A matrix formulated algorithm for solving parabolic equations with nonlocal boundary conditions

Transcendental Bernstein series for solving reaction–diffusion equations with nonlocal boundary conditions through the optimization technique

“…Following the steps of algorithm described in Table 1 According to Equation (4.1) and the above obtained values, the maximum absolute value is 1.82 × 10 −4 . This example also has been solved by using a matrix formulated algorithm proposed in [15]. In Table 2, we compared the results reported in [15] with the obtained absolute error values by using TBS for various points of (x, t) ∈ [0, 1] × [0, 1].…”

“…This example also has been solved by using a matrix formulated algorithm proposed in [15]. In Table 2, we compared the results reported in [15] with the obtained absolute error values by using TBS for various points of (x, t) ∈ [0, 1] × [0, 1]. Note that the states n 1 = n 2 = 3 and n 1 = n 2 = 5 in this method are corresponded with n = 3 and 5 in [15].…”

“…In Table 2, we compared the results reported in [15] with the obtained absolute error values by using TBS for various points of (x, t) ∈ [0, 1] × [0, 1]. Note that the states n 1 = n 2 = 3 and n 1 = n 2 = 5 in this method are corresponded with n = 3 and 5 in [15]. Regarding to the results in Table 2, it can be concluded that the TBS expansion provides the higher accuracy in comparison with the method presented in [15].…”

“…Therefore, their analytical and numerical solutions have been an interesting area of research for many years. Recently, several numerical methods such as Crank-Nicolson Hermite cubic orthogonal spline collocation method [1], polynomial-based mean weighted residuals methods [2], finite difference schemes [3][4][5][6][7], Laplace transform method [8], finite element method [9], wavelets method [10], Ritz-Galerkin method [11], spectral meshless radial point interpolation method [12], -method [13], implicit Euler method [14] and operational approach of the Tau method [15] have been proposed to solve systems including nonlocal boundary conditions. It is well known that, Bernstein polynomials (BPs) play an important role as basis functions for various numerical techniques for solving different mathematical systems.…”

“…In this study, we apply TBS for solving the following reaction-diffusion equation where p(x) > 0, q(x), f (x, t, u), k i (x), (i = 0, 1) and g i (x), (i = 0, 1, 2) are known continuous functions in their domains, and 0 and 1 are nonnegative constants [15]. We proposed an optimization method based on the TBS and operational matrices of derivatives together with the help of Lagrange multipliers method to approximate the solution of the reaction-diffusion equation in Equation (1.1).…”