$c=2$ 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%