Direct One-Step Method for Solving Third-Order Boundary Value Problems

Abstract

A direct explicit Runge-Kutta type (RKT) method via shooting technique to approximate analytical solutions to the third-order two-point boundary value problems (BVPs) with boundary condition type I and II are proposed. In this paper first, a three-stage fourth-order direct explicit Runge-Kutta type method denoted as RKT3s4 is constructed. A new algorithm of shooting technique for solving two-point BVPs for third-order ordinary differential equations (ODEs) is presented.

Error Analysis Using Three and Four Stage Eighth Order Embedded Runge-Kutta Method for Sixth Order Ordinary Differential Equation vvi(u)=f(u,v,v',v'',v''',viv)

“…the local truncation errors [24][25][26] of v p (u) where p is a derivative, i.e., 0 p v = … are calculated after substitution of the exact solution of (1) into (11)(12)(13)(14)(15)(16).…”

“…For instance, third-order ODEs are used in thin film flow problems [7][8][9][10], and fifth-order differential equations are used in fiber preservation transformations [11]. Beccar et al [12] and Abdulsalam et al [13] contributed towards analyzing various orders of ODEs. Malhotra et al [14] optimized the real-life problem, and Kaur et al [15] used an improvised concept for solving the differential equations.…”

Efficient Decomposition Shooting Method for Solving Third-Order Boundary Value Problems

“…From Table 6, we can see that EDSM is the most efficient method for solving example 3, compare with the results of the all methods in [24] [25] [27] and SRKM4.…”

“…From Table 8, we can see that EDSM is the most efficient method for solving example 4, comparison with the results of the all methods in [24] [27] [28] and SRKM4 when 10 m = . Also, the EDSM was more accurate when compared with the same methods at 20 m ≥ , where we obtained the absolute error value of zero for all values of  .…”

“…Example 3. Consider the third-order linear singularly perturbed BVP given by [24] Table 3. The absolute error for EDSM, SRKM4 and the methods described in [23] when 10 m = .…”