Abstract
Recently, direct methods that involve higher derivatives to numerically approximate higher order initial value problems (IVPs) have been explored, which aim to construct numerical methods with higher order and very high precision of the solutions. This article aims to construct a fourth and fifth derivative, three-point implicit block method to tackle general third-order ordinary differential equations directly. As a consequence of the increase in order acquired via the implicit block method of higher derivatives, a significant improvement in efficiency has been observed. The new method is derived in a block mode to simultaneously evaluate the approximations at three points. The derivation of the new method can be easily implemented. We established the proposed method’s characteristics, including order, zero-stability, and convergence. Numerical experiments are used to confirm the superiority of the method. Applications to problems in physics and engineering are given to assess the significance of the method.
Highlights
A wide variety of real life situations are represented by mathematical models as third order ordinary differential equations (ODEs), such as chemical engineering, biology, electromagnetic waves, quantum mechanics, the motion of rocket, and thin film flow [1,2,3,4]
In the classical way, solving higher order ODEs is done by reducing the equation into an equivalent system of first-order ODEs, but this process is too rigorous compared to the direct methods [9,10,11]
The following abbreviations will be used in the tables: ITPBO9: HCD: Adams Bashforth–Adams Moulton (ABAM): FSM: ILMM: ISHD: h: NS: AE: MAXE: Implicit three-point block direct method introduced in this paper of order nine
Summary
A wide variety of real life situations are represented by mathematical models as third order ordinary differential equations (ODEs), such as chemical engineering, biology, electromagnetic waves, quantum mechanics, the motion of rocket, and thin film flow [1,2,3,4]. The theoretical solutions for most of these equations are undefined; third-order ODEs have gained significant attention and the need to develop numerical methods with more accurate approximations is eminent [5,6,7,8]. But it is found that the implementation process of the direct methods is simpler and more accurate than the process of reduction [11]. In order to avoid the reduction effort, many researchers have proposed different methods to solve initial value problems (IVPs) of the ODEs directly [12,13,14,15,16]. To enhance the efficacy of numerical methods, many researchers developed block methods by producing the r-point of the approximate solutions simultaneously
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