A Parallel Algorithm for the Two-Dimensional Time Fractional Diffusion Equation with Implicit Difference Method

<table>The data dependence of 2D-TFDE of grid point <svg height="18.362499" id="M67" style="vertical-align:-5.76984pt" version="1.1" viewbox="0 0 22.549999 18.362499" width="22.549999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.113)"><path d="M619 482q0 -69 -40.5 -119t-96 -72.5t-118.5 -26.5h-44l-70 20l-31 -151q-14 -67 0.5 -83t89.5 -22l-5 -28h-287l8 28q65 7 81.5 22t29.5 83l79 398q12 56 0.5 70.5t-78.5 20.5l7 28h255q108 0 164 -43t56 -125zM524 478q0 141 -146 141q-25 0 -47 -8q-16 -6 -20.5 -13.5
t-10.5 -39.5l-43 -241q37 -13 83 -13q67 0 125.5 45t58.5 129z" id="x1D443"></path></g>
<g transform="matrix(.012,-0,0,-.012,10.875,15.188)"><path d="M244 607q0 -25 -15.5 -43t-37.5 -18q-19 0 -32 13t-13 35q0 21 15 41t39 20q20 0 32 -14t12 -34zM222 91q-29 -33 -79 -68t-75 -35q-13 0 -19 7.5t-6 31t10 65.5l62 253q5 26 -1 26q-21 0 -72 -43l-13 24q43 40 91 68t71 28q30 0 10 -78l-71 -274q-8 -30 3 -30
q16 0 76 48z" id="x1D456"></path></g><g transform="matrix(.012,-0,0,-.012,14.123,15.188)"><path d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" id="x2C"></path></g><g transform="matrix(.012,-0,0,-.012,16.836,15.188)"><path d="M394 607q0 -25 -15.5 -43t-39.5 -18q-19 0 -32 13t-13 35t16 41.5t40 19.5q20 0 32 -13.5t12 -34.5zM360 349l-57 -283q-38 -188 -122 -263q-72 -64 -114 -64q-16 0 -30 10.5t-14 22.5q0 24 21 45q15 13 26 6q30 -20 54 -20q41 0 65 95q37 147 86 415q6 34 6.5 42.5
t-5.5 11.5q-18 0 -79 -54l-14 25q48 48 96.5 79t71.5 31q29 0 9 -99z" id="x1D457"></path></g>
</svg> of time step <svg height="14.4125" id="M68" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 25.2125 14.4125" width="25.2125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.125)"><path d="M324 430l-26 -36l-112 -4l-55 -265q-13 -66 7 -66q13 0 44.5 20t50.5 40l17 -24q-38 -40 -85.5 -73.5t-87.5 -33.5q-50 0 -21 138l55 262h-80l-2 8l25 34h66l25 99l78 63l10 -9l-37 -153h128z" id="x1D461"></path></g>
<g transform="matrix(.012,-0,0,-.012,5.862,14.2)"><path d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2
q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" id="x1D45B"></path></g><g transform="matrix(.012,-0,0,-.012,11.836,14.2)"><path d="M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z" id="x2B"></path></g><g transform="matrix(.012,-0,0,-.012,18.82,14.2)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
</svg>.</table>

The Scientific World Journal

fig1

Figure 1

Figure 1: A Parallel Algorithm for the Two-Dimensional Time Fractional Diffusion Equation with Implicit Difference Method 