$c=2$ 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 0.56035906 0.56193917 0.042536% 0.240074% 0.522732% $t=1.000$ 0.58778527 0.58200532 0.58122589 0.58044618 0.983342% 1.115947% 1.248600% $h=0.1$ , $k=0.001$ , $r=4×{10}^{-4}$ , $n=10$ , $m=1000$ , 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% $t=0.100$ 0.47552826 0.47641108 0.47642000 0.47642892 0.185649% 0.187525% 0.189402% $t=1.000$ 0.58778527 0.58758916 0.58758877 0.58758838 0.033363% 0.033430% 0.033497% $h=0.01$ , $k=0.005$ , $r=1$ , $n=100$ , $m=200$ , 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 0.58749521 0.58749537 0.000004% 0.000004% 0.000025% $t=1.000$ 0.58778527 0.58777278 0.58777285 0.58777317 0.002124% 0.002112% 0.002057% $h=0.01$ , $k=0.001$ , $r=4×{10}^{-2}$ , $n=100$ , $m=1000$ , 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% 0.000000% 0.000000% $t=1.000$ 0.58778527 0.58778509 0.58778508 0.58778510 0.000030% 0.000031% 0.000029% $c=1$ $x=0.3$ $h=0.1$ , $k=0.1$ , $r=1$ , $n=10$ , $m=10$ , 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 0.77126826 0.77344309 0.042536% 0.240098% 0.522756% $t=1.000$ −0.80901701 −0.80729380 −0.80675574 −0.80621735 0.213001% 0.279508% 0.346056% $h=0.1$ , $k=0.001$ , $r={10}^{-4}$ , $n=10$ , $m=1000$ , 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% $t=1.000$ −0.80901701 −0.80894966 −0.80894953 −0.80894939 0.008325% 0.008342% 0.008358% $h=0.01$ , $k=0.01$ , $r=1$ , $n=100$ , $m=100$ , 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 0.80861804 0.000004% 0.000000% 0.000028% $t=1.000$ −0.80901701 −0.80894306 −0.80894315 −0.80894348 0.009141% 0.009129% 0.009089% $h=0.01$ , $k=0.001$ , $r={10}^{-2}$ , $n=100$ , $m=1000$ , Number of Grid Points = 100,000 $t=0.000$ 0.80901701 0.80901701 0.80901701 0.80901701 0.000000% 0.000000% 0.000000% $t=0.001$ 0.80901302 0.80901302 0.80901302 0.80901302 0.000000% 0.000000% 0.000000% $t=1.000$ −0.80901701 −0.80901289 −0.80901289 −0.80901289 0.000509% 0.000509% 0.000509%