$c=2$ Analytical Solution Approximate Solution Error FD CD BD FD CD BD $x=0.2$ $h=0.1$ , $k=0.05$ , $r=1$ , $n=10$ , $m=20$ , Number of Grid Points = 200 $t=0.000$ 0.58778527 0.58778527 0.58778527 0.58778527 0.000000% 0.000000% 0.000000% $t=0.050$ 0.55901701 0.55877922 0.55970419 0.56058575 0.042536% 0.122926% 0.280625% $t=1.000$ 0.58778527 0.36051918 0.36033826 0.36016582 38.664816% 38.695596% 38.724933% $h=0.1$ , $k=0.001$ , $r=4×{10}^{-4}$ , $n=10$ , $m=1000$ , Number of Grid Points = 10,000 $t=0.000$ 0.58778527 0.58778527 0.58778527 0.58778527 0.000000% 0.000000% 0.000000% $t=0.001$ 0.58777366 0.58777366 0.58777376 0.58777385 0.000000% 0.000016% 0.000032% $t=1.000$ 0.58778527 0.58184235 0.58184197 0.58184158 1.011069% 1.011135% 1.011201% $h=0.01$ , $k=0.005$ , $r=1$ , $n=100$ , $m=200$ , Number of Grid Points = 20,000 $t=0.000$ 0.58778527 0.58778527 0.58778527 0.58778527 0.000000% 0.000000% 0.000000% $t=0.005$ 0.58749523 0.58749521 0.58749522 0.58749523 0.000004% 0.000001% 0.000000% $t=1.000$ 0.58778527 0.55946067 0.55946068 0.55946069 4.818867% 4.818866% 4.818864% $h=0.01$ , $k=0.001$ , $r=4×{10}^{-2}$ , $n=100$ , $m=1000$ , Number of Grid Points = 100,000 $t=0.000$ 0.58778527 0.58778527 0.58778527 0.58778527 0.000000% 0.000000% 0.000000% $t=0.001$ 0.58777366 0.58777366 0.58777366 0.58777366 0.000000% 0.000001% 0.000000% $t=1.000$ 0.58778527 0.58201426 0.58201426 0.58201426 0.981822% 0.981823% 0.981822% $c=1$ $x=0.3$ $h=0.1$ , $k=0.1$ , $r=1$ , $n=10$ , $m=10$ , Number of Grid Points = 100 $t=0.000$ 0.80901701 0.80901701 0.80901701 0.80901701 0.000000% 0.000000% 0.000000% $t=0.100$ 0.76942090 0.76909362 0.77036673 0.77158027 0.042536% 0.122928% 0.280649% $t=1.000$ −0.80901701 −0.63430476 −0.63414648 −0.63399557 21.595621% 21.615186% 21.633840% $h=0.1$ , $k=0.001$ , $r={10}^{-4}$ , $n=10$ , $m=1000$ , Number of Grid Points = 10,000 $t=0.000$ 0.80901701 0.80901701 0.80901701 0.80901701 0.000000% 0.000000% 0.000000% $t=0.001$ 0.80901302 0.80901302 0.80901305 0.80901308 0.000000% 0.000004% 0.000008% $t=0.000$ −0.80901701 −0.80696433 −0.80696420 −0.80696406 0.253725% 0.253741% 0.253758% $h=0.01$ , $k=0.01$ , $r=1$ , $n=100$ , $m=100$ , Number of Grid Points = 10,000 $t=0.000$ 0.80901701 0.80901701 0.80901701 0.80901701 0.000000% 0.000000% 0.000000% $t=0.010$ 0.80861781 0.80861778 0.80861781 0.80861784 0.000004% 0.000000% 0.000004% $t=1.000$ −0.80901701 −0.78922993 −0.78922993 −0.78922994 2.445818% 2.445817% 2.445817% $h=0.01$ , $k=0.001$ , $r={10}^{-2}$ , $n=100$ , $m=1000$ , Number of Grid Points = 100,000