KM2016 / Parabolic / test_parab

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\begin{document}
\normalsize
\section*{Тестирование программы dif6}
Размерность пространства n=2.
Максимально допустимое число шагов по времени noftau=4
Вес разностной схемы sig=1.
Порядок квадратурной формулы Гаусса при интегрировании по времени ntime=3.
Порядок квадратурной формулы Гаусса при интегрировании по пространству nspace=3.
Начальный шаг по времени tau=0.01.
Поточечная оценка точности аппроксимации
в узловых точках по пространству дается выражениямим
\[\delta_{ij}=\left|1-\frac{\tilde u_{ij}}{u_{ij}}\right|,\]
где $u_{ij}=u(x_i,y_j,t)$, $\tilde u_{ij}$ --
результаты численной аппроксимации решения исходной задачи в точке $(x_i,y_j,t)$.

Результаты представлены в таблице.


\begin{table}
\caption{Момент времени $t=0.01$}
\begin{center}
\scriptsize
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{12}{|c|}{$t=0.01,\quad x_i=1$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&2&2.728&3.744&5.096&6.832&9&11.65&14.82&18.58&22.95&28\\
\hline
$\tilde u_{ij}$&2.036&2.745&3.759&5.112&6.85&9.02&11.67&14.85&18.6&22.96&27.92\\
\hline
$\delta_{ij}(\%)$&1.8&0.61&0.4&0.32&0.27&0.23&0.19&0.16&0.13&0.047&0.28\\
\hline
\multicolumn{12}{|c|}{$t=0.01,\quad x_i=1.2$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&2.728&3.456&4.472&5.824&7.56&9.728&12.38&15.55&19.3&23.68&28.73\\
\hline
$\tilde u_{ij}$&2.738&3.462&4.478&5.831&7.568&9.737&12.39&15.56&19.32&23.68&28.68\\
\hline
$\delta_{ij}(\%)$&0.37&0.17&0.13&0.12&0.1&0.093&0.083&0.073&0.058&0.017&0.16\\
\hline
\multicolumn{12}{|c|}{$t=0.01,\quad x_i=1.4$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&3.744&4.472&5.488&6.84&8.576&10.74&13.39&16.57&20.32&24.7&29.74\\
\hline
$\tilde u_{ij}$&3.753&4.477&5.493&6.846&8.583&10.75&13.4&16.58&20.33&24.7&29.71\\
\hline
$\delta_{ij}(\%)$&0.23&0.11&0.091&0.084&0.078&0.071&0.065&0.058&0.048&0.014&0.13\\
\hline
\multicolumn{12}{|c|}{$t=0.01,\quad x_i=1.6$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&5.096&5.824&6.84&8.192&9.928&12.1&14.74&17.92&21.67&26.05&31.1\\
\hline
$\tilde u_{ij}$&5.105&5.829&6.845&8.198&9.935&12.1&14.75&17.93&21.68&26.05&31.06\\
\hline
$\delta_{ij}(\%)$&0.18&0.09&0.077&0.073&0.07&0.066&0.061&0.056&0.046&0.015&0.12\\
\hline
\multicolumn{12}{|c|}{$t=0.01,\quad x_i=1.8$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&6.832&7.56&8.576&9.928&11.66&13.83&16.48&19.66&23.41&27.78&32.83\\
\hline
$\tilde u_{ij}$&6.842&7.566&8.582&9.934&11.67&13.84&16.49&19.67&23.42&27.79&32.79\\
\hline
$\delta_{ij}(\%)$&0.15&0.076&0.067&0.065&0.064&0.061&0.058&0.054&0.045&0.015&0.12\\
\hline
\multicolumn{12}{|c|}{$t=0.01,\quad x_i=2$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&9&9.728&10.74&12.1&13.83&16&18.65&21.82&25.58&29.95&35\\
\hline
$\tilde u_{ij}$&9.011&9.734&10.75&12.1&13.84&16.01&18.66&21.84&25.59&29.96&34.96\\
\hline
$\delta_{ij}(\%)$&0.12&0.065&0.058&0.058&0.058&0.057&0.055&0.052&0.044&0.016&0.11\\
\hline
\multicolumn{12}{|c|}{$t=0.01,\quad x_i=2.2$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&11.65&12.38&13.39&14.74&16.48&18.65&21.3&24.47&28.22&32.6&37.65\\
\hline
$\tilde u_{ij}$&11.66&12.38&13.4&14.75&16.49&18.66&21.31&24.48&28.24&32.61&37.61\\
\hline
$\delta_{ij}(\%)$&0.1&0.056&0.051&0.052&0.052&0.052&0.051&0.049&0.042&0.016&0.11\\
\hline
\multicolumn{12}{|c|}{$t=0.01,\quad x_i=2.4$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&14.82&15.55&16.57&17.92&19.66&21.82&24.47&27.65&31.4&35.78&40.82\\
\hline
$\tilde u_{ij}$&14.84&15.56&16.58&17.93&19.67&21.83&24.48&27.66&31.41&35.78&40.78\\
\hline
$\delta_{ij}(\%)$&0.087&0.048&0.044&0.045&0.046&0.047&0.047&0.045&0.04&0.015&0.11\\
\hline
\multicolumn{12}{|c|}{$t=0.01,\quad x_i=2.6$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&18.58&19.3&20.32&21.67&23.41&25.58&28.22&31.4&35.15&39.53&44.58\\
\hline
$\tilde u_{ij}$&18.59&19.31&20.33&21.68&23.42&25.59&28.24&31.41&35.16&39.53&44.53\\
\hline
$\delta_{ij}(\%)$&0.071&0.037&0.034&0.035&0.037&0.039&0.039&0.039&0.034&0.011&0.1\\
\hline
\multicolumn{12}{|c|}{$t=0.01,\quad x_i=2.8$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&22.95&23.68&24.7&26.05&27.78&29.95&32.6&35.78&39.53&43.9&48.95\\
\hline
$\tilde u_{ij}$&22.96&23.68&24.7&26.05&27.79&29.96&32.6&35.78&39.53&43.9&48.89\\
\hline
$\delta_{ij}(\%)$&0.043&0.0086&0.0055&0.0076&0.01&0.013&0.015&0.016&0.013&0.009&0.13\\
\hline
\multicolumn{12}{|c|}{$t=0.01,\quad x_i=3$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&28&28.73&29.74&31.1&32.83&35&37.65&40.82&44.58&48.95&54\\
\hline
$\tilde u_{ij}$&27.87&28.68&29.71&31.06&32.8&34.96&37.61&40.79&44.53&48.87&53.72\\
\hline
$\delta_{ij}(\%)$&0.48&0.18&0.13&0.12&0.11&0.1&0.095&0.089&0.095&0.16&0.53\\
\hline
\end{tabular}
\end{center}
\label{0}
\end{table}


\begin{table}
\caption{Момент времени $t=0.02$}
\begin{center}
\scriptsize
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{12}{|c|}{$t=0.02,\quad x_i=1$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&2&2.728&3.744&5.096&6.832&9&11.65&14.82&18.58&22.95&28\\
\hline
$\tilde u_{ij}$&2.057&2.758&3.771&5.125&6.864&9.036&11.69&14.87&18.62&22.96&27.89\\
\hline
$\delta_{ij}(\%)$&2.8&1.1&0.71&0.56&0.47&0.4&0.34&0.28&0.21&0.055&0.41\\
\hline
\multicolumn{12}{|c|}{$t=0.02,\quad x_i=1.2$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&2.728&3.456&4.472&5.824&7.56&9.728&12.38&15.55&19.3&23.68&28.73\\
\hline
$\tilde u_{ij}$&2.748&3.469&4.485&5.838&7.576&9.747&12.4&15.57&19.32&23.68&28.65\\
\hline
$\delta_{ij}(\%)$&0.74&0.38&0.29&0.25&0.22&0.19&0.17&0.14&0.11&0.016&0.25\\
\hline
\multicolumn{12}{|c|}{$t=0.02,\quad x_i=1.4$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&3.744&4.472&5.488&6.84&8.576&10.74&13.39&16.57&20.32&24.7&29.74\\
\hline
$\tilde u_{ij}$&3.76&4.482&5.498&6.852&8.589&10.76&13.41&16.59&20.34&24.7&29.68\\
\hline
$\delta_{ij}(\%)$&0.43&0.23&0.19&0.17&0.16&0.14&0.13&0.11&0.086&0.013&0.21\\
\hline
\multicolumn{12}{|c|}{$t=0.02,\quad x_i=1.6$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&5.096&5.824&6.84&8.192&9.928&12.1&14.74&17.92&21.67&26.05&31.1\\
\hline
$\tilde u_{ij}$&5.113&5.835&6.85&8.204&9.942&12.11&14.76&17.94&21.69&26.05&31.03\\
\hline
$\delta_{ij}(\%)$&0.33&0.18&0.15&0.14&0.14&0.13&0.12&0.11&0.082&0.014&0.2\\
\hline
\multicolumn{12}{|c|}{$t=0.02,\quad x_i=1.8$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&6.832&7.56&8.576&9.928&11.66&13.83&16.48&19.66&23.41&27.78&32.83\\
\hline
$\tilde u_{ij}$&6.85&7.572&8.587&9.941&11.68&13.85&16.5&19.68&23.43&27.79&32.77\\
\hline
$\delta_{ij}(\%)$&0.26&0.15&0.13&0.13&0.12&0.12&0.11&0.1&0.08&0.015&0.19\\
\hline
\multicolumn{12}{|c|}{$t=0.02,\quad x_i=2$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&9&9.728&10.74&12.1&13.83&16&18.65&21.82&25.58&29.95&35\\
\hline
$\tilde u_{ij}$&9.02&9.741&10.76&12.11&13.85&16.02&18.67&21.85&25.6&29.96&34.94\\
\hline
$\delta_{ij}(\%)$&0.22&0.13&0.11&0.11&0.11&0.11&0.11&0.097&0.078&0.016&0.18\\
\hline
\multicolumn{12}{|c|}{$t=0.02,\quad x_i=2.2$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&11.65&12.38&13.39&14.74&16.48&18.65&21.3&24.47&28.22&32.6&37.65\\
\hline
$\tilde u_{ij}$&11.67&12.39&13.41&14.76&16.5&18.67&21.32&24.49&28.24&32.61&37.58\\
\hline
$\delta_{ij}(\%)$&0.18&0.11&0.099&0.099&0.1&0.1&0.098&0.092&0.074&0.016&0.18\\
\hline
\multicolumn{12}{|c|}{$t=0.02,\quad x_i=2.4$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&14.82&15.55&16.57&17.92&19.66&21.82&24.47&27.65&31.4&35.78&40.82\\
\hline
$\tilde u_{ij}$&14.85&15.57&16.58&17.94&19.67&21.84&24.49&27.67&31.42&35.78&40.75\\
\hline
$\delta_{ij}(\%)$&0.15&0.093&0.083&0.085&0.087&0.088&0.088&0.084&0.069&0.014&0.17\\
\hline
\multicolumn{12}{|c|}{$t=0.02,\quad x_i=2.6$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&18.58&19.3&20.32&21.67&23.41&25.58&28.22&31.4&35.15&39.53&44.58\\
\hline
$\tilde u_{ij}$&18.6&19.32&20.33&21.69&23.42&25.59&28.24&31.42&35.17&39.53&44.5\\
\hline
$\delta_{ij}(\%)$&0.11&0.065&0.058&0.061&0.065&0.068&0.069&0.067&0.054&0.0027&0.18\\
\hline
\multicolumn{12}{|c|}{$t=0.02,\quad x_i=2.8$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&22.95&23.68&24.7&26.05&27.78&29.95&32.6&35.78&39.53&43.9&48.95\\
\hline
$\tilde u_{ij}$&22.96&23.68&24.69&26.05&27.79&29.96&32.61&35.78&39.53&43.88&48.83\\
\hline
$\delta_{ij}(\%)$&0.022&0.0067&0.0051&0.00064&0.0068&0.013&0.017&0.018&0.0076&0.048&0.25\\
\hline
\multicolumn{12}{|c|}{$t=0.02,\quad x_i=3$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&28&28.73&29.74&31.1&32.83&35&37.65&40.82&44.58&48.95&54\\
\hline
$\tilde u_{ij}$&27.8&28.64&29.68&31.04&32.77&34.94&37.59&40.76&44.5&48.81&53.57\\
\hline
$\delta_{ij}(\%)$&0.71&0.31&0.22&0.19&0.18&0.16&0.15&0.15&0.16&0.29&0.79\\
\hline
\end{tabular}
\end{center}
\label{1}
\end{table}


\begin{table}
\caption{Момент времени $t=0.03$}
\begin{center}
\scriptsize
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{12}{|c|}{$t=0.03,\quad x_i=1$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&2&2.728&3.744&5.096&6.832&9&11.65&14.82&18.58&22.95&28\\
\hline
$\tilde u_{ij}$&2.07&2.769&3.78&5.135&6.875&9.048&11.7&14.88&18.63&22.96&27.87\\
\hline
$\delta_{ij}(\%)$&3.5&1.5&0.97&0.76&0.63&0.53&0.45&0.37&0.27&0.05&0.48\\
\hline
\multicolumn{12}{|c|}{$t=0.03,\quad x_i=1.2$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&2.728&3.456&4.472&5.824&7.56&9.728&12.38&15.55&19.3&23.68&28.73\\
\hline
$\tilde u_{ij}$&2.757&3.477&4.492&5.846&7.585&9.756&12.41&15.58&19.33&23.68&28.64\\
\hline
$\delta_{ij}(\%)$&1.1&0.6&0.44&0.37&0.33&0.29&0.25&0.21&0.15&0.0074&0.32\\
\hline
\multicolumn{12}{|c|}{$t=0.03,\quad x_i=1.4$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&3.744&4.472&5.488&6.84&8.576&10.74&13.39&16.57&20.32&24.7&29.74\\
\hline
$\tilde u_{ij}$&3.767&4.488&5.504&6.858&8.596&10.77&13.42&16.6&20.34&24.7&29.67\\
\hline
$\delta_{ij}(\%)$&0.61&0.36&0.29&0.26&0.23&0.21&0.19&0.16&0.12&0.0049&0.26\\
\hline
\multicolumn{12}{|c|}{$t=0.03,\quad x_i=1.6$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&5.096&5.824&6.84&8.192&9.928&12.1&14.74&17.92&21.67&26.05&31.1\\
\hline
$\tilde u_{ij}$&5.119&5.84&6.856&8.209&9.948&12.12&14.77&17.95&21.7&26.05&31.02\\
\hline
$\delta_{ij}(\%)$&0.45&0.28&0.23&0.21&0.2&0.19&0.17&0.15&0.11&0.0065&0.25\\
\hline
\multicolumn{12}{|c|}{$t=0.03,\quad x_i=1.8$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&6.832&7.56&8.576&9.928&11.66&13.83&16.48&19.66&23.41&27.78&32.83\\
\hline
$\tilde u_{ij}$&6.857&7.577&8.593&9.947&11.69&13.86&16.51&19.68&23.43&27.79&32.75\\
\hline
$\delta_{ij}(\%)$&0.36&0.23&0.2&0.19&0.18&0.17&0.16&0.14&0.11&0.0086&0.24\\
\hline
\multicolumn{12}{|c|}{$t=0.03,\quad x_i=2$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&9&9.728&10.74&12.1&13.83&16&18.65&21.82&25.58&29.95&35\\
\hline
$\tilde u_{ij}$&9.027&9.747&10.76&12.12&13.85&16.03&18.68&21.85&25.6&29.96&34.92\\
\hline
$\delta_{ij}(\%)$&0.3&0.19&0.17&0.16&0.16&0.16&0.15&0.14&0.1&0.01&0.23\\
\hline
\multicolumn{12}{|c|}{$t=0.03,\quad x_i=2.2$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&11.65&12.38&13.39&14.74&16.48&18.65&21.3&24.47&28.22&32.6&37.65\\
\hline
$\tilde u_{ij}$&11.68&12.4&13.41&14.77&16.5&18.67&21.33&24.5&28.25&32.6&37.57\\
\hline
$\delta_{ij}(\%)$&0.25&0.16&0.14&0.14&0.14&0.14&0.14&0.13&0.098&0.011&0.22\\
\hline
\multicolumn{12}{|c|}{$t=0.03,\quad x_i=2.4$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&14.82&15.55&16.57&17.92&19.66&21.82&24.47&27.65&31.4&35.78&40.82\\
\hline
$\tilde u_{ij}$&14.85&15.57&16.59&17.94&19.68&21.85&24.5&27.68&31.43&35.78&40.74\\
\hline
$\delta_{ij}(\%)$&0.2&0.13&0.12&0.12&0.12&0.12&0.12&0.11&0.088&0.0058&0.22\\
\hline
\multicolumn{12}{|c|}{$t=0.03,\quad x_i=2.6$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&18.58&19.3&20.32&21.67&23.41&25.58&28.22&31.4&35.15&39.53&44.58\\
\hline
$\tilde u_{ij}$&18.6&19.32&20.34&21.69&23.43&25.6&28.25&31.43&35.17&39.52&44.47\\
\hline
$\delta_{ij}(\%)$&0.13&0.082&0.075&0.079&0.085&0.089&0.091&0.086&0.063&0.017&0.23\\
\hline
\multicolumn{12}{|c|}{$t=0.03,\quad x_i=2.8$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&22.95&23.68&24.7&26.05&27.78&29.95&32.6&35.78&39.53&43.9&48.95\\
\hline
$\tilde u_{ij}$&22.95&23.67&24.69&26.04&27.78&29.95&32.6&35.78&39.52&43.86&48.79\\
\hline
$\delta_{ij}(\%)$&0.013&0.033&0.023&0.013&0.0027&0.0062&0.013&0.013&0.0097&0.097&0.34\\
\hline
\multicolumn{12}{|c|}{$t=0.03,\quad x_i=3$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&28&28.73&29.74&31.1&32.83&35&37.65&40.82&44.58&48.95&54\\
\hline
$\tilde u_{ij}$&27.77&28.61&29.66&31.02&32.76&34.93&37.58&40.75&44.48&48.77&53.49\\
\hline
$\delta_{ij}(\%)$&0.83&0.4&0.28&0.24&0.22&0.2&0.19&0.18&0.22&0.38&0.94\\
\hline
\end{tabular}
\end{center}
\label{2}
\end{table}


\begin{table}
\caption{Момент времени $t=0.04$}
\begin{center}
\scriptsize
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{12}{|c|}{$t=0.04,\quad x_i=1$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&2&2.728&3.744&5.096&6.832&9&11.65&14.82&18.58&22.95&28\\
\hline
$\tilde u_{ij}$&2.08&2.778&3.789&5.143&6.884&9.058&11.71&14.89&18.63&22.96&27.85\\
\hline
$\delta_{ij}(\%)$&4&1.8&1.2&0.93&0.77&0.65&0.55&0.45&0.31&0.044&0.52\\
\hline
\multicolumn{12}{|c|}{$t=0.04,\quad x_i=1.2$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&2.728&3.456&4.472&5.824&7.56&9.728&12.38&15.55&19.3&23.68&28.73\\
\hline
$\tilde u_{ij}$&2.765&3.484&4.499&5.853&7.593&9.765&12.42&15.59&19.34&23.68&28.63\\
\hline
$\delta_{ij}(\%)$&1.3&0.81&0.6&0.5&0.43&0.38&0.33&0.27&0.18&0.0024&0.36\\
\hline
\multicolumn{12}{|c|}{$t=0.04,\quad x_i=1.4$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&3.744&4.472&5.488&6.84&8.576&10.74&13.39&16.57&20.32&24.7&29.74\\
\hline
$\tilde u_{ij}$&3.773&4.494&5.509&6.863&8.603&10.77&13.43&16.6&20.35&24.69&29.66\\
\hline
$\delta_{ij}(\%)$&0.77&0.49&0.39&0.34&0.31&0.28&0.25&0.21&0.14&0.0047&0.3\\
\hline
\multicolumn{12}{|c|}{$t=0.04,\quad x_i=1.6$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&5.096&5.824&6.84&8.192&9.928&12.1&14.74&17.92&21.67&26.05&31.1\\
\hline
$\tilde u_{ij}$&5.125&5.846&6.861&8.215&9.954&12.13&14.78&17.95&21.7&26.05&31.01\\
\hline
$\delta_{ij}(\%)$&0.56&0.37&0.3&0.28&0.26&0.24&0.22&0.19&0.13&0.0024&0.28\\
\hline
\multicolumn{12}{|c|}{$t=0.04,\quad x_i=1.8$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&6.832&7.56&8.576&9.928&11.66&13.83&16.48&19.66&23.41&27.78&32.83\\
\hline
$\tilde u_{ij}$&6.862&7.583&8.598&9.952&11.69&13.86&16.51&19.69&23.44&27.78&32.74\\
\hline
$\delta_{ij}(\%)$&0.44&0.3&0.26&0.24&0.23&0.22&0.21&0.18&0.13&0.00037&0.27\\
\hline
\multicolumn{12}{|c|}{$t=0.04,\quad x_i=2$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&9&9.728&10.74&12.1&13.83&16&18.65&21.82&25.58&29.95&35\\
\hline
$\tilde u_{ij}$&9.033&9.753&10.77&12.12&13.86&16.03&18.68&21.86&25.61&29.95&34.91\\
\hline
$\delta_{ij}(\%)$&0.36&0.25&0.22&0.21&0.21&0.2&0.19&0.17&0.12&0.0024&0.26\\
\hline
\multicolumn{12}{|c|}{$t=0.04,\quad x_i=2.2$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&11.65&12.38&13.39&14.74&16.48&18.65&21.3&24.47&28.22&32.6&37.65\\
\hline
$\tilde u_{ij}$&11.68&12.4&13.42&14.77&16.51&18.68&21.33&24.51&28.26&32.6&37.55\\
\hline
$\delta_{ij}(\%)$&0.3&0.21&0.19&0.18&0.18&0.18&0.18&0.16&0.11&0.0019&0.25\\
\hline
\multicolumn{12}{|c|}{$t=0.04,\quad x_i=2.4$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&14.82&15.55&16.57&17.92&19.66&21.82&24.47&27.65&31.4&35.78&40.82\\
\hline
$\tilde u_{ij}$&14.86&15.58&16.59&17.95&19.69&21.86&24.51&27.69&31.43&35.77&40.72\\
\hline
$\delta_{ij}(\%)$&0.23&0.16&0.15&0.15&0.15&0.15&0.15&0.14&0.099&0.0069&0.25\\
\hline
\multicolumn{12}{|c|}{$t=0.04,\quad x_i=2.6$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&18.58&19.3&20.32&21.67&23.41&25.58&28.22&31.4&35.15&39.53&44.58\\
\hline
$\tilde u_{ij}$&18.6&19.32&20.34&21.69&23.43&25.6&28.25&31.43&35.17&39.51&44.45\\
\hline
$\delta_{ij}(\%)$&0.14&0.091&0.084&0.09&0.098&0.1&0.11&0.098&0.062&0.041&0.28\\
\hline
\multicolumn{12}{|c|}{$t=0.04,\quad x_i=2.8$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&22.95&23.68&24.7&26.05&27.78&29.95&32.6&35.78&39.53&43.9&48.95\\
\hline
$\tilde u_{ij}$&22.94&23.67&24.69&26.04&27.78&29.95&32.6&35.78&39.52&43.84&48.75\\
\hline
$\delta_{ij}(\%)$&0.046&0.061&0.044&0.028&0.015&0.0026&0.0053&0.002&0.033&0.15&0.41\\
\hline
\multicolumn{12}{|c|}{$t=0.04,\quad x_i=3$}\\
\hline
$y_j$&1&1.2&1.4&1.6&1.8&2&2.2&2.4&2.6&2.8&3\\
\hline
$u_{ij}$&28&28.73&29.74&31.1&32.83&35&37.65&40.82&44.58&48.95&54\\
\hline
$\tilde u_{ij}$&27.75&28.6&29.65&31.01&32.75&34.92&37.57&40.74&44.46&48.73&53.44\\
\hline
$\delta_{ij}(\%)$&0.9&0.46&0.32&0.28&0.25&0.23&0.21&0.21&0.26&0.45&1\\
\hline
\end{tabular}
\end{center}
\label{3}
\end{table}
\end{document}