A new difference method for the numerical solution of fourth-order parabolic equations

B-splines methods with redefined basis functions for solving fourth order parabolic partial differential equations

“…Problem 2. We consider a fourth order non-homogeneous parabolic partial differential equation [19], @ 2 u @t 2 þ @ 4 u @x 4 ¼ ½24 À x 2 ð1 À xÞ 2 cos t; 0 6 x 6 1; t > 0: with initial conditions uðx; 0Þ ¼ x 2 ð1 À xÞ 2 ; u t ðx; 0Þ ¼ 0; ;0 6 x 6 1: and the boundary conditions at x = 0 and x = 1 of the form Table 3 Absolute errors in displacement u(x, t) (Problem 1).…”

“…Now, using (19) and substitute a = 120hq 2 , b == À 120(1 À h)q 2 in (44) and simplifying, it reduce in the form…”

Numerical Methods for Schrödinger-Type Problems

“…Jain et aI., [85] also considered this system and presented a multi-parameter family of ADI FDMs similar to those of [63]. For the case of d space variables, Saul'yev [145,146] formulated and analyzed FDMs based on the semi-explicit approach of Lees [97]. In [145], the direct approach is considered and a family of (semi-)explicit methods, containing the standard fully explicit method as a special case, is shown to be conditionally stable but, unlike the standard semi-explicit approach of Lees, unconditionally consistent.…”

“…For the case of d space variables, Saul'yev [145,146] formulated and analyzed FDMs based on the semi-explicit approach of Lees [97]. In [145], the direct approach is considered and a family of (semi-)explicit methods, containing the standard fully explicit method as a special case, is shown to be conditionally stable but, unlike the standard semi-explicit approach of Lees, unconditionally consistent. A similar approach is adopted in [146] for the problem reformulated as a system.…”