$c=2$

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

0.56062916

0.042536%

0.122926%

0.288391%

$t=1.000$

0.58778527

0.58631380

0.58608352

0.58585324

0.250341%

0.289518%

0.328696%

$h=0.1$ , $k=0.001$ , $r=4\times {10}^{-4}$ , $n=10$ , $m=1000$ , 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

0.58758962

0.58758923

0.58758884

0.033285%

0.033352%

0.033418%

$h=0.01$ , $k=0.005$ , $r=1$ , $n=100$ , $m=200$ , 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

0.58749532

0.000004%

0.000001%

0.000015%

$t=1.000$

0.58778527

0.58778260

0.58778269

0.58778287

0.000453%

0.000439%

0.000407%

$h=0.01$ , $k=0.001$ , $r=4\times {10}^{-2}$ , $n=100$ , $m=1000$ , Number of Grid Points = 100,000