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 = 0.0

0.58778527

0.58778527

0.58778527

0.58778527

0.000000%

0.000000%

0.000000%

t = 0.1

0.47552826

0.47507598

0.47552826

0.47598055

0.095113%

0.000000%

0.095113%

t = 0.2

0.18163562

0.18090380

0.18163562

0.18236743

0.402904%

0.000000%

0.402904%

t = 0.3

−0.18163567

−0.18236748

−0.18163567

−0.18090385

0.402904%

0.000000%

0.402904%

t = 0.4

−0.47552829

−0.47598058

−0.47552829

−0.47507601

0.095113%

0.000000%

0.095113%

t = 0.7

−0.18163557

−0.18090385

−0.18163567

−0.18236748

0.402850%

0.000054%

0.402958%

t = 0.8

0.18163571

0.18236743

0.18163562

0.18090380

0.402850%

0.000054%

0.402957%

t = 1.0

0.58778527

0.58778527

0.58778527

0.58778527

0.000000%

0.000000%

0.000000%

h = 0.1 , k = 0.001 , r = 4 × 10 4 , n = 10 , m = 1000 , Number of Grid Points = 10,000

t = 0.0

0.58778527

0.58778527

0.58778527

0.58778527

0.000000%

0.000000%

0.000000%

t = 0.1

0.47552826

0.47640902

0.47641793

0.47642683

0.185217%

0.187090%

0.188962%

t = 0.2

0.18163562

0.18450289

0.18451732

0.18453175

1.578586%

1.586531%

1.594477%

t = 0.3

−0.18163567

−0.17731856

−0.17730406

−0.17728957

2.376796%

2.384775%

2.392754%

t = 0.4

−0.47552829

−0.47194709

−0.47193802

−0.47192896

0.753101%

0.755006%

0.756912%

t = 0.7

−0.18163557

−0.19168562

−0.19169999

−0.19171436

5.533084%

5.540995%

5.548906%

t = 0.8

0.18163571

0.17007575

0.17006120

0.17004665

6.364366%

6.372376%

6.380387%

t = 1.0

0.58778527

0.58759009

0.58758969

0.58758930

0.033206%

0.033273%

0.033340%

h = 0.01 , k = 0.005 , r = 1 , n = 100 , m = 200 , Number of Grid Points = 20,000

t = 0.0

0.58778527

0.58778527

0.58778527

0.58778527

0.000000%

0.000000%

0.000000%

t = 0.1

0.47552826

0.47552782

0.47552826

0.47552871

0.000094%

0.000000%

0.000094%

t = 0.2

0.18163562

0.18163489

0.18163562

0.18163634

0.000398%

0.000000%

0.000398%

t = 0.3

−0.18163567

−0.18163639

−0.18163567

−0.18163494

0.000398%

0.000000%

0.000398%

t = 0.4

−0.47552829

−0.47552874

−0.47552829

−0.47552785

0.000094%

0.000000%

0.000094%

t = 0.7

−0.18163557

−0.18163494

−0.18163567

−0.18163639

0.000344%

0.000054%

0.000451%

t = 0.8

0.18163571

0.18163634

0.18163562

0.18163489

0.000344%

0.000054%

0.000451%

t = 1.0

0.58778527

0.58778527

0.58778527

0.58778527

0.000000%

0.000000%

0.000000%

h = 0.01 , k = 0.001 , r = 4 × 10 2 , n = 100 , m = 1000 , Number of Grid Points = 100,000

t = 0.0

0.58778527

0.58778527

0.58778527

0.58778527

0.000000%

0.000000%

0.000000%

t = 0.1

0.47552826

0.47553674

0.47553683

0.47553692

0.001783%

0.001802%

0.001821%

t = 0.2

0.18163562

0.18166320

0.18166335

0.18166349

0.015189%

0.015268%

0.015348%

t = 0.3

−0.18163567

−0.18159421

−0.18159407

−0.18159392

0.022823%

0.022903%

0.022982%

t = 0.4

−0.47552829

−0.47549407

−0.47549398

−0.47549389

0.007196%

0.007215%

0.007234%

t = 0.7

−0.18163557

−0.18173259

−0.18173273

−0.18173288

0.053416%

0.053495%

0.053575%