$c=2$

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%

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

$t=0.100$

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%

$h=0.01$ , $k=0.001$ , $r=4\times {10}^{-2}$ , $n=100$ , $m=1000$ , 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%