Metaheuristic algorithms for approximate solution to ordinary differential equations of longitudinal fins having various profiles

Highlights

• Approximate solutions to ordinary differential equations (ODEs) in engineering.

• Fourier series with the aid of metaheuristics used as suggested approximate method.

• The GA, PSO and HS are utilized for optimization purposes.

• Obtained approximate solutions by the proposed method confirm the results by others.

• Proposed approach offers acceptable accuracy for a wide range of ODEs.

Abstract

Differential equations play a noticeable role in engineering, physics, economics, and other disciplines. Approximate approaches have been utilized when obtaining analytical (exact) solutions requires substantial computational effort and often is not an attainable task. Hence, the importance of approximation methods, particularly, metaheuristic algorithms are understood. In this paper, a novel approach is suggested for solving engineering ordinary differential equations (ODEs). With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic methods, ODEs can be represented as an optimization problem. The target is to minimize the weighted residual function (error function) of the ODEs. The boundary and initial values of ODEs are considered as constraints for the optimization model. Generational distance and inverted generational distance metrics are used for evaluation and assessment of the approximate solutions versus the exact (numerical) solutions. Longitudinal fins having rectangular, trapezoidal, and concave parabolic profiles are considered as studied ODEs. The optimization task is carried out using three different optimizers, including the genetic algorithm, the particle swarm optimization, and the harmony search. The approximate solutions obtained are compared with the differential transformation method (DTM) and exact (numerical) solutions. The optimization results obtained show that the suggested approach can be successfully applied for approximate solving of engineering ODEs. Providing acceptable accuracy of the proposed technique is considered as its important advantage against other approximate methods and may be an alternative approach for approximate solving of ODEs.

Introduction

Mathematical formulation of most physical and engineering problems involves differential equations (DEs). The DEs have applications in all areas of science and engineering. Hence, it is important for engineers and scientists to know how to deal with the DEs and solve them. With regard to real life problems, which are highly nonlinear, many problems in engineering and science often include one or more ordinary differential equations (ODEs).

Analytical approaches are often inefficient in tackling ODEs. Therefore, approximate analytical methods are applied to obtain estimated solutions of ODEs. A number of analytical methods have been utilized to develop approximate analytical solutions for engineering problems, such as the variational iteration method (VIM) [1], [2], the homotopy analysis method (HAM) [3], the method of bilaterally bounded (MBB) [4], and the Adomian double decomposition method (ADM) [5], [6].

Differential transformation method (DTM), which is based on Taylor series expansion, as widely used approximate analytical method, was first introduced by Zhou [7] for solving linear and nonlinear initial value problems (IVPs) in electrical circuits. The DTM has been widely used to obtain accurate analytical solutions for nonlinear engineering problems [8], [9], [10], [11].

Recently, many studies have combined the concept of the DTM with finite difference approximation for increasing the capability of their approximate solution [12], [13], [14]. Further, approximate analytical procedures such as the VIM, homotopy perturbation method (HPM), and DTM have been combined with Padé approximation technique to overcome the disadvantages faced by these methods in certain cases [15].

In particular, many studies have applied approximation methods for solving various types of integro-differential equations (linear/nonlinear) [16], [17], [18], [19], [20]. However, each of these numerical approximation techniques has its own operational limitations that strictly narrow its functioning domain. Hence, it is possible that these approximate techniques fail to overcome a specific problem.

A few such instances are mentioned in the following. It was reported that the DTM was unable to produce physically reasonable data for the Glauert-jet Problem [11]. Moreover, for some specific parameter values, the HPM and VIM failed to provide accurate results for the motion of a solid particle in a fluid [21], [22]. Meanwhile, most of them are based on classical mathematical tools.

Despite there being a wide range of approximate methods for solving ODEs, there is a lack of a proper approach that meets most of the engineering demands having unconventional and nonlinear ODEs. It should be very interesting to solve linear and nonlinear ODEs having arbitrary boundaries and/or initial values.

Therefore, when analytical methods are not capable of solving differential equations or other types of equations in a logical given time, approximation methods are considered as the best solver. Among approximation methods, metaheuristic optimization algorithms, devised by observing the phenomena occurring in nature, have demonstrated their capabilities in finding near-optimal solutions to numerical real-valued problems [23], [24], [25], [26].

Nowadays, applications that use metaheuristic methods for finding approximate solution of ODEs have increased considerably. This includes genetic algorithms (GAs) [27], [28], particle swarm optimization (PSO) [4], genetic programming [29], and others [30], [31]; however, their approaches are different with each other in terms of applied strategy and base approximate function. For instance, in [4], different strategy named method of bilaterally bounded, has been employed along with the PSO.

Recently, the concept of Fourier series expansion has been used as a base approximate function for finding the approximate solution of ODEs [32]. For the sake of simplicity, in their approach, the weight function was set to unit weight function [32]. However, this assumption may not help us in obtaining better results for all types of ODEs.

Inspired by [32] and in order to improve the efficiency of the proposed approach, lately, Sadollah et al. [33] successfully applied weighted residual method for approximate solving of 10 ODE problems including mechanical vibration problems and more test ODE problems. Also, superiority of using new weight function (instead of using unit weight function) is shown and two metaheuristic algorithms (i.e., the PSO [34] and water cycle algorithm [35]) are utilized for optimization phase.

In this paper, using the concept of Fourier series as the base approximate function, other optimizers, including the harmony search (HS) [36], PSO [34], and GA [37] are applied for approximate solving of real life ODEs longitudinal heat transfer fins having various profiles (i.e., rectangular, trapezoidal, and concave parabolic profiles) and properties (i.e., 14 ODEs). In addition, least square weight (LSW) function is proposed for solving the ODEs instead of considering unit weight function.

The GA [37], known to be the most famous metaheuristic algorithms, has been applied for optimization phase. The reason of choosing PSO is that to apply the same optimizer to the considered engineering heat transfer problems as did others in the literature [4], [32], [33]. In case of HS algorithm, in light of its simple concept and coding, we selected this optimizer for having more comparisons.

Talking about the considered ODE problems, extended or finned surfaces are extensively utilized in engineering applications where there is a need to increase heat transfer between a hot primary surface and an adjoining coolant. A comprehensive literature review on different facets of extended surface heat transfer theory and experiment over the past several decades has been carried out by Kraus et al. [38].

Recently, Torabi et al. [39] investigated the approximate solving of convective-radiative longitudinal fins for different profiles and nonlinearities using the DTM. They compared their obtained approximate results with exact and numerical solutions. The approximated mathematical formulations for different profiles are given in their study [39].

Consequently, the nonlinear fin equations have been solved either numerically or using a variety of approximate analytical methods. In this paper, we modeled convective-radiative longitudinal fin problems as an optimization problem and determined their approximate solution using the proposed method.

The remainder of this paper is organized as follows: The next section describes problem formulation as a case study and its behavior in form of ODEs accompanied with its detail. In Section 3, suggested approximate approach for tackling considered problems (i.e., ODEs of longitudinal fins) and the applied optimization methods are explained in brief. In addition, the performance criteria for quantitative assessment among other methods are given in Section 3. Section 4 compares statistical optimization results using the suggested approach with the other approximation method. Graphical comparisons between the exact (numerical) and approximate solutions are demonstrated and the obtained approximate mathematical formulations are represented in this section. Finally, conclusions are drawn in Section 5.