A semi-Lagrangian approach for numerical simulation of coupled Burgers’ equations

Numerical solution for (2 + 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method

“…In Tables 1 and 2, we conduct numerical simulation for different Brownian motion with distinct values of x and t for the system considered in Example 4.1. Tables 3 and 4 present the numerical comparative study between VIM, CFRDTM, and FNDM in terms of absolute error for Equation 19. The numerical study has been conducted for coupled system considered in Example 4.2 as presented in Tables 5 and 6.…”

“…D n are the Adomian polynomials and they signifying the nonlinear terms. Using these terms, we can rewrite Equation 19 as…”

“…Þand v x; y; t ð Þ¼∑ ∞ n¼0 v n x; y; t ð Þ: TABLE 8 Comparative study between HPM, 32 CFRDTM, 33 and FNDM for the approximate solution v x; y; t ð Þof Example 4.2 at y = 0.3, α = 1, and β = 1 for n = 4 Thus, by comparing two sides of Equation 27, we can effortlessly get the recursive relation as Example 4.1 is presented in Figure 1A, and nature of the exact solution and absolute error for the corresponding equation is presented in Figure 1B,C, respectively. In the similar manner, the FNDM solution v x; y; t ð Þ, exact solution and efficiency of FNDM in terms of absolute error for Equation 19 are cited in Figure 2. Figures 3 and 4 are the response of obtained solutions for Example 4.1 with distinct Brownian motion and standard motion ( Figure 5).…”

“…The mathematical groundwork for derivatives of fractional order was laid through combined efforts of pioneers such are Liouville, Riemann, Caputo, Miller and Ross, Podlubny, and others. The theory of fractional‐order calculus has been related to practical projects, and it has been applied to chaos theory, signal processing, electrodynamics, fluid dynamics, finance, and other areas …”

“…The theory of fractional-order calculus has been related to practical projects, and it has been applied to chaos theory, 8 signal processing, 9 electrodynamics, 10 fluid dynamics, 11 finance, 12 and other areas. [13][14][15][16][17][18][19][20][21][22][23] The Burgers equations are the fundamental nonlinear partial differential equations that exist in distinct connected branches of science and technology. These equations can be seen as a simplified model of fluid dynamics, and it is used as a computational tool in order to deal with more complex problems.…”