Solutions of fractional order differential equations modeling temperature distribution in convective straight fins design

Abstract

In this paper, the problem of temperature distribution for convective straight fins with constant and temperature-dependent thermal conductivity is solved by using artificial neural networks trained by the biogeography-based heterogeneous cuckoo search (BHCS) algorithm. We have solved the integer and noninteger order energy balance equation in order to analyse the temperature distribution in convective straight fins. We have compared our results with homotopy perturbation method (HPM), variational iteration method (VIM), and homotopy perturbation Sumudu transform method (HPSTM). The results show that the ANN–BHCS algorithm gives better results than other analytical techniques. We have further checked the efficiency of the ANN–BHCS algorithm by using the performance metrics MAD, TIC, and ENSE. We have calculated the values of MAD, TIC, and ENSE for case 1 of the problem, and histograms of these metrics show the efficiency of our algorithm.

1 Introduction

A vast number of problems which model physical phenomena, for example heat transfer, involve the nonlinear function [1]. In mechanical engineering, heat transfer is a very common science because in various objects it can be required. The problems of improvement of heat transfer are solved on an extended surface which is known as fin. The heat transfer mechanism of fin is to conduct heat through its thermal conduction from the source of heat to the fin’s surface, and then heat is dissipated into the air through the effects of thermal radiation and convection. In addition to traditional uses, like heat exchangers, compressors, and engines with internal combustion, fins also demonstrate efficiency in the systems of heat rejection and cooling of electronic parts of space vehicles [2, 3]. A broad analysis on this subject was presented in [4] by Kern and Kraus. By using the homotopy perturbation technique, Domairry and Fazeli investigated the efficacy of the convective fins in [5]. The heat transfer dynamics of fins in space radiator and one-dimensional radiation fins are further investigated in [6–8]. The system of heat-rejecting comprising parallel tubes connected by web plates has been studied by Bartas and Sellers [9]. In [10], Hug and Aziz used a perturbation-based technique to find the closed form solutions for straight convective fin with thermal conductivity which is dependent on temperature. To find analytical solution for dimensionless temperature and investigate the efficiency of the fin having thermal conductivity dependent on temperature, Arslanturk [11, 12] used Adomian decomposition technique.

In recent years, studying the heat transfer on extended surfaces has become very pivotal with growing importance of performance of heat transfer fins having lower volumes, weights, initial and operating costs of the systems [11]. In addition to the developments in standard methods of numerical computation, a new methodology, which is known as homotopy perturbation Sumudu transformation technique, was proposed to analyze less or strongly nonlinear systems. HPSTM has recently been used for solution of nonlinear fractional equation of gas dynamics and some other physical phenomena [13]. Sumudu transform method, homotopy perturbation method (HPM), and He’s polynomials are combined to design the HPSTM technique [14]. Fractional calculus [15, 16] is a branch of applied mathematics which deals with arbitrary order differentiation and integration. It has found many applications in different areas of science and engineering over the last three decades [17–20]. The HPSTM is used for the solution of energy balance equation of fractional order [21]. More works on analytical and numerical techniques for the solving integer and fractional differential equations are available in [22–51].

In this work, we have designed a hybrid of ANNs and biogeography-based heterogeneous cuckoo search algorithm (BHCS) for the solution of integer and fractional order energy balance equation in order to analyze the temperature distribution in convective straight fins. We have named our algorithm the ANN–BHCS algorithm. We have analyzed seven cases of the problems and the results are compared with other techniques such as HPM, VIM, and HPSTM [6, 21, 52]. The results show that the ANN–BHCS algorithm is better than other techniques in terms of obtaining the solutions with high accuracy. We have further tested the efficiency of the ANN–BHCS algorithm by utilizing the performance metrics MAD, TIC, and ENSE.

Key contributions of our work are given below:

• We have designed a hybrid technique of ANNs and BHCS which is named ANN–BHCS algorithm, see Fig. 1. The ANN model is designed for the solution of integer and fractional order differential equations, see Fig. 3.

  Figure 1

  

  Graphical abstract for paper

• The problem of temperature distribution in convective straight fin is analyzed, see Fig. 2. Three integer order and four fractional order cases of the problem are considered.

• To validate the results obtained by the ANN–BHCS algorithm, we have compared it with analytical techniques such as HPM, VIM, and HPSTM.

• To check the quality of the solutions and efficiency of the algorithm, we have calculated three performance metrics which are mean absolute deviation (MAD), Theil’s inequality coefficient (TIC), and error in Nash–Sutcliffe efficiency ENSE.

2 Straight convective fins with temperature-dependent thermal conductivity

Consider a straight convective fin having thermal conductivity dependent on temperature, and it has an arbitrary constant area of cross section $A_{c}$, length b, and perimeter P, and the heat transfer coefficient is denoted by h. The fin is associated with the temperature T at the base surface, and extends into temperature $T_{a}$ of the fluid. The geometry of the straight fin is given in Fig. 2. The one-dimensional energy balance equation is given by [52, 53]

Figure 2

Geometry of the straight fin

$$ A_{c}\frac{d}{dx} \biggl[k(T) \frac{dT}{dx} \biggr]-Ph(T-T_{a})=0. $$

(1)

The thermal conductivity of fin’s material is considered as a linear function according to Eq. (2),

$$ k(T)=k_{a}\bigl[1+\lambda (T-T_{a}) \bigr], $$

(2)

where k is the parameter defining the variation of the thermal conductivity and $k_{a}$ is the thermal conductivity at the ambient fluid temperature of the fin.

Introducing the dimensionless parameters:

$$ \theta =\frac{T-T_{a}}{T_{b}-T_{a}}, \qquad \xi =\frac{x}{b}, \qquad \beta =\lambda (T_{b}-T_{a}) \quad \text{and}\quad \psi = \biggl( \frac{hPb^{2}}{k_{a}A_{c}} \biggr)^{1/2}. $$

(3)

Now Eq. (1) reduces to the following equation:

$$ \frac{d^{2}\theta }{d\xi ^{2}}+\beta \theta \frac{d^{2}\theta }{d\xi ^{2}}+\beta \biggl(\frac{d\theta }{d\xi } \biggr)^{2}- \psi ^{2}\theta =0;\quad 0\leq \xi \leq 1, $$

(4)

with the following boundary conditions:

$$ \theta {'}(0)=0\quad \text{and} \quad \theta (1)=1. $$

(5)

The computational domain $0\leq x\leq b$ is transformed to $0\leq \xi \leq 1$ by introducing the dimensionless parameters given in Eq. (3).

To understand the anomalous behavior of this system, we fractionalize the energy balance Eq. (4) into fractional order ($v>0$) as follows in order to find fin temperature in straight fins:

$$ \frac{d^{v}\theta }{d\xi ^{v}}+\beta \theta \frac{d^{2}\theta }{d\xi ^{2}}+\beta \biggl(\frac{d\theta }{d\xi } \biggr)^{2}- \psi ^{2}\theta =0;\quad 1< v\leq 2 \text{ and } 0\leq \xi \leq 1, $$

(6)

with the following boundary conditions as in Eq. (5).

3 Basic definitions

This section consists of some definitions and important relations from fractional calculus that have been used in the construction of ANN for FDEs. Fractional derivatives and integrals have been expressed in different ways in literature, i.e., Riemann–Liouville, Caputo, Erdélyi–Kober, Hadamard, Grünwald–Letnikov, and Riesz type etc. In standard fractional calculus, equivalence of these definitions for some functions has been given [15, 54, 55]. All of these definitions have their own importance and advantages in different kinds of problems in mathematics. Definitions of Riemann–Liouville and Caputo fractional derivatives are given below.

Definition 1

(The fractional order Riemann–Liouville integral and derivative)

The integral of fractional order $v>0$ can be written as [16]

$$\begin{aligned}& \bigl(I^{v} f \bigr) (x)=\frac{1}{\Gamma (\alpha )} \int _{0}^{x}(x- \tau )^{v-1} f(\tau ) \,d\tau , \end{aligned}$$

(7)

$$\begin{aligned}& \bigl(I^{0}f\bigr) (x)=f(x). \end{aligned}$$

(8)

Here, $I^{v}$ shows the fractional integral of order v. The fractional derivative of order $v>0$ is normally given as

$$ \bigl(D^{v} f \bigr) (x)= \biggl( \frac{d}{d x} \biggr)^{n} \bigl(I^{n-v} f \bigr) (x)\quad (n-1< v \leq n). $$

(9)

Here, $D^{v}$ represents the fractional derivative of order v and n is an integer.

Definition 2

(Caputo fractional derivatives)

There are some limitations of the definition of fractional derivatives given by Riemann–Liouville, when it is used for modeling of some real world phenomena related to differential equations of fractional order. Therefore, a modified definition for fractional differential operator $D^{v}$ is introduced by Caputo [16, 56]:

$$ \bigl(D^{v} f \bigr) (x)=I^{n-v} \frac{d^{n}}{d x^{n}} f(x)= \frac{1}{\Gamma (n-v)} \int _{0}^{x}(x-\tau )^{n-v-1} f^{(n)}(\tau ) \,d\tau\quad (n-1< v \leq n), $$

(10)

where $I^{v}$ is given in Eq. (3). Caputo integral operator is given by

$$ \bigl(I^{v} D^{v} f \bigr) (x)=f(x)- \sum_{k=0}^{n-1} f^{k}(0) \frac{x^{k}}{k !}\quad (n-1< v \leq n). $$

(11)

The ordinary derivative followed by a fractional integral gives the Caputo fractional derivative, while the calculation in reverse order gives the Riemann–Liouville derivative. Using Caputo fractional derivative, we can use the traditional homogeneous and nonhomogeneous initial/boundary conditions occurring in general applications. However, for homogeneous initial conditions, Riemann–Liouville and Caputo formulations coincide [16, 57].

Definition 3

(Mittag-Leffler function (MLF))

The Mittag-Leffler function (MLF) is one of the most important functions having widespread applications in fractional calculus. It plays an important role in the solution of differential equations of integer and fractional orders because of its exponential nature.

The classical MLF has the definition as given below [58]:

$$ E_{\alpha }(x)=\sum_{k=0}^{\infty } \frac{x^{k}}{\Gamma (\alpha k+1)} \quad (\alpha >0). $$

(12)

It becomes the exponential function when $\alpha =1$. The MLF function with two parameters α and β is as follows:

$$ E_{\alpha , \beta }(t)=\sum_{k=0}^{\infty } \frac{t^{k}}{\Gamma (\alpha k+\beta )}\quad (\alpha >0, \beta >0). $$

(13)

For $\beta =1$, it becomes a standard MLF function.

4 Solution methodology

4.1 ANN modeling

This section presents the mathematical modeling of artificial neural networks (ANN) for differential equations of fractional order. Neural networks modeling has already been implemented for solving integer order differential equations. Now, we model the ANN to find the solution for fractional order differential equations.

The exponential function is used as an activation function for ANN. It has the capability to approximate the functions and its fractional derivative is also calculated with terms represented by classical MLF. The fractional derivative of an exponential function can be written as

$$ \frac{d^{v}}{dx^{v}}e^{\lambda x}=x^{-v}E_{1,1-v}( \lambda x). $$

(14)

Approximate solution for the problem considered in this research and its v order derivative is given by

$$\begin{aligned}& \hat{\theta }(\xi )=\sum_{i=1}^{m} \alpha _{i}e^{\omega _{i}\xi +\beta }, \end{aligned}$$

(15)

$$\begin{aligned}& \frac{d^{v}}{d\xi ^{v}}\hat{\theta }(\xi )=\sum _{i=1}^{m}\alpha _{i}e^{ \beta _{i}} \xi ^{-v}E_{1,1-v}(\omega _{i}\xi ). \end{aligned}$$

(16)

Equations (15) and (16) are used to approximate a solution of the fractional order differential equation given in Eq. (6). The neural networks architecture for fractional differential equations is given in Fig. 3. The objective function for the problem considered in this paper is given by

Figure 3

ANN structure for the problem

$$ \min\quad E=E_{1}+E_{2}, $$

(17)

where $E_{1}$ and $E_{2}$ are given by

$$\begin{aligned}& E_{1}=\frac{1}{N+1}\sum _{m=0}^{N} \biggl( \frac{d^{v}\hat{\theta }}{d\xi ^{v}}+\beta \hat{\theta } \frac{d^{2}\hat{\theta }}{d\xi ^{2}}+\beta \biggl( \frac{d\hat{\theta }}{d\xi } \biggr)^{2}-\psi ^{2}\hat{\theta } \biggr)^{2}, \end{aligned}$$

(18)

$$\begin{aligned}& E_{2}=\frac{1}{2} \bigl(\hat{\theta } {'}(0)^{2}+\bigl(\hat{\theta }(1)-1 \bigr)^{2} \bigr). \end{aligned}$$

(19)

Here, $E_{1}$ is related to the differential equation and $E_{2}$ is related to the initial and boundary conditions. We try to find the weights $\alpha _{i}$, $\omega _{i}$, and $\beta _{i}$ in Eq. (15) such that $E_{1}$ and $E_{2}$ approach zero, then E will also approach zero. Hence the approximate solution $\hat{\theta }(\xi )$ will approach the exact solution $\theta (\xi )$.

4.2 Cuckoo search

Inspired by the cuckoo bird’s breeding behavior, a metaheuristic algorithm was developed which is called the cuckoo search algorithm [59]. The female bird lays eggs in other host birds’ nests and they unintentionally raise her brood. When the host bird finds the egg of the cuckoo bird in her nest, it either throws it out of the nest or starts making her own brood elsewhere [60].

In the cuckoo search algorithm, the solution is represented by the egg of the host bird and the new candidate solution is represented by cuckoo’s egg. There are three rules that are described for cuckoo search and those are [61]: (1) the cuckoo lays a single egg at a time and puts it in the host’s nest; (2) the nests that have a high quality egg, i.e., a better solution will go to the next generation; and (3) there is a fixed number of host nests, and the host bird can find an alien egg with certain probability.

Assuming $\boldsymbol{x}_{i}= (x_{i 1}, x_{i 2}, \ldots , x_{i D} )$ as the position for the ith egg (solution) then updated solution ${x_{i}}^{\mathrm{new}}$ is generated by Levy flights as given below:

$$ \begin{aligned} x_{i}^{\mathrm{new}} &=x_{i}^{\mathrm{old}}+\alpha (x_{l}-x_{g} ) \oplus \operatorname{Levy}(\beta ), \\ &=x_{i}^{\mathrm{old}}+\frac{0.01 u}{ \vert v \vert ^{1 / \beta }} (x_{i}-x_{g} ), \end{aligned} $$

(20)

where the product ⊕ is entry-wise multiplication; the Levy flight exponent is denoted by β; the step size for a cuckoo is determined by a positive parameter α; the best solution within the current population is denoted by $x_{g}$; u and v are random numbers:

$$\begin{aligned}& u \sim N \bigl(0, \sigma _{u}^{2} \bigr), \qquad v \sim N \bigl(0, \sigma _{v}^{2} \bigr), \end{aligned}$$

(21)

$$\begin{aligned}& \sigma _{u}= \biggl[ \frac{\sin (\pi \beta / 2) \cdot \Gamma (1+\beta )}{2^{(\beta -1) / 2} \beta \cdot \Gamma (\frac{1+\beta }{2} )} \biggr]^{1 / \beta }, \qquad \sigma _{v}=1, \end{aligned}$$

(22)

where Γ is used for gamma function, and β controls the value of $\sigma _{u}$. There is a discovery operator in CS which is used to replace the discovered nests with a probability (pa). The equation that is used to update the solution is given as follows:

$$ x_{i j}^{\mathrm{new}}= \textstyle\begin{cases} x_{i j}^{\mathrm{old}}+\mathrm{rand} \cdot (x_{r 1, j}(k)-x_{r 2 j}(k) ) & \text{if } P>p a, \\ x_{i j}^{\mathrm{old}}(k) & \text{else}, \end{cases} $$

(23)

where ${x_{ij}}^{\mathrm{new}}$ is the jth element of the ith solution ${x_{i}}^{\mathrm{new}}$; $x_{r1,j}$ and $x_{r2,j}$ are the jth elements of the two solutions $x_{r1}$ and $x_{r2}$, where r1 and r2 are two different integers in interval $[1, \mathit{NP}]$, where NP represents size of population, pa represents the discovery probability, P and rand are some random numbers that belong to the interval $[0, 1]$.

4.3 Biogeography-based optimization

Biogeography-based optimization (BBO) is an evolutionary algorithm which is inspired by different characteristics of species living in the islands [62]. In BBO, each habitat is considered as a candidate solution having some habitat’s suitability index (HSI), which is employed for measurement of the quality of a habitat. A habitat (solution) is represented by some suitability index variables (SIV). Two types of operators, i.e., migration and mutation, are used in BBO that are employed for the evolution of the population. In migration process, the solutions with high HSI share their characteristics with the solutions having low HSI and the solutions with low HSI accept new characteristics from the solutions with high HSI.

In BBO, population is randomly initialized with NP habitats (solutons). Each generation sorts the population from the best to the worst and immigration and emigration rates λ and μ respectively are assigned to each habitat:

$$ \textstyle\begin{cases} \lambda _{i}=I (1-\frac{S_{i}}{N P} ), \\ \mu _{i}=E \frac{S_{i}}{N P}, \end{cases} $$

(24)

where immigration (I) and emigration (E) rates are such that $I=E=1$; $S_{i}$ represents the number of species of the habitats and $S_{i}=\mathit{NP}-i$. Accordingly, for the best solution the $S_{i}$ value is $\mathit{NP}-1$, and for the second best solution the $S_{i}$ value is $\mathit{NP}-2$, and for the worst solution the $S_{i}$ value is 0.

The migration mixes the features within the population that modifies the solutions. After migration, to modify the solutions, BBO also uses the mutation operator.

4.4 Heterogeneous cuckoo search algorithm based on BBO

CS and BBO are hybridized because CS uses the Levy flights to modify the solutions as it is good at exploration, and BBO modifies the solutions using the migration operator as it is good at exploitation. Combining the exploration and exploitation, a hybrid metaheuristic algorithm is developed which is known as BBO-based heterogeneous cuckoo search (BHCS) algorithm. The proposed BHCS algorithm has two main stages that are the heterogeneous cuckoo search and the discovery based on biogeography. The details of these two stages are explained in the next section.

4.4.1 Heterogeneous cuckoo search strategy

At first stage, the BHCS algorithm uses the Levy flights and quantum mechanism based heterogeneous cuckoo search. This strategy is inspired by quantum mechanism and was first presented in [60, 63]. The rules to update the solutions by heterogeneous cuckoo search are given as follows [60, 63]:

$$ x_{i}^{\mathrm{new}}= \textstyle\begin{cases} x_{i}^{\mathrm{old}}+\alpha \cdot (x_{i}-x_{g} ) \oplus \operatorname{Levy}(\beta ) & \frac{2}{3}< s r \leqslant 1\ (\mathrm{a}), \\ \bar{x}+L \cdot (\bar{x}-x_{i}^{\mathrm{old}} ) & \frac{1}{3}< s r \leqslant \frac{2}{3}\ (\mathrm{b}), \\ x_{i}^{\mathrm{old}}+\varepsilon \cdot (x_{g}-x_{i}^{\mathrm{old}} ) & \text{else (c)}, \end{cases} $$

(25)

where $L=\delta \ln (1 / \eta )$, $\varepsilon =\delta \exp (\eta )$, $x_{g}$ is used for the best solution at the current iteration; $\bar{x}=\frac{1}{N P} \sum_{i=1}^{N P} x_{i}$ represents the mean of all solutions; sr and η are random numbers in the interval $[0, 1]$. Equation (25) shows that heterogeneous cuckoo search employs three equations to update the solutions with the same probabilities. The first equation is related to Levy flights in original cuckoo search and the second and third equations to update the solutions are based on quantum mechanism. Updating the solutions using heterogeneous rules diversifies the search and follows the direction towards the real global region.

4.4.2 Biogeography-based discovery operator

At the second stage, new solutions are generated using a discovery operator. When the host bird finds an alien egg with probability pa, it abandons the old nest and starts making a new nest based on the migration operator.

Initially, solutions are listed from the best to worst, and an immigration rate μ is assigned to each solution:

$$ \mu _{l}=E \frac{S_{i}}{N P}, $$

(26)

where $E=1$ represents the maximum emigration rate; $S_{i}=\mathit{NP}-i$ represents the number of species in solutions.

In biogeography-based discovery operator, those solutions having best fitness value share their characteristics with other solutions, which helps to enhance the exploitation.

4.4.3 Overall BHCS algorithm

The BHCS algorithm uses a cascading structure for the implementation of its two steps. The cooperation between heterogeneous search strategy and biogeography-based discovery operator can efficiently balance the exploitation and exploration.

5 Performance metrics

We have performed 100 simulations on all four problems to establish the stability, adaptability, and certainty of the BHCS algorithm. For this purpose, we have determined the mean absolute deviation (MAD) in solutions, root-mean-square error (RMSE), error in Nash–Sutcliffe efficiency (ENSE), Theil’s inequality coefficient (TIC), and Nash–Sutcliffe efficiency (NSE). The analytical definition of these indexes are provided in Eqs. (27)–(30),

$$\begin{aligned}& \mathit{MAD}=\frac{1}{N}\sum _{m=1}^{N}{ \vert \theta _{m}- \hat{\theta }_{m} \vert }, \end{aligned}$$

(27)

$$\begin{aligned}& \mathit{TIC}= \frac{\sqrt{\frac{1}{N}\sum_{m=1}^{N}{(\theta _{m}-\hat{\theta }_{m})^{2}}}}{ (\sqrt{\frac{1}{N}\sum_{m=1}^{N}{\theta _{m}^{2}}}+\sqrt{\frac{1}{N}\sum_{m=1}^{N}{\hat{\theta }_{m}^{2}}} )}, \end{aligned}$$

(28)

$$\begin{aligned}& \mathit{NSE}=1-{ \frac{\sum_{m=1}^{N} {(\theta _{m}-\hat{\theta }_{m})^{2}}}{\sum_{m=1}^{N}{(\theta _{m}-\bar{\theta }_{m})^{2}}}},\quad \bar{\theta }_{m}=\frac{1}{N}\sum_{m=1}^{N}{ \theta _{m}}, \end{aligned}$$

(29)

$$\begin{aligned}& \mathit{ENSE}=1-\mathit{NSE}. \end{aligned}$$

(30)

6 Results and discussion

In this paper, the biogeography-based heterogeneous cuckoo search (BHCS) algorithm is used to train the ANN model for the solution of fractional differential equations. We have considered the fractional form of the energy balance equation (6) in order to find the temperature in straight fins. We have considered the cases with integer and fractional orders. The problems are solved using the ANN modeling given in Eqs. (15) and (16). In our ANN network, we have taken 10 neurons with 30 unknown weights α, ω, and β. The training is performed over the interval $[0,1]$ with a step size of 0.1. The domain has 11 grid points.

6.1 Case 1

In the first case, we have taken $\beta =0$, $v=2$. Using these values, Eq. (6) becomes

$$ \frac{d^{2}\theta }{d\xi ^{2}}-\psi ^{2}\theta =0;\quad 0\leq \xi \leq 1, $$

(31)

with boundary conditions

$$ \theta {'}(0)=0,\qquad \theta (1)=1. $$

(32)

The fitness function for Eqs. (31) and (32) is given as

$$ \min\quad E=\frac{1}{11}\sum _{m=0}^{10} \biggl( \frac{d^{2}\hat{\theta }}{d\xi ^{2}}-\psi ^{2}\hat{\theta } \biggr)^{2}+ \frac{1}{2} \bigl(\bigl(\hat{\theta } {'}(0)\bigr)^{2}+\bigl(\hat{ \theta }(1)-1\bigr)^{2} \bigr). $$

(33)

Equation (33) is minimized for $\psi =0.2$, $\psi =0.5$, and $\psi =0.8$ using the ANN–BHCS algorithm. The minimum fitness values obtained for the cases with $\psi =0.2$, $\psi =0.5$, and $\psi =0.8$ are 7.4829E−13, 1.5272E−16, and 2.8616E−12 respectively. Weights obtained by the ANN–BHCS algorithm for different values of ψ in case 1 are given in Fig. 4.

Figure 4

Weights obtained by the ANN–BHCS algorithm for case 1

Using the weights given in Fig. 4, the series solutions for case 1 are given as follows:

Solution for $\psi =0.2$:

$$ \begin{aligned} \hat{\theta }(\xi )={}&0.143377926259098e^{(-0.0418794570500770*\xi +0.603339509800116)} \\ &{}-1.50409601849347e^{(-0.581908044758456*\xi -2.01551159960342)} \\ &{}+0.287241489951004e^{(-0.671405234875654*\xi -0.731521145322519)} \\ &{}-0.0627038559905543e^{(-1.36851490007073*\xi -0.436732558847139)} \\ &{}+2.97799690160653e^{(0.208371755956944*\xi -2.36837490446120)} \\ &{}+0.394989568829285e^{(-0.586047360504350*\xi -1.99633165510105)} \\ &{}+1.04859467368134e^{(0.261805873347917*\xi -4.83414069981780)} \\ &{}+0.918091810798712e^{(-1.44365419755033*\xi -3.41866339844307)} \\ &{}+2.19401669184716e^{(0.251280939194522*\xi -2.99313408698317)} \\ &{}+1.84383655001332e^{(-0.239279689235954*\xi -1.69054475191618)}, \end{aligned} $$

(34)

Solution for $\psi =0.5$:

$$ \begin{aligned} \hat{\theta }(\xi )={}&10e^{(0.499970067232250*\xi -3.11686717118239)} \\ &{}+10e^{(10*\xi -10)}-10e^{(10*\xi -10)}+10e^{(10*\xi -10)} \\ &{}+9.99494624581934e^{(0.527213929554713*\xi -9.99999999999726)} \\ &{}+2.86432831656622e^{(-0.500001463874875*\xi -1.86559907908470)} \\ &{}+10e^{(9.99999999998309*\xi -10)}-9.99999999999504e^{(10*\xi -10)} \\ &{}-1.54342599995394e-05e^{(-9.98889167939289*\xi -10)}-10e^{(10*\xi -10)}, \end{aligned} $$

(35)

Solution for $\psi =0.8$:

$$ \begin{aligned} \hat{\theta }(\xi )={}&-2.84540527255357e^{(-3.21845338461347*\xi -6.53112157800192)} \\ &{}-3.80413421079954e^{(-3.74181618183772*\xi -8.93065320014428)} \\ &{}+2.00944346021810e^{(0.799869640284646*\xi -1.62836178434453)} \\ &{}-0.441669875127631e^{(-0.981705252103056*\xi -4.67920234633275)} \\ &{}-0.900148712340136e^{(0.296185270518098*\xi -5.15015758661538)} \\ &{}+6.63668690521927e^{(-0.779416608705294*\xi -2.89499226031466)} \\ &{}+0.167565735886103e^{(-1.23922653155550*\xi -0.390700545289821)} \\ &{}-3.22533792775341e^{(0.822874709544892*\xi -5.18290223782813)} \\ &{}+3.19253397853502e^{(-3.39413566604332*\xi -6.69094305479483)} \\ &{}-4.23009948394512e^{(-1.23074833318238*\xi -3.75559355870230)}. \end{aligned} $$

(36)

Exact and approximate solutions for $\psi =0.2$, $\psi =0.5$, and $\psi =0.8$ obtained by HPM, VIM, HPSTM, and ANN–BHCS algorithm are given in Tables 1, 2 and 3 respectively. The absolute errors in approximate solutions for $\psi =0.2$, $\psi =0.5$, and $\psi =0.8$ are given in Tables 4, 5 and 6 respectively. The tables show that the absolute errors in solutions obtained by the ANN–BHCS algorithm are less than those of the HPM, VIM, and HPSTM, which shows that the ANN–BHCS algorithm gives better solutions than other techniques. The exact and approximate solutions obtained by the ANN–BHCS algorithm for different values of ψ are also plotted in Fig. 5(a). The figure shows that the solutions obtained by the ANN–BHCS algorithm are very close to the exact solution. From Fig. 5(a), we can see that the dimensionless temperature θ decreases as the value of thermo-geometric fin parameter ψ increases. The absolute errors in solutions for different values of ψ are plotted in Fig. 5(b). The absolute errors in solutions for $\psi =0.2$, $\psi =0.5$, and $\psi =0.8$ are in the range 2.25E−08 to 7.71E−09, 4.19E−09 to 8.80E−09, and 2.34E−07 to 4.25E−07 respectively. Convergence of the fitness values for different values of ψ is given in Fig. 5(c). Histogram plots of the values of performance metrics are given in Figs. 6 and 7. In Fig. 6, the histograms for MAD and TIC values are given, which shows that more than 90% of the values are very close to zero. Figure 7 shows that more than 90% of the fitness and ENSE values are very close to zero, which shows the accuracy of the ANN–BHCS algorithm.

Figure 5

Results obtained by the ANN–BHCS algorithm for case 1

Figure 6

Histograms of MAD and TIC for case 1

Figure 7

Histograms of ENSE and fitness values for case 1

Table 1 Exact and approximate solutions for case 1 with $\psi =0.2$

Table 2 Exact and approximate solutions for case 1 with $\psi =0.5$

Table 3 Exact and approximate solutions for case 1 with $\psi =0.8$

Table 4 Absolute errors in solutions for case 1 with $\psi =0.2$

Table 5 Absolute errors in solutions for case 1 with $\psi =0.5$

Table 6 Absolute errors in solutions for case 1 with $\psi =0.8$

6.2 Case 2

In the second case, we have taken $\psi =0.5$ and $v=2$. Using these values, Eq. (6) becomes

$$ \frac{d^{2}\theta }{d\xi ^{2}}+\beta \theta \frac{d^{2}\theta }{d\xi ^{2}}+\beta \biggl(\frac{d\theta }{d\xi } \biggr)^{2}-(0.5)^{2} \theta =0;\quad 0\leq \xi \leq 1, $$

(37)

with boundary conditions

$$ \theta {'}(0)=0,\qquad \theta (1)=1. $$

(38)

The fitness function for Eqs. (37) and (38) is given as

$$ \begin{aligned} \min\quad E={}&\frac{1}{11}\sum _{m=0}^{10} \biggl( \frac{d^{2}\hat{\theta }}{d\xi ^{2}}+\beta \hat{\theta } \frac{d^{2}\hat{\theta }}{d\xi ^{2}}+\beta \biggl( \frac{d\hat{\theta }}{d\xi } \biggr)^{2}-(0.5)^{2}\hat{\theta } \biggr)^{2} \\ &{}+ \frac{1}{2} \bigl(\bigl(\hat{\theta } {'}(0) \bigr)^{2}+\bigl(\hat{\theta }(1)-1\bigr)^{2} \bigr). \end{aligned} $$

(39)

Equation (39) is minimized for different values of β using the ANN–BHCS algorithm. A total of 100 simulations were performed for different values of β. The minimum fitness values for $\beta =-0.5,-0.3,-0.1,0.1,0.3$, and 0.5 are 7.6027E−10, 2.8034E−11, 6.4330E−12, 1.3345E−10, 2.8926E−12, and 3.7965E−12 respectively. Weights obtained to minimize the fitness function for different values of β are plotted in Fig. 8. Using the weights given in Fig. 8, series solutions of case 2 for different values of β are given as follows:

Figure 8

Weights obtained by ANN–BHCS algorithm for case 2

Solution for $\beta =-0.5$:

$$ \hat{\theta }(\xi )=1.3350e^{(-0.7910*\xi -0.6658)}+ \cdots+0.96921e^{(-3.0480* \xi -2.5067)}, $$

(40)

Solution for $\beta =-0.3$:

$$ \hat{\theta }(\xi )=0.8911e^{(3.2514*\xi -9.0745)}+ \cdots-2.8323e^{(1.5411* \xi -9.8469)}, $$

(41)

Solution for $\beta =-0.1$:

$$ \hat{\theta }(\xi )=-1.4834e^{(-1.9619*\xi -1.9046)}+ \cdots+8.8526e^{(-2.0635* \xi -3.6484)}, $$

(42)

Solution for $\beta =0.1$:

$$ \hat{\theta }(\xi )=-0.6059e^{(-0.8974*\xi -8.3929)}+ \cdots+0.3641e^{(0.6138* \xi -0.4443)}, $$

(43)

Solution for $\beta =0.3$:

$$ \hat{\theta }(\xi )=0.2523e^{(-0.5144*x-3.6000)}+ \cdots-0.4030e^{(0.9104*x-5.1234)}, $$

(44)

Solution for $\beta =0.5$:

$$ \hat{\theta }(\xi )=0.0511e^{(0.5214*\xi -1.8989)}+ \cdots-0.3029e^{(-1.6382* \xi -0.6656)}. $$

(45)

Numerical solutions for case 2 with different values of thermal conductivity β are given in Table 7. Solutions are also plotted in Fig. 9(a). In this case, the thermo-geometric fin parameter is taken as $\psi =0.5$ and the thermal conductivity β is varied from −0.5 to 0.5 with a step size of 0.2. From Table 7 and Fig. 9(a), we see that the dimensionless temperature θ increases with increase in the value of β. The accuracy of the ANN–BHCS algorithm can be seen from convergence of the fitness values and histograms of fitness values in Fig. 9.

Figure 9

Results obtained by ANN–BHCS algorithm for case 2

Table 7 Solutions of case 2 for different values of β

6.3 Case 3

In the third case, we have taken $v=2$, $\psi =1.5$, and β is varied from −0.5 to 0.5 with step size 0.2. For this case, Eq. (6) takes the form

$$ \frac{d^{2}\theta }{d\xi ^{2}}+\beta \theta \frac{d^{2}\theta }{d\xi ^{2}}+\beta \biggl(\frac{d\theta }{d\xi } \biggr)^{2}-(1.5)^{2} \theta =0;\quad 0\leq \xi \leq 1, $$

(46)

with boundary conditions

$$ \theta {'}(0)=0,\qquad \theta (1)=1. $$

(47)

The fitness function for Eqs. (46) and (47) is given as

$$ \begin{aligned} \min\quad E={}&\frac{1}{11}\sum _{m=0}^{10} \biggl( \frac{d^{2}\hat{\theta }}{d\xi ^{2}}+\beta \hat{\theta } \frac{d^{2}\hat{\theta }}{d\xi ^{2}}+\beta \biggl( \frac{d\hat{\theta }}{d\xi } \biggr)^{2}-(1.5)^{2}\hat{\theta } \biggr)^{2} \\ &{}+ \frac{1}{2} \bigl(\bigl(\hat{\theta } {'}(0) \bigr)^{2}+\bigl(\hat{\theta }(1)-1\bigr)^{2} \bigr). \end{aligned} $$

(48)

The ANN–BHCS algorithm is used to minimize the fitness function for different values of β. The minimum fitness values obtained by the ANN–BHCS algorithm for $\beta =-0.5$, −0.3, −0.1, 0.1, 0.3, and 0.5 are 2.5336E−07, 8.0939E−09, 3.0616E−10, 3.6515E−11, 4.3335E−10, and 3.9055E−09 respectively. Weights obtained to minimize the fitness function for different problems are plotted in Fig. 10. Using these weights, the series solutions of different values of β are given as follows:

Figure 10

Weights obtained by ANN–BHCS algorithm for case 3

Solution for $\beta =-0.5$:

$$ \hat{\theta }(\xi )=0.3204e^{(0.3247*\xi +0.3378)}+ \cdots+0.6760e^{(0.8112* \xi +0.9999)}, $$

(49)

Solution for $\beta =-0.3$:

$$ \hat{\theta }(\xi )=1.8800e^{(-1.3968*\xi -2.0628)}+ \cdots+8.9815e^{(10.0578* \xi -17.6447)}, $$

(50)

Solution for $\beta =-0.1$:

$$ \hat{\theta }(\xi )=15.3357e^{(6.1567*\xi -19.5578)}+ \cdots+17.6613e^{(2.9498* \xi -8.3558)}, $$

(51)

Solution for $\beta =0.1$:

$$ \hat{\theta }(\xi )=4.8310e^{(-10.9679*\xi -19.0044)}+ \cdots-19.9914e^{(2.6348* \xi -8.1145)}, $$

(52)

Solution for $\beta =0.3$:

$$ \hat{\theta }(\xi )=1.3616e^{(5.4673*\xi -11.9129)}+ \cdots+3.8582e^{(-5.2181* \xi -7.6499)}, $$

(53)

Solution for $\beta =0.5$:

$$ \hat{\theta }(\xi )=0.0896e^{(4.4270*\xi -7.1525)}+ \cdots-19.6139e^{(2.0733* \xi -5.9705)}. $$

(54)

Numerical solutions of case 3 for different values of β are given in Table 8 and Fig. 11(a). Convergence and histogram plots of fitness values are given in Fig. 11. For all the values of β considered in case 3, most of the fitness values are very close to zero, which shows the accuracy of the ANN–BHCS algorithm. From Fig. 11(a), it is clear that the dimensionless temperature θ increases as the values of thermal conductivity β increase.

Figure 11

Results obtained by ANN–BHCS algorithm for case 3

Table 8 Solutions of case 3 for different values of β

6.4 Case 4

In this case, we have taken $v=1.75$, $\psi =0.5$, and β is varied from −0.5 to 0.5 with step size of 0.2. Using these values, Eq. (6) becomes

$$ \frac{d^{1.75}\theta }{d\xi ^{1.75}}+\beta \theta \frac{d^{2}\theta }{d\xi ^{2}}+\beta \biggl(\frac{d\theta }{d\xi } \biggr)^{2}-(0.5)^{2} \theta =0;\quad 0\leq \xi \leq 1, $$

(55)

with boundary conditions

$$ \theta {'}(0)=0,\qquad \theta (1)=1. $$

(56)

The fitness function for Eqs. (55) and (56) is given by

$$ \begin{aligned} \min\quad E={}&\frac{1}{11}\sum _{m=0}^{10} \biggl( \frac{d^{1.75}\hat{\theta }}{d\xi ^{1.75}}+\beta \hat{\theta } \frac{d^{2}\hat{\theta }}{d\xi ^{2}}+\beta \biggl( \frac{d\hat{\theta }}{d\xi } \biggr)^{2}-(0.5)^{2}\hat{\theta } \biggr)^{2} \\ &{}+ \frac{1}{2} \bigl(\bigl(\hat{\theta } {'}(0) \bigr)^{2}+\bigl(\hat{\theta }(1)-1\bigr)^{2} \bigr). \end{aligned} $$

(57)

The ANN–BHCS algorithm is used to minimize the fitness function (57) for different values of thermal conductivity β. The minimum of fitness values for $\beta =-0.5, -0.3, -0.1, 0.1, 0.3$, and 0.5 are 6.4804E−08, 2.7340E−09, 2.3225E−10, 7.0940E−10,5.1839E−11, and 4.8342E−11 respectively. Weights obtained by the ANN–BHCS algorithm to minimize fitness functions for different values of β are given in Fig. 12. Using these weights, the series solutions for different values of β are given as follows:

Figure 12

Weights obtained by ANN–BHCS algorithm for case 4

Solution for $\beta =-0.5$:

$$ \hat{\theta }(\xi )=8.1511e^{(0.6314*\xi -2.8504)}+ \cdots-10.5854e^{(2.2229* \xi -8.0687)}, $$

(58)

Solution for $\beta =-0.3$:

$$ \hat{\theta }(\xi )=0.0851e^{(-3.5320*\xi -1.5651)}+ \cdots-3.3255e^{(0.2556* \xi -6.5176)}, $$

(59)

Solution for $\beta =-0.1$:

$$ \hat{\theta }(\xi )=11.5319e^{(-0.5860*\xi -8.0212)}+ \cdots-5.3904e^{(-5.4688* \xi -19.5416)}, $$

(60)

Solution for $\beta =0.1$:

$$ \hat{\theta }(\xi )=-19.4309e^{(-2.3198*\xi -11.2920)}+ \cdots+5.6906e^{(-2.3634* \xi -5.7827)}, $$

(61)

Solution for $\beta =0.3$:

$$ \hat{\theta }(\xi )=-0.0232e^{(11.2897*\xi -19.9726)}+ \cdots-9.9720e^{(-3.3227* \xi -10.7209)}, $$

(62)

Solution for $\beta =0.5$:

$$ \hat{\theta }(\xi )=-6.4741e^{(-11.6490*\xi -10.8501)}+ \cdots+2.8678e^{(-6.3008* \xi -8.9989)}. $$

(63)

Numerical solutions of case 4 for different values of β are given in Table 9 and Fig. 13(a). From Fig. 13(a), it is clear that the dimensionless temperature θ increases as the value of thermal conductivity β goes from −0.5 to 0.5. Convergence of the fitness values for case 4 is given in Fig. 13(b). Histograms of the fitness values for different values of β are given in Figs. 13(c)–13(h). The figures show that more than 90% of the fitness values are very close to zero, which shows the efficiency of the ANN–BHCS algorithm.

Figure 13

Results obtained by ANN–BHCS algorithm for case 4

Table 9 Solutions of case 4 for different values of β

6.5 Case 5

In this case, we have taken $v=1.75$, $\psi =1.5$, and β is varied from −0.5 to 0.5 with step size of 0.2. Using these values, Eq. (6) becomes

$$ \frac{d^{1.75}\theta }{d\xi ^{1.75}}+\beta \theta \frac{d^{2}\theta }{d\xi ^{2}}+\beta \biggl(\frac{d\theta }{d\xi } \biggr)^{2}-(1.5)^{2} \theta =0; \quad 0\leq \xi \leq 1, $$

(64)

with boundary conditions

$$ \theta {'}(0)=0, \qquad \theta (1)=1. $$

(65)

The fitness function for Eqs. (64) and (65) is given by

$$ \begin{aligned} \min\quad E={}&\frac{1}{11}\sum _{m=0}^{10} \biggl( \frac{d^{1.75}\hat{\theta }}{d\xi ^{1.75}}+\beta \hat{\theta } \frac{d^{2}\hat{\theta }}{d\xi ^{2}}+\beta \biggl( \frac{d\hat{\theta }}{d\xi } \biggr)^{2}-(1.5)^{2}\hat{\theta } \biggr)^{2} \\ &{}+ \frac{1}{2} \bigl(\bigl(\hat{\theta } {'}(0) \bigr)^{2}+\bigl(\hat{\theta }(1)-1\bigr)^{2} \bigr). \end{aligned} $$

(66)

The ANN–BHCS algorithm is used to minimize the fitness function (66) for different values of β. The minimum fitness values for $\beta =-0.5, -0.3, -0.1, 0.1, 0.3$, and 0.5 are 5.5365E−05, 1.1040E−07, 3.6501E−09, 4.5598E−08, 1.9501E−08, and 9.3420E−09 respectively. Weights obtained to minimize the fitness function for different values of β are given in Fig. 14. Series solutions of case 5 with different values of β are given as follows:

Figure 14

Weights obtained by ANN–BHCS algorithm for case 5

Solution for $\beta =-0.5$:

$$ \hat{\theta }(\xi )=-19.8513e^{(-18.7508*\xi -8.0998)}+ \cdots+19.9362e^{(-13.2450* \xi -7.2254)}, $$

(67)

Solution for $\beta =-0.3$:

$$ \hat{\theta }(\xi )=3.2064e^{(-1.6155*\xi -3.8584)}+ \cdots+19.7640e^{(-12.2351* \xi -8.2594)}, $$

(68)

Solution for $\beta =-0.1$:

$$ \hat{\theta }(\xi )=1.9312e^{(-0.1831*\xi -4.6329)}+ \cdots+1.0286e^{(-8.5327* \xi -6.4404)}, $$

(69)

Solution for $\beta =0.1$:

$$ \hat{\theta }(\xi )=-14.1166e^{(-13.8929*\xi -19.4453)}+ \cdots+19.9964e^{(-1.7336* \xi -14.4155)}, $$

(70)

Solution for $\beta =0.3$:

$$ \hat{\theta }(\xi )=1.5998e^{(-1.3810*\xi -13.6786)}+ \cdots+0.7404e^{(-12.4352* \xi -5.3138)}, $$

(71)

Solution for $\beta =0.5$:

$$ \hat{\theta }(\xi )=9.2817e^{(-13.1731*\xi -8.1456)}+ \cdots+5.5888e^{(-2.2593* \xi -4.2874)}. $$

(72)

Numerical solutions of case 5 for different values of β are given in Table 10 and Fig. 15(a). From Fig. 15(a), it is clear that the dimensionless temperature θ increases as the value of thermal conductivity β goes from −0.5 to 0.5. Convergence of the fitness values for case 5 is given in Fig. 15(b). Histograms of the fitness values for different values of β are given in Figs. 15(c)–15(h). The figures show that more than 90% of the fitness values are very close to zero, which shows the efficiency of the ANN–BHCS algorithm.

Figure 15

Results obtained by ANN–BHCS algorithm for case 5

Table 10 Solutions of case 5 for different values of β

6.6 Case 6

In this case, we have taken $v=1.5$, $\psi =0.5$, and β is varied from −0.5 to 0.5 with step size of 0.2. Using these values, Eq. (6) becomes

$$ \frac{d^{1.5}\theta }{d\xi ^{1.5}}+\beta \theta \frac{d^{2}\theta }{d\xi ^{2}}+\beta \biggl(\frac{d\theta }{d\xi } \biggr)^{2}-(0.5)^{2} \theta =0;\quad 0\leq \xi \leq 1, $$

(73)

with boundary conditions

$$ \theta {'}(0)=0,\qquad \theta (1)=1. $$

(74)

The fitness function for Eqs. (73) and (74) is given by

$$ \begin{aligned} \min\quad E={}&\frac{1}{11}\sum _{m=0}^{10} \biggl( \frac{d^{1.5}\hat{\theta }}{d\xi ^{1.5}}+\beta \hat{\theta } \frac{d^{2}\hat{\theta }}{d\xi ^{2}}+\beta \biggl( \frac{d\hat{\theta }}{d\xi } \biggr)^{2}-(0.5)^{2}\hat{\theta } \biggr)^{2} \\ &{}+ \frac{1}{2} \bigl(\bigl(\hat{\theta } {'}(0) \bigr)^{2}+\bigl(\hat{\theta }(1)-1\bigr)^{2} \bigr). \end{aligned} $$

(75)

The ANN–BHCS algorithm is used to minimize the fitness function (75) for different values of β. The minimum fitness values for $\beta =-0.5, -0.3, -0.1, 0.1, 0.3$, and 0.5 are 1.1723E−06, 2.0482E−07, 3.9230E−08, 2.9632E−09, 2.2827E−10, and 4.3199E−11 respectively. Weights obtained to minimize the fitness function for different values of β are given in Fig. 16. Series solutions of case 6 with different values of β are given as follows:

Figure 16

Weights obtained by ANN–BHCS algorithm for case 6

Solution for $\beta =-0.5$:

$$ \hat{\theta }(\xi )=42.9653e^{(-15.1854*\xi -10.0023)}+ \cdots-3.2385e^{(4.2429* \xi -9.0045)}, $$

(76)

Solution for $\beta =-0.3$:

$$ \hat{\theta }(\xi )=-5.3645e^{(0.2792*\xi -2.1669)}+ \cdots+0.5784e^{(0.6436* \xi -0.1355)}, $$

(77)

Solution for $\beta =-0.1$:

$$ \hat{\theta }(\xi )=6.2815e^{(-5.9369*\xi -11.4495)}+ \cdots-17.7978e^{(-10.9295* \xi -8.1251)}, $$

(78)

Solution for $\beta =0.1$:

$$ \hat{\theta }(\xi )=-17.1800e^{(2.8172*\xi -16.2736)}+ \cdots+0.9790e^{(0.3888* \xi -0.5252)}, $$

(79)

Solution for $\beta =0.3$:

$$ \hat{\theta }(\xi )=1.0322e^{(-6.3497*\xi -4.6384)}+ \cdots+4.8820e^{(-0.8769* \xi -3.3310)}, $$

(80)

Solution for $\beta =0.5$:

$$ \hat{\theta }(\xi )=-0.7394e^{(-12.6785*\xi -7.3373)}+ \cdots+7.4517e^{(-0.8547* \xi -3.7189)}. $$

(81)

Numerical solutions of case 6 for different values of β are given in Table 11 and Fig. 17(a). From Fig. 17(a), it is clear that the dimensionless temperature θ increases as the value of thermal conductivity β goes from −0.5 to 0.5. Convergence of the fitness values for case 6 is given in Fig. 17(b). Histograms of the fitness values for different values of β are given in Figs. 17(c)–17(h). The figures show that more than 90% of the fitness values are very close to zero, which shows the efficiency of the ANN–BHCS algorithm.

Figure 17

Results obtained by ANN–BHCS algorithm for case 6

Table 11 Solutions of case 6 for different values of β

6.7 Case 7

In this case, we have taken $v=1.5$, $\psi =1.5$, and β is varied from −0.5 to 0.5 with step size of 0.2. Using these values, Eq. (6) becomes

$$ \frac{d^{1.5}\theta }{d\xi ^{1.5}}+\beta \theta \frac{d^{2}\theta }{d\xi ^{2}}+\beta \biggl(\frac{d\theta }{d\xi } \biggr)^{2}-(1.5)^{2} \theta =0; \quad 0\leq \xi \leq 1, $$

(82)

with boundary conditions

$$ \theta {'}(0)=0, \qquad \theta (1)=1. $$

(83)

The fitness function for Eqs. (82) and (83) is given by

$$ \begin{aligned} \min\quad E={}&\frac{1}{11}\sum _{m=0}^{10} \biggl( \frac{d^{1.5}\hat{\theta }}{d\xi ^{1.5}}+\beta \hat{\theta } \frac{d^{2}\hat{\theta }}{d\xi ^{2}}+\beta \biggl( \frac{d\hat{\theta }}{d\xi } \biggr)^{2}-(1.5)^{2}\hat{\theta } \biggr)^{2} \\ &{}+ \frac{1}{2} \bigl(\bigl(\hat{\theta } {'}(0) \bigr)^{2}+\bigl(\hat{\theta }(1)-1\bigr)^{2} \bigr). \end{aligned} $$

(84)

The BHCS algorithm is used to minimize the fitness function (84) for different values of β. The minimum fitness values for $\beta =-0.5, -0.3, -0.1, 0.1, 0.3$, and 0.5 are 3.2749E−04, 6.1684E−06, 1.0813E−07, 6.9620E−08, 3.3707E−08, and 6.1241E−08 respectively. Weights obtained to minimize the fitness function for different values of β are given in Fig. 18. Series solutions of case 6 with different values of β are given as follows:

Figure 18

Weights obtained by ANN–BHCS algorithm for case 7

Solution for $\beta =-0.5$:

$$ \hat{\theta }(\xi )=0.0011e^{(-18.7295*\xi -12.1889)}+ \cdots+0.0036e^{(10.6494* \xi -4.9802)}, $$

(85)

Solution for $\beta =-0.3$:

$$ \hat{\theta }(\xi )=2.1083e^{(14.4756*\xi -16.5181)}+ \cdots+19.4106e^{(-14.7538* \xi -16.7618)}, $$

(86)

Solution for $\beta =-0.1$:

$$ \hat{\theta }(\xi )=13.2030e^{(0.9806*\xi -4.6034)}+ \cdots+6.9143e^{(-4.5284* \xi -5.0134)}, $$

(87)

Solution for $\beta =0.1$:

$$ \hat{\theta }(\xi )=-12.5240e^{(8.4684*\xi -13.8879)}+ \cdots+4.0159e^{(1.5309 \xi -2.9223)}, $$

(88)

Solution for $\beta =0.3$:

$$ \hat{\theta }(\xi )=8.6030e^{(-19.6605*\xi -9.1743)}+ \cdots+10.7660e^{(-11.8943* \xi -6.3870)}, $$

(89)

Solution for $\beta =0.5$:

$$ \hat{\theta }(\xi )=0.7823e^{(4.4278*\xi -8.2779)}+ \cdots-1.8288e^{(-3.3932* \xi -6.4575)}. $$

(90)

Numerical solutions for case 7 are given in Table 12. Solutions are also plotted in Fig. 19(a) for different values of β. The solution figures show that dimensionless temperature θ increases with increasing the values of thermal conductivity from −0.5 to 0.5. Convergence of the fitness values for case 7 is given in Fig. 19(b). Histograms for the fitness values for different values of β are plotted in Figs. 19(c)–19(h).

Figure 19

Results obtained by ANN–BHCS algorithm for case 7

Table 12 Solutions of case 7 for different values of β

7 Sensitivity analysis of parameters

In this section, we analyze the sensitivity of different parameters which are number of neurons, population size, and discovery probability pa. The ODE in case 1 with $\psi =0.2$ is solved for sensitivity analysis of parameters. Solutions and absolute errors for different values of parameters are given in Tables 13, 14, 15, 16, 17, 18. We have solved the problem for 3, 5, and 10 neurons with fixed population size of 50 and discovery probability $pa=0.3$, and the results show that the ANN–BHCS algorithm gives better solution for 10 neurons. The problem is solved for different values of discovery probability $pa=0.01, 0.15$, and 0.3, and the number of neurons and population size were fixed as 10 and 50 respectively. For $pa=0.3$, the ANN–BHCS algorithm gives better solution. Similarly, the problem is solved for different population sizes, and the results show that for 50 population size the algorithm gives better results.

Table 13 Solutions obtained for different number of neurons

Table 14 Absolute errors (AE) for different number of neurons

Table 15 Solutions obtained for different values of discovery probability (pa)

Table 16 Absolute errors (AE) for different values of discovery probability (pa)

Table 17 Solutions obtained for different population sizes

Table 18 Absolute errors (AE) for different population sizes

8 Conclusion

In this paper, we have used ANN-based biogeography-based heterogeneous cuckoo search algorithm (ANN–BHCS) to analyze the problem of temperature distribution for convective straight fins. The ANN–BHCS algorithm is an efficient technique for the solution of fractional differential equations. We have considered seven cases of the problem, in which three cases are of integer order and four cases are of fractional order. In the first case, we have solved integer order energy balance equation for different values of thermo-geometric fin parameter ψ. The series solutions for case 1 are given in Eqs. (34), (35) and (36). The comparison of exact and numerical solutions obtained by HPM, VIM, HPSTM, and ANN–BHCS for case 1 is given in Tables 1–6. Solutions obtained for case 1 are also plotted in Fig. 5(a). The results show that the ANN–BHCS algorithm gives better solutions than other techniques. The efficiency of the algorithm is also obvious from the histograms of MAD, TIC, and ENSE values for case 1 in Figs. 6 and 7. From the second to seventh case, we have taken $\beta =-0.5, -0.3, -0.1, 0.1, 0.3, 0.5$ for all the cases. In the second and third case, we have considered integer order energy balance equation. Series solutions for the second and third case are given in Eqs. (40)–(45) and (49)–(54). Numerical solutions for the second and third case are given in Tables 7 and 8. Solution plots and histograms for fitness values of second and third case are given in Figs. 9 and 11. For both the cases, the dimensionless temperature increases for the values of $\beta =-0.5, -0.3, -0.1, 0.1, 0.3, 0.5$. The cases from the fourth to seventh are of fractional order. The series solutions for all the cases are given in results section. Numerical solutions from the fourth to seventh case are given in Tables 9–12. Solution plots and histograms of fitness values for the fourth to seventh cases are given in Figs. 13, 15, 17 and 19. The results show that the ANN–BHCS algorithm can efficiently solve the integer and fractional order differential equations.