame}

%////////////////////////////////////////////////////////////

\begin{frame}
\begin{itemize}
\item Fourier visualized the problem in terms of three components:heat transport in space, heat storage with in a small element of the solid , and boundary conditions.
\item The differential equation itself pertained only to the interior of the flow domain. The interaction of the interior with the exterior across the boundary was handled in terms of "boundary conditions".
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{History}
\begin{itemize}
\item Fourier was clearly aware of the earlier work of Bernouli, Euler, and Lagrange relating to solutions in the form of trigonometric series.
\item He was also aware that Euler, D'Almbert, and Lagrange viewed trigonometric series with great suspicion.
\item Fourier boldly applied the method of separation of variable and generated solutions in terms of infinite trigonometric series.
\item Fourier abandoned the action at a distance approach around 1804 and made a bold departure convention, which eventually led to his masterpiece, the transient heat conduction equation.
\item Fourier differential equation mathematically describes the rate at which temperature is changing at any location in the interior of the solid as a function of time.
\item Physically the equation describes the conservation of heat energy per unit volume over an infinitesimally small volume of the solid centered at the point of interest.
\item Later he also generated solutions in the form of integrals that are come to be known as Fourier integrals.
\item In the last part of his 1807 work,Fourier also presented some results pertaining to heat conduction in a cylindrical annulus, a sphere and a cube.
\end{itemize}
\end{frame}

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\begin{frame}
\begin{itemize}
\item The concept of heat equation has inspired the mathematical formulation of many other physical processes in terms of diffusion.
\item Fourier's method began to be applied to analyze problems in many fields besides heat transfer such as in electricity, chemical diffusion,investigation of flow of blood through capillary veins,fluids in porous media,genetics and economics. It also inspired a great deal of research into the theory of differential equation.
\item In numerically integrating Fourier's equation, the common practice is to approximate the spatial and temporan gradients with finer and finer discretization of space and time.
\end{itemize}
\end{frame}

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\begin{frame}
\section{Heat Equation}
\frametitle{Heat Equation}
\begin{itemize}
\item Consider a bar of length 'L' lying along the x-axis, so that x=0 and x=L are the ends of bar.
\item We assume that the bar is of homogeneous material, is straight and has uniform cross section.
\item T(x,t) is the temperature at a position x and time t, which is constant on any given cross-section and depends on the horizontal position along the x-axis and time.
\end{itemize}
\begin{figure}[!h]
\centering
\includegraphics[width = 5cm, keepaspectratio]{fig}
\end{figure}
\end{frame}

%/////////////////////////////////////////////////////////////////

\begin{frame}
\frametitle{Heat Equation}
\begin{itemize}
\item The heat equation is given by
\begin{equation}
\frac{\partial T(x,t)}{\partial t} = \alpha \frac{\partial^2 T(x,t)}{\partial x^2}
\end{equation}
\item It describes a fundamental physical balance: the rate at which heat flows into any portion of the bar is equal to the rate at which heat is absorbed into that portion of the bar.
\item $\alpha$ = $\frac{K}{C_p \rho}$ is called thermal diffusivity.
\item The left hand side of equation (1) refereed as absorption term and the right hand side is refereed as flux term.
\end{itemize}
\end{frame}

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\begin{frame}
\frametitle{Heat Equation}
\begin{itemize}
\item Let us define a dimensionless function
\begin{equation}
u(x,t) = \frac{T \left(xL, \frac{L^2}{\alpha} t \right) - T_0}{T_0}
\end{equation}
where, x $\in$ [0,1]
\item The function u(x,t) follows Dirichlet boundary conditions i.e. u(0, t) = u(1,t) = 0.
\item Using equation (2) in equation (1)
\begin{equation}
\frac{\partial u(x,t)}{\partial t} = \frac{\partial^2 u(x,t)}{\partial x^2}
\end{equation}
\end{itemize}
\end{frame}

%/////////////////////////////////////////////////////////////////

\begin{frame}
\section{Discretization}
\frametitle{Discretization}
The solution of equation (16) will be computed in the interval x $\in$ [0,1] for t $\in$ [0,$t_f$]. The problem is defined on a two dimensional discrete lattice. the differential equation will be approximated by finite difference equations. \\
the lattice is defined by $N_x$ spatial points $x_i$ $\in$ [0,1] \\
\begin{equation}
x_i = 0 + (i-1) \Delta x, i = 1,2,....,N_x
\end{equation}
Where the $N_x$ -1 intervals have the same width
\begin{equation}
\Delta x = \frac{1-0}{N_x - 1}
\end{equation}
The $N_t$ time points $t_j$ $\in$ [0,$t_f$]
\begin{equation}
t_j = 0 + (j-1) \Delta t, j = 1, 2, ....,N_t
\end{equation}
Where the $N_t$-1 time intervals have the same duration
\begin{equation}
\Delta t = \frac{t_f - 0}{N_t - 1}
\end{equation}
The ends of the intervals are
\begin{equation*}
x_1 = 0, x_{N_x} = 1, t_1 = 0, t_{N_t} = t_f
\end{equation*}
\end{frame}

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\begin{frame}
\frametitle{Discretization for diffusion in rod}
Let, u(i,j) = u($x_i,t_j$), then the derivatives can be replaced by finite differences,
\begin{equation}
u_t = \frac{\partial u(x,t)}{\partial t} = \lim_{\Delta x\to\infty} \frac{u \left(x_i, t_j + \Delta t \right) - u \left(x_i, t_j \right)
}{\Delta t}
\end{equation}
Therefore we can approximate
\begin{equation}
\frac{\partial u(x,t)}{\partial t} \approx \frac{u \left(x_i, t_j + \Delta t \right) - u \left(x_i, t_j \right)
}{\Delta t} \equiv \frac{1}{\Delta t} (u_{i,j+1} - u_{i,j})
\end{equation}
\end{frame}

%////////////////////////////////////////////////////////////

\begin{frame}
\frametitle{Discretization for diffusion in rod}
\begin{equation}
\frac{\partial^2 u(x,t)}{\partial x^2} = \frac{1}{(\Delta x)^2} (u_{{i+1},j} - 2 u_{ij} + u_{{i-1},j})
\end{equation}
Equating equations (20) and (28)
\begin{equation}
\frac{1}{\Delta t} (u_{i,j+1} - u_{i,j}) = \frac{1}{(\Delta x)^2} (u_{{i+1},j} - 2 u_{ij} + u_{{i-1},j})
\end{equation}
\begin{equation}
u_{i,j+1} = u_{ij} + \frac{\Delta t}{(\Delta x)^2} (u_{{i+1},j} - 2 u_{ij} + u_{{i-1},j})
\end{equation}
The second term of right hand side of this equation contains only the nearest neighbors $u_{i \pm 1, j}$ of the lattice point a given time $t_j$. Therefore it can be used for all i = 2 .........$N_x$ - 1. The relation (30) is not needed for the points i = 1 and i = $N_x$, since the values $u_{1,j}$ = $u_{Nx, j}$ = 0 are kept constant.\\
the parameter $ \frac{\Delta t}{(\Delta x)^2}$ is called Courant parameter. In order to have a time evolution with out instabilities, it is necessary to have $ \frac{\Delta t}{(\Delta x)^2}$ $<$ $\frac{1}{2}$
\end{frame}

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\begin{frame}
\frametitle{Discretization for diffusion in circle}
The discretization is same as in the case of thin rod. In order to take into account the cyclic topology we take u = 1 in equation(30)
\begin{equation}
u_{1,j+1} = u_{1,j} + \frac{\Delta t}{(\Delta x)^2} (u_{2,j} - 2 u_{1,j} + u_{{N_x},j})
\end{equation}
And
\begin{equation}
u_{{n_x},j+1} = u_{i,j} + \frac{\Delta t}{(\Delta x)^2} (u_{1,j} - 2 u_{{N_x},j} + u_{{N_x - 1},j})
\end{equation}
Since the neighbor to the right of the point $x_{N_x}$ is the oint $x_1$ and the neighbor to the left of the point $x_1$ is the point $x_{N_x}$. Similarly the equation(30) can be applied to the rest of the points i = 2,.........,${N_x}$ - 1.\\
The variable probability = $\sum_{i = 1}^{N_x} u_{ij}$, should conserved and is equal to 1. The variable r2 = $\sum_{i=1}^{N_x} (x_i - x{n_x}/2))^2 u_{ij}$ is the expectation value of the distance squared from the initial position. For small enough time it follow
\begin{equation}
<r>^2 = <(x - x_0)^2 (t) = \int_{-\infty}^{\infty} (x-x_0)^2 u(x,t) \sim 2t
\end{equation}
\end{frame}

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\begin{frame}
\section{Fortran code}
\frametitle{The fortran program for heat diffusion in thin rod }
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{rodcode}
\caption{Fortran codes for the solution of heat equation in thin rod}
\end{figure}
\end{frame}

%///////////////////////////////////////////////////////////////

\begin{frame}
\frametitle{The fortran program for heat diffusion in thin circle }
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{circlecode1}
\caption{Fortran codes for the solution of heat equation in thin rod}
\end{figure}
\end{frame}

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\begin{frame}
\frametitle{The fortran program for heat diffusion in thin circle }
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{circlecode2}
\caption{Fortran codes for the solution of heat equation in thin rod}
\end{figure}
\end{frame}

%////////////////////////////////////////////////////////////

\begin{frame}
\section{Result and discussion}
\frametitle{Result and discussion}
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{rod11}
\caption{The function u(x,t) for $N_x$ = 8, $N_t$ = 200 and $t_f$ = 0.6}
\end{figure}
\end{frame}

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\begin{frame}
\frametitle{Result and discussion }
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{rod12}
\caption{The function u(x,t) for $N_x$ = 8, $N_t$ = 200 and $t_f$ = 0.6}
\end{figure}
\end{frame}

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\begin{frame}
\frametitle{Result and discussion }
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{rod13}
\caption{The function u(x,t) for $N_x$ = 8, $N_t$ = 200 and $t_f$ = 0.6}
\end{figure}
\end{frame}

%//////////////////////////////////////////////////////////////

\begin{frame}
\frametitle{Result and discussion }
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{rod21}
\caption{The plot of function u(x,t) with distance for different values of $t_j$}
\end{figure}
\end{frame}

%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

\begin{frame}
\frametitle{Result and discussion }
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{rod31}
\caption{Plot of u(x,t) for $N_x$ = 10, $N_t$ = 100, $t_f$ = 0.16 taking zlabel upto 1}
\end{figure}
\end{frame}

%//////////////////////////////////////////////////////////

\begin{frame}
\frametitle{Result and discussion }
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{rod32}
\caption{Plot of u(x,t) for $N_x$ = 10, $N_t$ = 100, $t_f$ = 0.16 taking zlabel upto 1}
\end{figure}
\end{frame}

%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

\begin{frame}
\frametitle{Result and discussion }
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{rod41}
\caption{Plot of u(x,t) for $N_x$ = 10, $N_t$ = 100, $t_f$ = 0.16 taking zlabel upto 0.2}
\end{figure}
\end{frame}

%//////////////////////////////////////////////////////////

\begin{frame}
\frametitle{Result and discussion }
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{rod42}
\caption{Plot of u(x,t) for $N_x$ = 10, $N_t$ = 100, $t_f$ = 0.16 taking zlabel upto 0.2}
\end{figure}
\end{frame}

%/////////////////////////////////////////////////////////////

\begin{frame}
\frametitle{Result and discussion }
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{rod43}
\caption{Plot of u(x,t) for $N_x$ = 10, $N_t$ = 100, $t_f$ = 0.16 taking zlabel upto 0.2}
\end{figure}
\end{frame}

%\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

\begin{frame}
\frametitle{Result and discussion }
\begin{figure}[!h]
\centering
\includegraphics[width = 7cm, keepaspectratio]{rod44}
\caption{Plot of u(x,t) for $N_x$ = 10, $N_t$ = 100, $t_f$ = 0.16 taking zlabel upto 0.2}
\end{figure}
\end{frame}

%//////////////////////////////////////////////////////////

\begin{frame}
\frametitle{Result and discussion for circle}
\begin{figure}[!h]
\centering
\includegraphics[width = 12cm, keepaspectratio]{circle11}
\caption{The function u(x,t) as a function of $t_j$ for $N_x$ =8, $N_t$ = 200 and $t_f$ = 0.6}
\end{figure}
\end{frame}

%/////////////////////////////////////////////////////////////

\begin{frame}
\frametitle{Result and discussion for circle}
\begin{figure}[!h]
\centering
\includegraphics[width = 12cm, keepaspectratio]{circle21}
\caption{the expectation value $<r^2>_j$ as a function of $t_j$}
\end{figure}
\end{frame}

%////////////////////////////////////////////////////////////

\begin{frame}
\begin{itemize}
\item After the compilation of the program we obtain a data file d.dat.
\item The data are then plotted with gnuplot as shown in figure(2),figure(3) and figure(4).
\item Figure (2),figure(3) and figure (4) are the different snapshots of the plot of u(x,t) with time in x-axis and distance in y-axis for $N_x$ = 8, $N_t$ = 200 and $t_f$ = 0.6.
\item The plots show that ,the temperature distribution have sinusoidal variation with distance and exponentially with time.
\item The plot of function u(x,t) with distance for different values of $t_j$ is shown figure(5).
\end{itemize}
\end{frame}

%///////////////////////////////////////////////////////////

\begin{frame}
\begin{itemize}
\item From figure (5), it is seen that the function u(x,t) varies sinusoidally with position,however it decreases with increase in j.
\item figure (13) is the plot of $u_{{N_x}/2,j}$, $u_{{N_x}/4,j}$ and $u_{1,j}$ as a function of $t_j$.
\item Here it is observed that for large time we obtain uniform diffusion.
\item Figure (14) is the plot of the expectation value $<r^2>_j$ as a function of $t_j$ for $N_x$ = 10, $N_t$ = 100 and $t_f$ = 0.4.
\item The solid line is the straight line 2t.
\item From this figure the asymptotic relation of $<r^2>_j$ with $t_j$ is confirmed.
\end{itemize}
\end{frame}

%.................................................................

\begin{frame}
\frametitle{References}
\section{References}
\begin{thebibliography} {}
\bibitem{1} Narasimhan,T. N.(1999).Fourier's Heat Conduction Equation: History, Influence and Connections.\emph{Reviews of Geophysics},
(37),151-172
\bibitem{2} Anagnostopoulos,K.N. (2014).\emph{A Practical Introduction to Computational Physics and Scientific Computing}. Athens,Greece:Konstantinos N. Anognostopoulos
\bibitem{3} Chapman,S.J. (2018).\emph{Fortran for Scientist and Engineers}. New York,USA:Mc Graw Hill Education
\bibitem{4}$ Wikipedia contributors. (2019, February 22). Heat equation. In Wikipedia, The Free Encyclopedia. Retrieved 04:25, March 22, 2019, from https://en.wikipedia.org/w/index.php?ti ... =884580138 $
\end{thebibliography}
\end{frame}

%............................................................

\begin{frame}
\frametitle{Acknowledgment}
\section{Acknowledgment}
\begin{itemize}
\item Prof. Dr. Narayan Adhikari \\
Project Supervisor
\item Assoc. Prof. Dr. Ishwar Koirala \\
Ph.D Supervisor
\item Prof. Dr.Binil Aryal \\
Head of Department (CDP)
\item CDP Family
\item Institute of Engineering, T.U.
\item Everest Engineering College
\item Friend and Family
\end{itemize}
\end{frame}

%.....................................................................

\begin{frame}
\Huge{\centerline{Thank You}}
\end{frame}

%-------------------------------------------------------------------

\end{document}

[Bartman]

The closing brace of the \mode command is missing:

Code: Select all

\mode<presentation>{\usetheme{AnnArbor}}

Some off topic comments:

graphicx is already loaded by the beamer class.

\date{\today} is not needed to get the current date.

The package csquotes offers the command \enquote to put quotes.

Font size commands like \large or \Huge have no argument. Instead use for example

Code: Select all

{\large enlarged text\par}

Use cleveref to avoid manual cross-references to equations, images or source code.

I don't know the reason, why you insert source code in the fortran language as an image, but one of the packages listings or minted can be a useful replacement. In this case, don't forget the fragile option of the frame environment.

I don't understand, why the content of the entry about wikipedia contributors in the bibliography is set in mathematical mode.