Direct integration of the third-order two point and multipoint Robin type boundary value problems

Abstract

This numerical study exclusively focused on the direct two point diagonally multistep block method of order four (2DDM4) in the form of Adams-type formulas. The proposed predictor–corrector scheme was applied in this study to compute two equally spaced numerical solutions for the third-order two point and multipoint boundary value problems (BVPs) subject to Robin boundary conditions concurrently at each step. The optimization of the computational costs was taken into consideration by not resolving the equation into a set of first-order differential equations. Instead, its implementation involved the use of shooting technique, which included the Newton divided difference formula employed for the iterative part, for the estimation of the initial guess. Apart from studying the local truncation error, the study also included the method analysis, including the order, stability, and convergence. The results of eight numerical problems demonstrated and highlighted competitive computational cost attained by the scheme, as compared to the existing method.

Introduction

A wide range of applications in science and engineering leads to the mathematical formulation of higher order differential equations mainly applied to model a physical phenomenon. In fluid flow studies, as discussed in [21] and [17], these differential equations have commonly occurred. Concerning that, the third-order differential equation was emphasized in the current study given by

$$y^{‴}\left(x\right)=f(x,y,y^{\prime },y^{″}),a_1\le x\le a_3$$

subject to the set of three Robin type boundary conditions

$$c_{1,1}{y}^{″}\left(a_1\right)+c_{1,2}{y}^{\prime }\left(a_1\right)+c_{1,3}y\left(a_1\right)=B_1,$$$$c_{2,1}{y}^{″}\left(a_2\right)+c_{2,2}{y}^{\prime }\left(a_2\right)+c_{2,3}y\left(a_2\right)=B_2,$$$$c_{3,1}{y}^{″}\left(a_3\right)+c_{3,2}{y}^{\prime }\left(a_3\right)+c_{3,3}y\left(a_3\right)=B_3,$$

where $a_1\le a_2\le a_3$, $a_1<a_3$; the coefficients of $c_{1,i}$, $c_{2,i}$, $c_{3,i}$, $a_i$ and $B_i$ for $i=1,2,3$ are all real constants. The theory on existence and uniqueness for a solution on (1)–(4) is referred based on the discussion in [16] and [23] in order to verify the Lipschitz conditions and to assure the function $f$ is continuous in $[a_1,a_3]\times R^3$.

For variables $\mathbf{u,v}=(y,y^{\prime },y^{″})$, the function, $f$ satisfy the hypothesis on Lipschitz conditions which guarantees that there exists a positive constant $M_k$, $k=0,\dots ,2$ such that

$$|f(x,\mathbf{u})-f(x,\mathbf{v})|\le \sum _{k=0}^2{M}_k|{\mathbf{u}}^{\left(\mathbf{k}\right)}-{\mathbf{v}}^{\left(\mathbf{k}\right)}|$$

for any $\mathbf{u,v}\in R^3$. It is worthwhile to mention here that the proposed method presented in this study will give a special attention on solving problems correspond to scalar form of (1)–(4).

Accordingly, a well-known conventional method to solve Eq. (1) involves resolving the equations into an equivalent first-order system. Although this conventional method is typically recommended given its capability of computing good numerical solutions, its process involves additional human and machine efforts. Nonetheless, there are various methods of solving (1) with non-Robin boundary conditions, which were applied in numerous past studies. For examples quartic nonpolynomial spline [10], homotopy analysis method [19], differential transformation method [4] and a combination of fixed iterative method and Green’s functions [2].

Meanwhile, recent studies focused on computing the approximate solution for Robin-type, mainly on the second-order problems. For instance, Anakira et al. [5] used the strategy of partitioning the domain into several subdomains using multistage optimal homotopy analysis method (MOHAM) to compute the particular solution. On the other hand, several other studies examined the multistep block method with a predictor–corrector scheme [24] and a new Falkner-type block method with the third derivative [28] to facilitate the direct approximate solution of second-order BVPs with Robin boundary conditions. To date, the most recent study attempted to solve (1) with two point and multipoint Robin boundary conditions using the Adomian decomposition approach [27]. Meanwhile, [13] and [14] constructed a continuous linear multistep method (LMM) for solving third-order boundary value problems associated with mixed boundary conditions including Robin type.

Alternatively, several studies successfully proved that the direct method improves many computational element costs as well as the aspects of accuracy, efficiency and timing. Focusing on solving (1) with initial value problems (IVPs), the advantages of the direct method were further supported by several recent studies by Mehrkanoon [22] (using the three-step block method) and Majid et al. [20] (using the two-point block method), in the form of Adams-type formula. Besides that, the two-point fully implicit block method of order four in predictor–corrector scheme was also found to be applicable in solving (1), subject to the Dirichlet and Neumann boundary conditions [1]. Omar et al. [25] also employed the predictor–corrector block method with Taylor series to directly handle the third-order differential equations. Furthermore, Awoyemi [7] developed a continuous multistep collocation method using power series to solve the third-order IVPs. Apart from that, Pandey [26] presented a finite difference method to solve the third-order BVPs. Meanwhile, several other studies [12] and [15] applied the hybrid block methods to directly solve special type of third-order differential equations.

Hence, the significant findings of previous studies propelled this study to provide direct solutions for (1) with Robin boundary conditions through the incorporation of two-point diagonally multistep block method via shooting technique. In this study, the Newton divided difference interpolation approach was employed to re-adjust the initial guesses of the shooting technique.