Neural Process Lett DOI 10.1007/s11063-017-9711-6

Determining Approximate Solutions of Nonlinear Ordinary Differential Equations Using Orthogonal Colliding Bodies Optimization Arnapurna Panda1 · Sabyasachi Pani1

© Springer Science+Business Media, LLC 2017

Abstract The solution of nonlinear Ordinary Differential Equations (ODE) finds potential applications in physics, economics, computing and engineering. Conventional approaches used for solving ODE are effective in case of 1st order or 2nd order problems. With increase in order the complexity associated with the problem increases. Thus instead of going for an exact solution, determining an approximate solution is also helpful. In this paper, solving ODE is handled as an optimization problem. Popular Fourier Series expansion is used as an approximation function. A hybrid algorithm Orthogonal Colliding Bodies Optimization (OCBO) is formulated by assembling good features of orthogonal array (exploration of solution in search space) and Colliding Bodies Optimization (bodies after collision quickly move to a position with minimal energy level on the search space). The coefficients of the Fourier series are computed with OCBO. Simulation studies are carried out to determine solution of popular practically used ODEs: Bernoulli Equation for flowing fluids, IntegroDifferential equation, Brachistochrone equation for gravity, current response of an oscillatory Tank circuit, voltage and current decay with time in an electrical circuit. Simulations are also carried out on three benchmark ODEs used for modelling the biological processes. Comparative analysis demonstrated the superior approximation of the proposed approach over Orthogonal PSO, water cycle algorithm and Harmonic search. Keywords Ordinary differential equations · Orthogonal array · Colliding bodies optimization · Orthogonal PSO · Water cycle algorithm

B

Arnapurna Panda [email protected] Sabyasachi Pani [email protected]

1

School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Bhubaneswar, Odisha 751007, India

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1 Introduction Solving ordinary differential equations(ODE) finds extensive applications in physics (formulation of Bernoulli equation), electrical engineering (characteristics of voltage/current variation with time in an circuit), mechanical engineering (vibrations in mass-spring systems) and many more. The ODE differs from partial differential equations (PDE) in terms of considering the derivative i.e. in ODE derivative is taken with a single independent variable where as in PDE it takes into account of multiple variables [1]. In this manuscript solving ODE is formulated as an optimization problem. The aim is to determine approximate solution for any order ODE. Earlier approaches for determining the approximate solution of differential equation(DE) involve numerical approximation methods: Taylor polynomials [2], Taylor expansion approach [3,4], Homotopy-perturbation method [5]. These methods have the limitation for convergence to local optimum while approximating multi-modal non-linearities. Therefore, nature inspired meta-heuristic optimization algorithms are used to determine the near global optimal solutions. In these algorithms a group of agent interact among themselves and work in an collaborative manner to find out the solution of the problem. Literature survey reveals some of these popular algorithms have been used for solving the DE includes: genetic programming [6], genetic algorithm [7], particle swarm optimization [8,9], cuckoo search and water cycle algorithm [10], Harmonic Search [11]. The Fourier Series is wide popular for approximating any linear/nonlinear functions. Babaei [9] used Fourier Series to determine of approximate solution of ODE. The optimal value of coefficients of Fourier series were computed with the particle swarm optimization. Recently Sadollah et al. [10] proposed the use of cuckoo search and water cycle algorithm to obtain the coefficients more accurately. The use of orthogonal array to generate offsprings in nature inspired algorithms have been effective for several real life applications. The algorithms employing this concept include Orthogonal Genetic Algorithm (OGA) [12], Orthogonal differential evolution (ODE) [13,14], Orthogonal particle swarm optimization (OPSO) [15,16]. In most of the cases it is observed that the accuracy of optimization is more for the solutions obtained with orthogonal array. Attempts have also recently been made to reduce the computational complexity of OPSO [17]. Kaveh and Mahdavi [18] in 2014, proposed a new nature inspired meta-heuristic algorithm known as Colliding Bodies Optimization (CBO). It is influenced by collision between bodies in a surface which leads to their moment to achieve a minimum energy position on the surface. The advantage of CBO over other meta-heuristics is: it’s parameter free and it does not use memory to save good solutions. It has been applied to solve structural design problems [19], resource allocation problem [19], identification of Hammerstein plant [20]. The enhanced version of the CBO (ECBO) [19] is reported for optimal design of Reinforced Concrete Structures. In this manuscript Fourier Series is used as an approximator to determine solution of ODE. An hybrid algorithm orthogonal colliding bodies optimization (OCBO) is formulated to accurately compute the coefficient of Fourier Series. Simulations are carried out for determining approximate solution of five real life nonlinear ODE problems: three belongs to science (Bernoulli Equation for flowing fluids, Integro-Differential equation, Brachistochrone equation for gravity) and two belongs to electrical engineering (current response of an oscillatory Tank Circuit, Voltage and current decay in an electrical circuit). The results obtained with the proposed method is compared with those obtained by OPSO, water cycle algorithm (WCA) and Harmonic search algorithm (HS).

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Determining Approximate Solutions of Nonlinear Ordinary…

The ODEs plays a significant role in modelling the biological processes. Some real applications are malaria epidemics [26], disease of respiratory disorders [27], disease of hematopoiesis [27], disease of HIV [31]. Initially after use of stochastic gradient based methods, Mendes first applied simulated annealing (SA) [31] and evolutionary programming (EP) [32] for parameter estimation (to determine several rate constants of the mechanism) in irreversible inhibition of HIV proteinase. Then Moles et al. [33] used evolution strategies (ES) to estimate 36 parameters of a nonlinear biochemical dynamic model. All these methods though ensure solution they are their computational involvement is extensive. In 2007, Peifer and Timmer [34] proposed a multiple shooting method to estimate the parameters of ODEs for biochemical processes which has lower computational burden. Qian et al. [35] developed a nonlinear ODE model for yeast protein synthesis. In this model they opted Genetic programming (GP) to determine the structure of the model and employ Kalman filtering to determine the parameters in every iteration. Zhan et al. [36] proposed two models for determining parameters of known structure dynamical biological systems. One is using spline theory and Nonlinear Programming (NLP) with which the parameter estimation is treated as an optimization problem. Other one is use of differential evolution (DE) algorithm to estimate the parameter of ODE. With further analysis Zhan et al. [37] reported to improve the identification quality by using S-system representation. With certain assumptions they have applied it to yeast fermentation analysis. Inspired by the recent trends of research of ODEs role in the biological processes, in this manuscript we have used the OCBO algorithm to estimate the parameters of three benchmark ODEs used in biological modelling given by Shakeri and Dehghan in [29]. The performance of the proposed parameter estimation are compared with that obtained by OPSO, WCA and HS based models. The paper is organized as follows. Section 2 describes the problem formulation of determining the approximate solution of ODE with Fourier series. The creation of orthogonal array and corresponding pseudo code for implementation is given in Sect. 3. Formulation of hybrid algorithm OCBO and its step wise implementation is described in Sect. 4. For obtaining approximate solution of ODE, the steps involve in computing Fourier series coefficients with proposed OCBO are narrated in Sect. 5. The real life ODE problems for analysis, comparative algorithms, simulation environment and performance evaluation criteria are presented in Sect. 6. Discussions on the obtained results are highlighted in Sect. 7. Concluding remarks on the proposed methodology are given in Sect. 8.

2 Approximate Solution of Ordinary Differential Equations with Fourier Series Let z be a function of x. Then an ODE of order n is expressed by   dz d 2 z dn z F x, z, , 2,..., n = 0 dx dx dx     or F x, z, z , z , . . . , z n = 0

(1)

The Eq. (1) is subject to the boundary conditions: z(x0 ) = z 0 , .... .. z(xn ) = z n ,





z (x0 ) = z 0 , .... ..   z (xn ) = z n

(2)

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or the initial conditions 



(n−1)

z(x0 ) = z 0 , z (x0 ) = z 0 , . . . z (n−1) (x0 ) = z 0

(3)

Following [23] the actual solution z(x) can be approximated using Fourier series as below:     T   0) 0) (4) z(x) ≈ Z apx (x) = a0 + am cos mπ(x−x + bm sin mπ(x−x N N m=1

where T is the number of terms associate with sin and cos functions. The N is the length of the interval solution where [x0 , xn ] is the bound of the interval. The derivatives of Eq. (4) are given by   T  mπ  mπ(x − x0 )   z (x) ≈ Z apx (x) = am sin − m=1 N N    mπ  mπ(x − x0 ) bm cos + N N

    2 T mπ mπ(x − x0 )   z (x) ≈ Z apx (x) = − am cos m=1 N N    mπ 2 mπ(x − x0 ) bm sin − N N ::

   mπ n T mπ(x − x0 ) nπ n n + am cos z (x) ≈ Z apx (x) = m=1 N N 2 (5)    mπ n mπ(x − x0 ) nπ bm sin + + N N 2 In order to generate approximate solution of (1), the Eqs. (17)–(5) are substituted in Eq. (1), which creates an error function E(x) given by     n E(x) = Fˆ x, Z apx , Z apx , Z apx , . . . , Z apx (6) The aim is to determine the coefficients of Fourier Series i.e. a0 , am and bm such that the value of error function E(x) approaches zero value. The meta-heuristic algorithm are used to optimize the coefficient value such that the error becomes minimal.

3 Orthogonal Design Combination In meta-heuristic based approach to solve optimization problems, the ‘Orthogonal Design Combination’ play a vital role as it scatters the solutions uniformly in the search space which enhance the exploration capabilities of the algorithm. Suppose there are N factors and Q levels, that means N number of vectors are there and each vector has Q components. The general combination of N factors and Q levels is Q N which is very vast and difficult for calculation of fitness value. With Orthogonal design combination the resulting orthogonal array of N factors and Q levels is L M (Q N ) where M is the number of combinations of levels and L represents the Latin square. The L M (Q N ) is a matrix of M rows and N columns which is a representative set of all combinations Q N . It can be represented as L M (Q N ) = [xi, j ] M X N where the jth factor in ith combination is represented by xi, j . The xi, j takes value in 1, 2, . . . , Q. The combination is orthogonal due to some specific properties. If any two columns of L M (Q N ) are swapped then the resulting L M (Q N ) is also an orthogonal array. Secondly if

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Determining Approximate Solutions of Nonlinear Ordinary…

we delete some columns of L M (Q N ) then the resulting matrix is also a orthogonal array with less number of columns. To understand the ‘Orthogonal Design Combination’ consider an example: Lets consider four factors and three levels, that means four variables having three components each. In general total number of possible combination is 34 = 81. With Orthogonal Design Combination results in an array L 16 (34 ) which have 16 combinations. Thus here instead of 81 only 16 combinations are to be tested. The representative 16 combinations are shown in Fig. 1.

3.1 Construction of Orthogonal Array Construction of orthogonal array has been well described in the literature [15,25]. A simple procedure is outlined in Algorithm 1 for creating the array. The columns of orthogonal array xi, j denoted by x j is categorized into two parts: Primary columns and Secondary columns. The primary columns are j = 1, 2, (Q 2 − 1)/(Q − 1) + 1, . . . , (Q P −1)/(Q −1)+1. Here Q is odd. The remaining columns are termed as secondary columns. The procedure to create the primary columns and secondary columns are highlighted in Algorithm 1. Algorithm 1 Creating Orthogonal Array L M (Q N ) 1: M ← Q P , P is a positive integer; 2: 3: 4: 5: 6: 7: 8:

P

−1 N ← QQ−1 ; loop: (1 ≤ i ≤ M) and (1 ≤ j ≤ N ) Step 1: Construct the Primary Columns. for p = 1 to P do p−1 j = Q Q−1−1 + 1; for i = 1 to Q P do xi, j = i−1 mod Q; P− p

Q

9: end for 10: end for 11: Step 2: Construct the Secondary Columns. 12: for p = 2 to P do 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23:

p−1

j = Q Q−1−1 + 1; for u = 1 to j − 1 do for v = 1 to Q − 1 do   x j+(u−1)(Q−1)+v = xu × v + x j mod Q; end for end for end for Step 3: Increment xi, j j ← j + 1; i ← i + 1; goto loop.

4 Orthogonal Colliding Bodies Optimization (OCBO) OCBO is a hybrid algorithm which contains good feature of Orthogonal Array and characteristics of Colliding Bodies Optimization (CBO). The Orthogonal Array explained in Sect. 3 elaborated the creation of new solutions by making orthogonal combination of the existing

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Fig. 1 Example of orthogonal design combination

solutions. Thus it is helpful for exploring and determining the optimal solution by making suitable combination. The CBO [19] is based on the natural phenomenon of collision between bodies with which they try to achieve a position with minimal energy level on the search space. As shown in Fig. 2 a body with mass m 1 moving with velocity u 1 collide with another

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Determining Approximate Solutions of Nonlinear Ordinary…

Fig. 2 Collision between two bodies of mass m 1 and m 2 moving with velocities u 1 and u 2 . With progressive collision bodies attend positions corresponding to minimal energy level

body m 2 with velocity u 2 . During collision conservation of momentum and kinetic energy   takes place. After collision mass m 1 achieve a velocity u 1 and m 2 move with velocity u 2 . The CBO does not use memory for saving good solution. Another advantage is it does not posses any internal parameter tuning.

4.1 Implementation Steps of OCBO The motivation here is to design orthogonal array between bodies positions attended before collision and after collision. It is beneficial to obtain new positions which may explore the possibilities of attending minimum energy level. The detailed procedure is outlined as follows: Objective Minimize a function Q = f (w1 , w2 , . . . , w D ) in the range [wmin , wmax ]. Where D is the dimensionality which also reflect the number of variable to minimize simultaneously. Step wise procedure for determining optimal solution: 1. Initialize population Initialize B number of Colliding Bodies (CB) which represent the population. Each CB is of dimension D. The bodies position are given by ⎡ → ⎢ ⎢ W= ⎢ ⎣

⎤ ⎡ W1 w1,1 w1,2 ⎢ w2,1 w2,2 W2 ⎥ ⎥ ⎢ .. ⎥ = ⎢ .. .. . ⎦ ⎣ . . WB w B,1 w B,2

⎤ . . . w1,D . . . w2,D ⎥ ⎥ ⎥ .... .. : ⎦ . . . w B,D

(7)

where Wi = Wmin + r nd(Wmax − Wmin ) i = 1, 2, . . . , B

(8)

Here Wmin = [min(w1 ), min(w2 ), . . . min(w D )] and Wmax = [max(w1 ), max(w2 ), . . . max(w D )]. The r nd is a random number in range [0, 1]. 2. Compute the fitness of each body The fitness of ith body is calculated by substituting it’s position value in the fitness function given by Q i = f (Wi ) ∀i = 1, 2, . . . , B

(9)

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3. Compute the mass of each body The mass of each body is calculated as 1 Qi

mi = B

1 j=1 Q j

∀i = 1, 2, . . . , B

(10)

4. Initial velocities of bodies The bodies are sorted in according to their mass in a decreasing order. They are classified into two groups: – Stationary bodies: The upper half of bodies having higher mass are termed as stationary bodies and their velocities before collision are considered as as zero. u i = 0; ∀i = 1, 2, . . . ,

B 2

(11)

– Moving bodies: The lower half of bodies are movable and their velocities before collision are B + 1, . . . , B (12) u i = Wi − Wi− B ; ∀i = 2 2 5. Bodies velocity after collision The velocity of each stationary body after collision is :   m i+ B + m i+ B u i+ B B  2 2 2 ui = ; ∀i = 1, . . . (13) m i + m i+ B 2 2

Velocity of every moving body after collision is:   m − m B i i− 2 u i B  ; ∀i = + 1, . . . , B ui = m i + m i− B 2

(14)

2

The  is known as coefficient of restitution. 6. Bodies position after collision The positions of stationary bodies after collision are 

Winew = Wi + r nd ⊗ u i ∀i = 1, . . . ,

B 2

(15)

The moving bodies positions after collision are 

Winew = Wi− B + r nd ⊗ u i ∀i = 2

B + 1, . . . , B 2

(16)

The r nd represent random numbers in range [-1, 1]. The ‘⊗’ denotes term by term multiplication. 7. Design Orthogonal Positions The updated position Winew and old position Wi are used to create orthogonal vectors. Creating the vectors are described in algorithm 1 of earlier Sect. 3.1. Total number of bodies for orthogonal combination is Q = 2B and each with dimension N = D. The orthogonal combination produces a Latin Square of M × N given by     L M Q N = αi, j M×N where M = Q log N (N +1)

(17)   The details to obtain L M Q N is described in Sect. 3. 8. Upgrade each body position with it’s best Orthogonal Position   – Compute the fitness of all orthogonal combinations L M Q N .

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Determining Approximate Solutions of Nonlinear Ordinary… 

– Find out the Winew obtained from orthogonal combinations which is superior than  Winew . Replace xinew by the orthogonal position xinew . 9. Calculate coefficient of restitution (COR) The COR is computed as  =1−

gen gen max

(18)

where gen is the present generation and gen max is the maximum number of generation. 10. Termination Criteria The steps from 2 to 9 are repeated till a satisfactory minimal function value is achieved or the algorithm is set to run for a maximum number of generation.

5 Computing the Fourier Series Coefficients with Proposed OCBO The steps to compute the Fourier Series coefficients with proposed OCBO algorithm are narrated below: 1. Determine the number of terms for approximation in Fourier Series It is dependent on the non-linearity associated with the ODE. For complex non-linearity more number of terms are required for approximation. Based on the application the optimal value of number of sin-cos expansion required may be set by hit and trail method by the programmer. ex- If the number terms is 3 then total number of variable to optimize is 7 (a0 along with 3 sine and 3 cosine component). Therefore the dimension at the initialization D = 7. → 2. Initialization of bodies Initialize a W = B × D dimensional matrix that represent ‘B’ bodies each with ‘D’ dimension as described in Eqs. (7)–(8). 3. Bodies fitness evaluation For each of the body the error function E(x) in (6) represent the fitness. Suppose E(x) = a0 + a1 cos(π x) + b1 sin(π x) (19) →

Then each row of W representing a possible solution (i.e. a body) [a0 , a1 , b1 ]. With the →

progress in OCBO the positions of bodies in W are updated such that E(x) becomes minimum. Then the implementation steps 3–10 described in Sect. 4 are continued. Finally the obtained solution i.e. minimal value of E(x) and corresponding coefficient values are reported. Once the training is complete for twenty independent runs the performance evaluation of the obtained approximate solution is carried out using the criteria defined in Sect. 6.4 and obtained results are reported.

6 Simulation Analysis 6.1 Test Problems: Differential Equations Arising in Science and Engineering Problems – Example 1: Solving Bernoulli Equation

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The Bernoulli Equation is famous and deals with the conservation of energy in the flowing fluids. It is represented by second order ODE [24] given by 



z + (z )2 − 2e(−z) = 0 z(0) = 0, z(1) = 0

(20)

The true solution for the above ODE obtained with analytical method is z(x) = ln((x − 0.5)2 + 0.75)

(21)

For approximate solution using Fourier Series the number of expansion using sin-cos is taken as 3. The solution space is assumed to be in the range [0, 1] as earlier taken in the literature [9,10]. The approximate solution takes the form Z apx (x) = a0 + a1 cos(π x) + b1 sin(π x) + a2 cos(2π x) + b2 sin(2π x) + a3 cos(3π x) + b3 sin(3π x)

(22)

– Example 2: Solving Integro-Differential Equation The Integro-differential equation is mathematically given by   x  1x ≥0 z + 2z + 5 , z(0) = 0 z(t)dt = 0 x <0 0

(23)

The solution interval is in range 0 to π. The exact solution of this integral value problem is 1 −x e sin(2x) (24) 2 For effective approximate solution using Fourier Series the number of terms in the sin-cos expansion is set as six. The approximate solution obtained is in the form z(x) =

Z apx (x) = a0 + a1 cos (x) + b1 sin (x) + a2 cos (2x) + b2 sin (2x) + a3 cos (3x) + b3 sin (3x) + a4 cos (4x) + b4 sin (4x) + a5 cos (5x) + b5 sin (5x) + a6 cos (6x) + b6 sin (6x)

(25)

– Example 3: Solving Brachistochrone Equation This nonlinear ODE is mathematically represented by 

[1 + (z )2 ]z = k 2 z(0) = 0, z(1) = 2

(26)

where k 2 is unknown constant. The true solution mathematically is a parametric equation of cycloid  2 x(θ ) = k2 (θ − sin(θ )) (27) 2 z(θ ) = k2 (1 − cos(θ )) The analytical value of constant k 2 is 4.81. This constant value is also taken as an additional parameter and approximated by the optimization algorithm. In the sin-cos expansion number of terms taken is 5. The approximate expansion consist of 11 variable. Thus total number of parameter to optimize including k 2 is 12. The approximate solution is Z apx (x) = a0 + a1 cos (π x) + b1 sin (π x) + a2 cos (2π x) + b2 sin (2π x) + a3 cos (3π x) + b3 sin (3π x) + a4 cos (4π x) + b4 sin (4π x) + a5 cos (5π x) + b5 sin (5π x)

123

(28)

Determining Approximate Solutions of Nonlinear Ordinary… Fig. 3 A tank circuit in Ex. 4 which produces damped oscillations

– Example 4: Oscillatory circuit or Tank circuit In electrical engineering the circuit that produces voltage/current oscillations of any desired frequency is popularly known as Oscillatory circuit or Tank circuit. A simple tank circuit comprise of a switch (S), capacitor (C)and an inductor (L) is shown in Fig. 3. The capacitor is initially charged from a d.c. source. Then the switch is closed. The circuit behave as an oscillator and generate damped oscillations till the current decreases gradually and becomes zero. The frequency of oscillation is f =

2π

1 √

LC

(29)

The damped oscillation of current in the circuit is represented by second order differential equation given by   z + 0.3z + z − 1 = 0 (30)  z(0) = 0, z (0) = 0 The exact solution for this ODE using analytical calculation is given by ⎞ ⎛ ⎞ √ √ ⎛ √ −3 391 391x 391x cos sin 391 20 20 ⎠−⎝ ⎠     z(x) = 1 + ⎝ 3x 3x 20 20 e e

(31)

The number of expansions in Fourier series is set to six. The time interval is taken from 0 to 15 to observe the damping oscillations. The approximate solution is obtained in the form πx  πx   x Z apx (x) = a0 + a1 cos + a2 cos 2π 15  + b1 sin   2π x 15  3π 15 x + b2 sin 15 + a3 cos 15x + b3 sin 3π 15   4π x   4π x  5π x (32) + a4 cos 15 + b4 sin 15 + a5 cos 15       x x 6π x + a6 cos 6π + b5 sin 5π 15 15 + b6 sin 15 – Example 5: Voltage and Current Decay in a circuit In electrical engineering circuits the voltage (V) and current (I) decay with time (t) and their behavior is expressed in terms of ODE. One example of such ODE is given by dV = 2I − V dt dI = −I − V dt

(33) (34)

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with I(0)=2, V(0)=2 are the initial conditions. The exact solutions are given by √ √ √ −t cos( 2t) V (t) = 2 2e−t sin( 2t) + 2e √ √ √ I (t) = 2e−t cos( 2t) − 2e−t sin( 2t)

(35)

The time interval for solution is considered between 0 to 1.5 seconds. Te approximate solution for the voltage and current are approximated with three sin-cos expansions given by       πt πt 2πt Vapp (t) = a0 + a1 cos + b1 sin + a2 cos 1.5 1.5 1.5       2πt 3πt 3πt + a3 cos + b3 sin (36) + b2 sin 1.5 1.5 1.5       πt πt 2πt + b1 sin + a2 cos Iapp (x) = a0 + a1 cos 1.5 1.5 1.5       2πt 3πt 3πt + b2 sin + a3 cos + b3 sin (37) 1.5 1.5 1.5

6.2 Test Problems: Differential Equations Arising in Mathematical Biology The differential equations are extensively used in modeling several biological phenomenons. Some important applications are: effect of incubation delays on malaria epidemics [26], effect of CO2 concentration in a mammal causing a dynamic disease of respiratory disorders [27], density of mature cells in blood circulation leading to disease of hematopoiesis [27], the effect of concentration of a chemical component and temperature resulting an exothermic/irreversible reaction [28]. Shakeri and Dehghan in [29] proposed a generalized form of differential equation expressing the characteristics of all the above model given by u (m) (t) =

J m−1

μ jk (t)u (k) (α jk t + β jk ) + g(u) + f (t)

(38)

j=0 k=0

where g(u) = bu +

l

di u γi

(39)

i=1

The di ∈ R, γi > 1 and γi ∈ N , l ∈ N . The α jk ∈ [0, 1] and β jk ∈ R. The u (m) is the mth derivative of function u. The Eq. (38) is associated with initial conditions m−1

cik u (k) (0) = λi , i = 0, 1, . . . , m − 1

(40)

k=0

In [30] Yildirim et al. applied variational iteration method for a reliable analysis of this generalized differential equation for mathematical biology. In this paper the proposed Fourier series-OCBO based approximation model is used to determine the effective solutions of this generalized differential equation. – Example 6: First Order Differential Equation The first order differential equation used in [29] for modelling the biological processes is given by   1 t  u (t) = −2tu(t − ) − t 2 u(t) + 2 sin(t)u + u 3 (t) + f (t) (41) 2 2

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Determining Approximate Solutions of Nonlinear Ordinary… Table 1 Comparative results of Ex. 1, Ex. 2 and Ex. 3: obtained coefficients of Fourier Series after optimized with OCBO, OPSO, WCA and HS (Reported best values after 20 independent runs) Ex. Ex. 1

Ex. 2

Ex. 3

where

Coefficients

OCBO

OPSO

WCA

HS

a0

2.2512E-02

2.2436E-02

2.2508E-02

2.2528E-02

a1

8.3125E-05

8.3150E-05

8.3129E-05

8.3142E-05

b1

−0.3290

−0.3328

−0.3250

−0.3188

a2

−2.2490E-02

−2.2468E-02

−2.2550E-02

−2.2608E-02

b2

−1.6735E-04

−1.6750E-04

−1.6727E-04

−1.6742E-04

a3

−8.3096E-05

−8.3202E-05

−8.3114E-05

−8.3150E-05

b3

3.6882E-03

3.6850E-03

3.6871E-03

3.6864E-03

a0

1.9998E-02

2.0004E-02

1.9992E-02

1.9989E-02

a1

1.1000E-02

1.1022E-02

1.0097E-02

1.0092E-02

b1

3.8360E-02

3.8367E-02

3.8354E-02

3.8351E-02

a2

−7.9945E-04

−7.9951E-04

−7.9942E-04

−7.9938E-04

b2

1.1888E-01

1.1886E-01

1.1876E-01

1.1879E-01

a3

−3.6250E-02

−3.6254E-02

−3.6245E-02

−3.6239E-02

b3

4.7975E-02

4.7987E-02

4.7968E-02

4.7970E-02

a4

−4.2281E-02

−4.2285E-02

−4.2277E-02

−4.2268E-02

b4

6.5365E-02

6.5371E-02

6.5358E-02

6.5365E-02

a5

2.7610E-02

2.7616E-02

2.7608E-02

2.7598E-02

b5

4.9890E-02

4.9902E-02

4.9892E-02

4.9896E-02

a6

2.0755E-02

2.0761E-02

2.0748E-02

2.0742E-02

b6

−4.1470E-03

−4.1478E-03

−4.1467E-03

−4.1456E-03

a0

0.8683

0.8690

0.8680

0.8677

a1

−1.0000

−1.0020

−0.9996

−0.9985

b1

0.5370

0.5374

0.5366

0.5362

a2

0.0630

0.0636

0.0627

0.0622

b2

0.2754

0.2758

0.2749

0.2750

a3

−0.0370

−0.0375

−0.0369

−0.0366

b3

0.0884

0.0890

0.0882

0.0878

a4

0.0690

0.0702

0.0682

−0.0678

b4

0.0895

0.0912

0.0884

0.0881

a5

0.0370

0.0376

0.0368

0.0363

b5

−0.0184

−0.0188

−0.0182

−0.0175

k2

4.8224

4.8418

4.8556

4.8810

    t t 7 f (t) = − e− 2 −2 −t 4 − t 3 + 23 t 2 − 23 t − 23 − 2te− 2 − 4 −t 2 + 45  2  2 3  3t t t + e− 8 −6 − 16 − 4t + 1 + 2 sin(t)e− 4 −2 − t4 − 2t + 1

(42)

with initial condition u(0) = e−2 . The exact solution is given by   t u(t) = −t 2 + 1 − t e− 2 −2

(43)

123

A. Panda, S. Pani Table 2 Comparative results for Ex. 4 and Ex. 5: obtained coefficients of Fourier Series after optimized with OCBO, OPSO, WCA and HS (reported best values after 20 independent runs ) Ex. Ex. 4

Ex. 5

Coefficients

OCBO

OPSO

WCA

HS 0.8852

a0

0.8865

0.8880

0.8861

a1

−0.1648

−0.1664

−0.1652

−0.1645

b1

0.2529

0.2538

0.2533

0.2522

a2

0.3277

0.3280

0.3278

0.3272

b2

0.2070

0.2055

0.2068

0.2080

a3

0.1610

0.1598

0.1606

0.1622

b3

−0.4635

−0.4642

−0.4637

−0.4640

a4

−0.6636

−0.6639

−0.6637

−0.6634

b4

−0.1966

−0.1963

−0.1965

−0.1960

a5

−0.5220

−0.5223

−0.5218

−0.5216 0.1912

b5

0.1910

0.1910

0.1909

a6

−0.0235

−0.0240

−0.0238

−0.0234

b6

0.0926

0.0930

0.0926

0.0928

a0

0.6620

0.6623

0.6618

0.6615

a1

0.8185

0.8187

0.8179

0.8177

b1

1.3270

1.3277

1.3265

1.3268

a2

0.4820

0.4822

0.4821

0.4815

b2

8.5915E-02

8.5921E-02

8.5909E-02

8.5911E-02

a3

3.7430E-02

3.7433E-02

3.7436E-02

3.7428E-02

b3

−0.1135

−0.1133

−0.1137

−0.1139

a0

0.8385

0.8392

0.8381

0.8378

a1

1.3625

1.3633

1.3621

1.3616

b1

−1.0562

−1.0567

−1.0557

−1.0553

a2

−9.8535E-02

−9.8542E-02

−9.8533E-02

−9.8528E-02

b2

−0.4600

−0.4611

−0.4597

−0.4591

a3

−0.1025

−0.1027

−0.1021

−0.1019

b3

1.1920E-02

1.1931E-02

1.1915E-02

1.1912E-02

Voltage

Current

The approximate solution is represented in the form u apx (t) = a0 + a1 cos (πt) + b1 sin (πt) + a2 cos (2πt) + b2 sin (2πt) + a3 cos (3πt) + b3 sin (3πt) + a4 cos (4πt) + b4 sin (4πt) + a5 cos (5πt) + b5 sin (5πt)

(44)

– Example 7: Second Order Differential Equation The second order differential equation from [29] for analyzing the biological processes is represented by     t 1   + (t − 1)u(t) − 2u + u 2 (t) + f (t) (45) u (t) = e−t u t − 5 3

123

Determining Approximate Solutions of Nonlinear Ordinary… Table 3 Comparative results of biological systems Ex. 6, Ex. 7 and Ex. 8: obtained coefficients of Fourier Series after optimized with OCBO, OPSO, WCA and HS (Reported best values after 20 independent runs) Ex. Ex. 6

Ex. 7

Ex. 8

Coefficients

OCBO

OPSO

WCA

HS

a0

0.02360

0.02364

0.02361

0.02357

a1

0.08133

0.08128

0.08131

0.08126

b1

0.00602

0.00604

0.00603

0.00602

a2

0.00680

0.00682

0.00676

0.00675

b2

0.01821

0.01821

0.01819

0.01818

a3

0.03461

0.03463

0.03460

0.03461

b3

−0.00570

−0.00568

−0.00572

−0.00573

a4

−0.00383

−0.00380

−0.00384

−0.00387

b4

−0.02195

−0.02192

−0.02196

−0.02195

a5

−0.00694

−0.00694

−0.00696

−0.00699 0.00151

b5

0.00152

0.00154

0.00152

a0

0.33341

0.33342

0.33340

0.33339

a1

−0.04320

−0.04318

−0.04321

−0.04324 0.12042

b1

0.12046

0.12050

0.12043

a2

0.07530

0.07534

0.07529

0.07528

b2

−0.01342

−0.01340

−0.01344

−0.01347

a3

−0.02242

−0.02238

−0.01345

−0.01346

b3

−0.03873

−0.03870

−0.03875

−0.03876

a4

−0.01390

−0.01388

−0.01392

−0.01393

b4

0.01371

0.01375

0.01368

0.01367

a5

0.00410

0.00414

0.00405

0.00406

b5

0.00282

0.00284

0.00277

0.00277

a0

0.55030

0.55028

0.55032

0.55035

a1

−0.35231

−0.35228

−0.35233

−0.35235

b1

0.34333

0.34328

0.34335

0.34335

a2

−0.05042

−0.05036

−0.05045

−0.05048

b2

−0.02280

−0.02280

−0.02282

−0.02286

a3

−0.19565

−0.19562

−0.19568

−0.19570

b3

−0.00610

−0.00606

−0.00613

−0.00612

a4

−0.00033

−0.00030

−0.00034

−0.00034 0.14310

b4

0.14310

0.14305

0.14311

a5

0.04635

0.04634

0.04635

0.04638

b5

−0.00060

−0.00060

−0.00061

−0.00061

where   1 t  1   t  2 t 2 t f (t) = − 41 sin  t 3 sin1 3 cos  t 2 − 49 cos t 2 1 t   13 − + − 2 sin 3 − 3 cos 2 t + 9 sin  3 + 4 cos 2       1 1 − 16 sin 2t − 10 + sin 9t + 23 cos 6t − e−t 16 cos 3t − 15 with initial conditions



(46)



3u(0) + 6u (0) = 2  −2u(0) + u (0) = − 21

(47)

123

A. Panda, S. Pani

(a) 0.05 Exact OCBO OPSO WCA HS

0 −0.05

Z(x)

−0.1 −0.15 −0.2 −0.25 −0.3

(b)

0

0.2

0.4

x

0.6

0.8

1

0.3

Exact OCBO OPSO WCA HS

0.25 0.2

Z(x)

0.15 0.1 0.05 0 −0.05 −0.1

0

0.5

1

1.5

2

2.5

3

3.5

x

Fig. 4 Response matching of the true solution and approximate solutions obtained by OCBO and comparative algorithms: a Example 1 b Example 2

The exact solution is given by u(t) =

    t 1 t 1 sin + cos 2 3 3 2

(48)

The approximate solution using proposed approach here also takes the form defined in Eq. (44). – Example 8: Third Order Differential Equation The third order exothermic reaction from [29] used for modelling the biological processes is represented by     t t t     u (t) = u (t) + et−1 u −u + f (t) (49) 3 3 2

123

Determining Approximate Solutions of Nonlinear Ordinary…

(a) 2.5 2

Z(x)

1.5 1 0.5

Exact OCBO OPSO WCA HS

0 −0.5

0

0.2

0.4

x

0.6

0.8

(b) 1.8

1

Exact OCBO OPSO WCA HS

1.6 1.4 1.2

Z(x)

1 0.8 0.6 0.4 0.2 0 −0.2

0

5

10

x

15

Fig. 5 Response matching of the true solution and approximate solutions obtained by OCBO and comparative algorithms: a Example 3 b Example 4

where

 8 7 f (t) = − et−1 − 2187 t +

2 5 27 t

+

5 4 81 t

−

16 3 27 t

+ 13 t 2 − 43 t + 3

 (50)

+

893 7 48 t

−

with initial conditions

5847 5 16 t

−

305 4 48 t

+ 374t 3

+

235 2 4 t

−

290 3 t

+9

⎧   ⎨ u(0) − u (0) − 2u (0) = 5   u (0) − u (0) = 7 ⎩   2u (0) + 3u (0) = −6

(51)

u(x, t) = −t 8 + 3t 6 + t 5 − 4t 4 + t 3 − 2t 2 + 3t

(52)

The exact solution is given by

123

A. Panda, S. Pani

(a) 2.5

Exact OCBO OPSO WCA HS

Vapp(t)

2

1.5

1

0.5

0

0

0.5

1

1.5

time (t)

(b) 2.5

Exact OCBO OPSO WCA HS

2

Iapp(t)

1.5 1 0.5 0 −0.5 −1

0

0.5

time (t)

1

1.5

Fig. 6 Response matching of the true solution and approximate solutions obtained by OCBO and comparative algorithms for Example 5: a Approximation Voltage b Approximation Current

Here also the approximate solution by the proposed method is represented in the form defined in Eq. (44).

6.3 Comparative Algorithms The performance of the proposed OCBO is compared with Orthogonal PSO [15], Water Cycle Algorithm (WCA) [10] and Harmonic Search [11]. Sadollah et al. in 2015 [10], have reported the superior performance of WCA for solving ODE over cuckoo search and particle swarm optimization(PSO). In literature [11], the Harmonic Search has reported better performance than genetic algorithm and PSO for determining approximate solutions of ODE of longitudinal fins (having rectangular, trapezoidal and concave parabolic profiles). In all four algorithms the initial population size is taken for Ex. 1–Ex. 2 is 50, Ex. 3–Ex. 5 is 200. The maximum number of iteration allowed to run for Ex. 1–Ex. 3 is 500, Ex. 4 is

123

Determining Approximate Solutions of Nonlinear Ordinary…

(a) 0.15

Exact OCBO OPSO WCA HS

0.1

U(t)

0.05

0

−0.05

−0.1

0

0.2

0.4

t

0.6

0.8

1

(b) 0.46 0.44

U(t)

0.42 0.4 0.38 0.36

Exact OCBO OPSO WCA HS

0.34 0.32

0

0.2

0.4

t

0.6

0.8

1

Fig. 7 Response matching of the true solution and approximate solutions obtained by OCBO and comparative algorithms for biological systems: a Example 6 b Example 7

100, Ex. 5 is 200, Ex. 6 is 100, Ex. 7 is 500 and Ex. 8 is 200. The total number of fitness evaluation from Ex. 1–Ex. 3 are 1,00,000, for Ex. 4 is 20,000, for Ex. 5 is 40,000, for Ex. 6 is 20,000, for Ex. 7 is 1,00,000 and Ex. 8 is 40,000. The rest parameter settings for WCA and HS are kept same as given in [10] and [11] respectively. The settings for orthogonal array using proposed OCBO are elaborated in Sect. 3. The parameter settings for OPSO is kept same as given in [15]. As all the algorithms used in this context are meta-heuristic in nature therefore twenty independent runs of each algorithm is considered. The best, average, worst and standard deviation values are considered during performance evaluation.

6.4 Simulation Environment The simulation studies are carried out in MATLAB R2011 platform. All the four algorithms are set to run in a Dell Latitude E5430 laptop with Intel i7-3540M 3GHz CPU, 8GB RAM in Windows 8.1 (64-bit) environment.

123

A. Panda, S. Pani 1.2 1

U(t)

0.8 0.6 0.4 0.2

Exact OCBO OPSO WCA HS

0 −0.2

0

0.2

0.4

0.6

0.8

1

t

Fig. 8 Response matching of the true solution and approximate solutions obtained by OCBO and comparative algorithms for biological system Example 8

6.5 Performance Evaluation In order to evaluate the performance of the algorithms the following criteria are considered: – Response matching The response matching between true analytical solution and obtained approximate solution reveals the closeness of the solutions. – Generalized distance It is the Euclidean distance between points lying on the exact and approximate solution [10]: " # S # GD = $ (53) (z − Z )2 s

s

s=1

where z s represent points true analytical solution, Z s are points approximate solution and S is the total number of points in both the curves. The minimum value of G D represents the approximate solution is nearest to the true solution. – Estimation of the Coefficients The best values of Fourier Coefficients obtained by the comparative algorithms reveals an near effective value of the approximation to the ODE. – Run time The average run time for fixed number of fitness evaluation is a measure of computational complexity of the algorithm.

7 Results and Discussion The results obtained over 20 independent runs by proposed OCBO, OPSO, WCA and HS are reported here for five examples of science-engineering problems described in Sect. 6.1 and three examples of biological modelling problems given in Sect. 6.2. The approximated value of Fourier coefficients obtained for science ODE problems (Ex. 1–Ex. 3) are reported in Table 1 and ODEs for electrical engineering problems are reported in Table 2. Similarly the approximated value of coefficients obtained for biological modelling problems are presented

123

Determining Approximate Solutions of Nonlinear Ordinary… Table 4 Comparative results of Generalized Distance obtained for 20 independent runs with OCBO, OPSO and WCA and HS Ex.

GD value

Ex. 1

Best

2.86E-06

3.26E-06

3.55E-06

4.26E-06

Average

5.42E-06

6.10E-06

6.49E-06

8.62E-06

Ex. 2

Ex. 3

Ex. 4

Ex. 5

OCBO

OPSO

WCA

HS

Worst

8.22E-06

8.86E-06

9.71E-06

12.51E-06

SD

2.21E-06

2.43E-06

2.45E-06

3.20E-06

Best

9.18E-05

1.02E-04

2.35E-04

4.12E-04

Average

3.51E-04

5.57E-04

7.83E-04

8.83E-04

Worst

5.99E-04

7.83E-04

1.21E-03

1.35E-03

SD

1.67E-04

1.96E-04

2.12E-04

2.54E-04

Best

6.62E-04

8.76E-04

1.14E-03

1.74E-03

Average

1.21E-03

1.85E-03

2.56E-03

3.12E-03

Worst

3.65E-03

4.98E-03

6.22E-03

7.31E-03

SD

6.11E-04

8.25E-04

1.03E-03

1.62E-03

Best

3.66E-03

3.80E-03

3.80E-03

4.12E-03

Average

4.54E-03

4.62E-03

4.64E-03

7.56E-03

Worst

7.66E-03

8.02E-03

8.10E-03

10.11E-03

SD

9.08E-04

1.04E-03

1.08E-03

2.21E-03

Voltage Best

4.08E-04

4.18E-04

4.25E-04

5.74E-04

Average

7.62E-03

8.12E-03

9.52E-03

1.24E-02

Worst

1.59E-02

1.98E-02

2.17E-02

2.63E-02

SD

2.38E-03

3.12E-03

5.34E-03

6.88E-03

Current

Ex. 6

Ex. 7

Ex. 8

Best

2.61E-03

3.24E-03

3.75E-03

4.32E-03

Average

9.25E-03

1.14E-02

1.23E-02

1.45E-02

Worst

1.42E-02

1.94E-02

2.73E-02

2.94E-02

SD

3.45E-03

4.12E-03

6.48E-03

6.75E-03

Best

6.22E-04

7.35E-04

8.14E-04

8.94E-04

Average

7.42E-04

8.54E-04

9.45E-04

9.88E-04

Worst

2.68E-03

3.12E-03

3.57E-03

3.99E-03

SD

1.18E-04

1.75E-04

2.16E-04

2.46E-04

Best

7.68E-04

8.07E-04

9.13E-04

9.84E-04

Average

9.32E-04

9.86E-04

1.23E-03

1.72E-03

Worst

2.43E-03

2.40E-03

2.68E-03

2.85E-03

SD

1.57E-04

1.74E-04

1.94E-04

2.10E-04

Best

1.25E-03

1.71E-03

2.53E-03

2.91E-03

Average

2.20E-03

2.80E-03

3.64E-03

4.05E-03

Worst

3.12E-03

3.91E-03

4.81E-03

5.42E-03

SD

1.02E-03

1.11E-03

1.23E-03

1.18E-03

123

A. Panda, S. Pani Table 5 Comparative results of run time obtained for 20 independent runs with OCBO, OPSO, WCA and HS

Ex.

Comp. time

OCBO

Ex. 1

Best

12.4510

15.3222

9.0202

10.1240

Average

12.6870

15.5260

9.2042

10.3250

Worst

10.4246

Ex. 2

Ex. 3

Ex. 4

Ex. 5

Ex. 6

Ex. 7

Ex. 8

OPSO

WCA

HS

12.8010

15.7500

9.3721

SD

0.1778

0.2140

0.1760

0.1642

Best

15.9212

18.6234

11.4116

12.8073

Average

15.9844

18.7015

11.4730

12.9124

Worst

16.1006

18.8425

11.5488

13.0112

SD

0.0816

0.1226

0.0798

0.1304

Best

14.7314

17.4014

10.6225

11.9700

Average

14.7862

17.4621

10.6310

11.9762

Worst

14.8405

17.5123

10.6375

11.9805

SD

0.0658

0.0712

0.0850

0.0755

Best

16.6324

19.0210

12.1022

13.2120

Average

16.7514

19.2631

12.2310

13.5024

Worst

16.8620

19.4100

12.3532

13.8218

SD

0.1148

0.1964

0.1255

0.2506

Best

15.7344

18.5017

11.3008

12.5842

Average

15.7912

18.5869

11.3120

12.5961

Worst

15.8413

18.6210

11.3224

12.6210

SD

0.0714

0.0902

0.0821

0.0935

Best

14.9115

17.7526

10.9025

12.2133

Average

14.9743

17.7920

10.9504

12.3025

Worst

15.0520

17.8621

10.9805

12.3843

SD

0.0591

0.0635

0.0412

0.0762

Best

15.2325

17.9624

11.4110

12.5312

Average

15.2947

18.0328

11.4562

12.6024

Worst

12.6840

15.3504

18.1004

11.4987

SD

0.0621

0.0658

0.0425

0.0780

Best

17.5328

20.4020

14.0020

15.7125

Average

17.5836

20.4621

14.0436

15.7720

Worst

17.6440

20.5203

14.0866

15.8262

0.0524

0.0584

0.0412

0.0645

SD

in Table 3. In all the eight examples it is observed that the obtained coefficients by the meta-heuristic algorithms are nearby close to each other. The plot of true analytical response and the proposed approximate response obtained with Fourier Series coefficient (values obtained by the four meta-heuristic algorithms reported in Tables 1 and 2) for all the five examples of science-engineering problems are shown in Figs. 4, 5 and 6 (Fig. 4 for Ex. 1–2, Fig. 5 for Ex. 3–4 and Fig. 6 representing approximate voltage and current value in Ex. 5). Similarly the response matching for the three biological modelling problems are presented in Figs. 7 and 8 (Fig. 7 for Ex. 6–7 and Fig. 8 corresponds to Ex. 8). Close response matching between the obtained curves with respect to the exact solution justifies the significance of the proposed method. In most of the cases on a close look

123

Determining Approximate Solutions of Nonlinear Ordinary…

at the curves it is observed that the red line (OCBO response) is close to the exact analytical curve, the blue line (OPSO response) marginally overshoots it. The green (WCA response) and dotted line (HS response) are slightly lower than the true response (under-fitting). The similar trend is also observed in Tables 1, 2 and 3. In these problems true parameters are unknown. But comparison between most parameters (in Tables 1, 2 and 3) reveals higher values for OPSO and lower values for WCA and HS compared to the OCBO values. The best, average, worst and standard deviation values of G D achieved over 20 independent runs are presented in Table 4. In all the eight examples, among the four algorithms the OCBO based learning results in lower G D values which are highlighted in bold letters. Except Ex. 4 in all the rest cases mostly the superiority in terms of G Dare OCBO followed by OPSO, WCA and HS. In case of Ex. 4 the GD values of OPSO and WCA are approximately equal, though from Fig. 5b the blue line overshoots where as green line under-fits. Therefore in this case the approximate average distance between elements of green and blue curve are same from the black solid curve (exact) though they lie on both sides of the exact solution. The computational complexity is not a matter in this case as the applications taken here are offline in nature. However to know the comparative value of complexity run time of the four algorithms obtained over the 20 independent runs are presented in Table 5. In all the eight examples it is observed that by combining the orthogonal array the overall computational time of the algorithm increases. It is due to the time required to generate orthogonal combination. Thus the run time of WCA is lower followed by HS, than that of the orthogonal algorithms (OCBO and OPSO). Earlier it has been reported that the CBO has lower computational complexity than the PSO for solving multi-objective [21] and constrained optimization problems [22]. In this case also after combining with the orthogonal array the complexity of OCBO in all the five examples reported lower than OPSO.

8 Conclusion In this paper an attempt is made to determine approximate solution of nonlinar ordinary differential equation (ODE). Fourier series expansion is used as the approximator and it’s coefficients are optimized by meta-heuristic algorithms. A hybrid algorithm OCBO is derived by combining orthogonal array with colliding bodies optimization. Simulation studies were carried out on three popular applications of ODE in science, two applications in electrical engineering and three applications on biological process modelling. Comparative analysis is carried out with results obtained by Orthogonal PSO, water cycle algorithm and Harmonic Search over twenty independent runs. Superior performance of OCBO is reported over counter part algorithms in terms of close response matching of the approximated solution with the exact analytical solution, lower generalized distance value. It is also observe that adding orthogonal array enhance the accuracy but simultaneously computational complexity of the process also increase slightly due to incorporation of additional steps of creating the array. Thus complexity of WCA and HS are lower. However in the eight examples the proposed OCBO has lower complexity than the OPSO algorithm. Compliance with Ethical Standards Conflict of interest The authors declare that they have no conflict of interest. Human Participants or Animals Further, this manuscript does not contain any studies with human participants or animals performed by any of the authors.

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