An Efficient Third-Derivative Hybrid Block Method for the Solution of Second-Order BVPs

Abstract

A new one-step hybrid block method with two-point third derivatives is developed to solve the second-order boundary value problems (BVPs). The mathematical derivation of the proposed method is based on the interpolation and collocation methods. The theoretical properties of the proposed method, such as consistency and convergence, are well analysed. Some BVPs with different boundary conditions are solved to demonstrate the efficiency and feasibility of the suggested method. The numerical results of the proposed method are much closer to the exact solutions and more competitive than other numerical methods in the available literature.

An adaptive optimized Nyström method for second‐order IVPs

“…Data in Table 7 and Figure 6 also ascertain the viability and effectiveness of the proposed ONM. Exact and approximate solutions of the ONM for (23) utilizing h min = 10 −14 and h max = 1 are plotted in Figure 6.…”

“…where 𝛿x = (x N+1 −x 0 ) N+1 , N is the internal number of spatial nodes, and x 1 = x 0 + 𝛿x, … , x N = x 0 + N𝛿x, x N+1 = x 0 + (N + 1)𝛿x. Taking N = 19 and using (24) on the grid points x i = i 20 , i = 1, … , N, we get 𝑦 1 , 𝑦 2 , 𝑦 3 , … , 𝑦 N by solving the following system of differential equations after replacing 𝜕 2 𝑦 𝜕x 2 (x i , t) in ( 24) into (23),…”

“…Consider the time dependent semi-discretization of the problem given in 25 We solved (23) by discretizing the second-order spatial derivative, leaving the time variable continuous, and then applying the ONM, following a procedure as in the method of lines. The 𝜕 2 𝑦 𝜕x 2 in ( 23) is discretized by…”

“…In order to enhance the previously mentioned techniques, scholars like Kalogiratou et al, 15 Ramos and Rufai, [16][17][18][19][20][21] Amodio and Brugnano, 22 and Rufai 23 have derived and implemented block methods for solving the class of problems in (1) directly. The advantage of block methods over the Runge-Kutta type and predictor-corrector methods is that they can be less expensive regarding the number of functions evaluated and CPU time.…”

A New Hybrid Block Method for Solving First-Order Differential System Models in Applied Sciences and Engineering

“…Some of these approximation techniques comprise embedded Runge-Kutta type methods, general linear methods, Numerov-type methods, spline methods, finite difference, Nyström methods, space-time numerical methods, block methods, and various collocation methods. These methods are extensively discussed in various research papers and books, such as [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].…”