A variable step-size fourth-derivative hybrid block strategy for integrating third-order IVPs, with applications

Abstract

In this paper, an efficient fourth-derivative two-step hybrid block strategy (FDTHBS) to get the approximate solution of a third-order IVP with applications to problems in Fluid Dynamics and Engineering is constructed. The mathematical derivation of the proposed FDTHBS is based on the interpolation and collocation of the exact solution and its derivatives at the selected equidistant grid and off-grid points. The theoretical characteristics of the proposed method are analysed. An embedding-like procedure is considered and executed in variable step-size mode to get better performance of the newly developed strategy. Some test problems, including the wellknown Blasius equation and three different types of non-linear thin-film flow problems, are integrated numerically to ascertain the superior impact of our developed error estimation and control strategy. It is worth concluding that the proposed technique is not only efficient in term of CPU time, but also minimizes errors and support the analytical results.

Modified Fourth Derivative Block Method and its direct applications to third-order initial value problems

“…As previously stated, the zero-stability of a numerical technique refers to how the numerical method behaves when h → 0. h > 0 is a common occurrence in practice (see Refs. [22,25,27]). We need a definition of stability other than zero-stability to assess whether a numerical technique will generate acceptable results for a given value of h > 0.…”

“…This problem has recently appeared [27] and [28]. The solution of numerical test problem 1 is considered within [0, 1] for h = 0.1.…”

“…The plot of the absolute errors with MFDFBM on the logarithm scale are compared with those of FDTHBS in Ref. [27] and ITBDM in Ref. [28] are shown in Figure 2.…”

“…which represents the well-known fluid wet surface drainage problem according to [27] and ν a > 0 denotes the small film thickness. The solution obtained for ν a = 0.1 in the interval [0, 5] over ten iteration.…”

An adaptive optimized Nyström method for second‐order IVPs

“…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.…”