$c=2$

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%

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

0.58184235

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%

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

0.000001%

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