Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation

Abdulrazzaq T. Abed; Azzam S. Y. Aladool

doi:10.3934/naco.2021008

Article Contents

2022, Volume 12, Issue 2: 321-337. Doi: 10.3934/naco.2021008

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Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation

Abdulrazzaq T. Abed and
Azzam S. Y. Aladool^,

Department of Mathematics, College of Education for Pure Science, University of Mosul, Mosul, Iraq

^* Corresponding author: Azzam S. Y. Aladool
^* Corresponding author: Azzam S. Y. Aladool

Received: December 2020

Revised: February 2021

Early access: March 2021

Published: June 2022

Abstract / Introduction Full Text(HTML) Figure(7) / Table(6) Related Papers Cited by

Abstract

Ordinary differential equations are converted into a constrained optimization problems to find their approximate solutions. In this work, an algorithm is proposed by applying particle swarm optimization (PSO) to find an approximate solution of ODEs based on an expansion approximation. Since many cases of linear and nonlinear ODEs have singularity point, Padé approximant which is fractional expansion is employed for more accurate results compare to Fourier and Taylor expansions. The fitness function is obtained by adding the discrete least square weighted function to a penalty function. The proposed algorithm is applied to 13 famous ODEs such as Lane Emden, Emden-Fowler, Riccati, Ivey, Abel, Thomas Fermi, Bernoulli, Bratu, Van der pol, the Troesch problem and other cases. The proposed algorithm offer fast and accurate results compare to the other methods presented in this paper. The results demonstrate the ability of proposed approach to solve linear and nonlinear ODEs with initial or boundary conditions.

Keywords:

Linear and nonlinear ordinary differential equations,
Particle swarm optimization,
Padé approximant,
Discrete least square weighted function.

Mathematics Subject Classification: Primary: 34Bxx, 65Lxx, 65Kxx, 41A21; .

Citation:

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Figure 1. shows a plot of the resulting approximate solution with the corresponding exact solutions.

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Figure 2. Shows numerical error between the approximate solution and exact solution arising from the proposed algorithm in four examples.in NLIVP5, NLIVP6, NBVP1 and NBVP3 respectively.

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Figure 3. Show the convergence of PSO-PEA-DLSWF algorithm in 500 iteration.

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Figure 4. Comparison of expansions: PEA, TEA and FEA to consumed time.

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Figure 5. shows the simulating of example (NLBVP3) in terms of errors, convergence and stability.

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Figure 6. Comparison between the fitness function: WRF, LSWF and DLSWF with consumed time.

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Figure 7. shows the simulating of example (NLBVP1) in terms of errors, convergence and stability.

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Table 1. Illustrates different type of the ordinary differential equations, with its conditions, domain and exact solution.

ODE NAME	ODE EQUATION	I.C. or B.C DOMAIN	EXACT SOLUTION
Simple ODEs1 LIVP1	$y'=\frac{y}{x}+1$	$y(1)=0$ $x\in[1,2]$	$y=xln(x)$
Simple ODEs2 LIVP2	$y''-y=e^x$	$y(0)=0$ $y'(0)=1$ $x\in[0,1]$	$y=0.25(e^x+2xe^x-e^{-x})$
Simple Harmonic LBVP1	$y''+25y=0$	$y(0)=1$ $y(\frac{\pi}{10})=0$ $x\in[0,\frac{\pi}{10}]$	$y=\cos(5x)$
Riccati NLIVP1	$y'=1+2y-y^2$	$y(0)=0$ $x\in[0,1]$	$y=1+\sqrt{2} \tanh{(\sqrt{2}x}+\frac{1}{2} ln(\frac{\sqrt{2}-1}{\sqrt{2}+1}))$
Abel NLIVP2	$y'=1-x^2 y+y^3$	$y(0)=0$ $x\in[0,1]$	$y=x$
Ivey NLIVP3	$y''=\frac{(y' )^2}{y}-\frac{2y'}{x}-2y^2$	$y(1)=1$ $y'(1)=-2$ $x\in[1,2]$	$y=\frac{1}{x^2}$
Lane- Emden NLIVP4	$y''+2\frac{y'}{x}=-4(2e^y+e^{\frac{y}{2}})$	$y(0)=0$ $y'(0)=0$ $x\in[0,1]$	$y=-2ln(1+x^2)$
Emden-Fowler NLIVP5	$y''+6\frac{y'}{x}+14y+4ylny=0$	$y(0)=1$ $y'(0)=0$ $x\in[0,1]$	$y=e^{-x^2}$
Van Der Pol NLIVP6	$y''+y=0.05(1-y^2 )y'$	$y(0)=0$ $y'(0)=0.5$ $x\in[0,1]$	Sol. By ode45 solver in MATLAB
Bratu-Gelfand NLBVP1	$y''-\pi^2 e^y=0$	$y(0)=0$ $y(1)=0$ $x\in[0,1]$	$y=-2ln(\sqrt{2}\cos(\pi(\frac{x}{2}-\frac{1}{4})))$
Thomas-Fermi NLBVP2	$y''=\sqrt{\frac{y^3}{x}}$	$y(1)=144$ $y(2)=18$ $x\in[1,2]$	$y=\frac{144}{x^3}$
Bernoulli NLBVP3	$y''+(y')^2=2e^{-y}$	$y(0)=0$ $y(1)=0$ $x\in[0,1]$	$y=\log{(0.75+(x-0.5)^2)}$
Troesch Problem NLBVP4	$y''=\sinh{y}$	$y(0)=0$ $y(1)=1$ $x\in[0,1]$	Sol. By bvp4c solver in MATLAB

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Table 2. the values of parameters and inputs used in all example.

parameter	$ \phi_1 $	$ \phi_2 $	$ \kappa $	$ \rho $	$ \sigma $	$ h $	$ TOL $	$ K_m \forall m $	$ Maxit $	$ nPop $
value	2.05	2.05	1	0.8	0.4	0.01	1e-03	1 or 10	500	200

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Table 3. Search space and nVar sitting of each example.

Equation	PEA			TEA			FEA
	VarMin	VarMax	nVar	VarMin	VarMax	nVar	VarMin	VarMax	nVar
LIVP1	-1	1	10	-2	2	10	-2	2	9
LIVP2	-2	2	10	-2	2	10	-2	2	11
LBVP1	-3	3	10	-3	3	10	-1	1	11
NLIVP1	-2	2	10	-2	2	10	-2	2	9
NLIVP2	0	1	10	0	1	10	-1	1	11
NLIVP3	0	1	10	-3	3	10	-2	2	9
NLIVP4	-2	2	10	-2	2	10	-1	1	11
NLIVP5	-1	1	10	-1	1	10	-1	1	9
NLIVP6	-1	1	10	-2	2	10	-0.5	0.5	9
NLBVP1	-1	1	10	-3	3	10	-1	1	9
NLBVP2	0	1	10	-900	900	10	-200	200	9
NLBVP3	-2	2	10	-3	3	10	-1	1	9
NLBVP4	-4	4	10	-1	1	10	-1	1	9

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Table 4. shows the values of variables founded by using the PSO-PEA-DLSWF algorithm.

Eq.	Coff,	$ m=1 $	$ m=2 $	$ m=3 $	$ m=4 $	$ m=5 $
LIVP1	$ \alpha_m $	0.449646621	-0.955885566	-0.662451518	0.5990158869	0.5696745764
	$ \beta_m $	-0.383029549	1.000000000	0.9928258613	0.2090703809	-0.01223876
LIVP2	$ \alpha_m $	1.5905e-08	-2.000000000	-1.99993129	-1.999993802	1.320889355
	$ \beta_m $	-2.000000000	-1.028027457	-0.640309044	1.893006788	-0.627961135
LBVP1	$ \alpha_m $	-0.23700822	0.2939052397	1.7497711122	-0.815400817	-0.28105974
	$ \beta_m $	-0.237008298	0.2940587929	-1.172283084	1.983810204	-2.525130497
NLIVP1	$ \alpha_m $	-6.595e-07	-2.000000000	-1.969510526	-2.000000000	-1.99980317
	$ \beta_m $	-1.999999998	0.1334264437	-1.999346637	0.2828697917	-1.132644395
NLIVP2	$ \alpha_m $	0.00000000	1.00000000	1.00000000	0.9159523195	1.00000000
	$ \beta_m $	1.00000000	1.00000000	0.9159523195	1.00000000	0.00000000
NLIVP3	$ \alpha_m $	1.00000000	0.501789937	0.3963523812	0.00000000	0.00000000
	$ \beta_m $	0.00000000	0.00000000	1.00000000	0.501789937	0.3963523812
NLIVP4	$ \alpha_m $	-3.556e-10	1.1605e-09	1.371544377	1.705016485	1.664659425
	$ \beta_m $	-0.699013232	-0.795001437	-1.237894199	-0.487132482	-0.20116878
NLIVP5	$ \alpha_m $	-1.0000000	-0.999999714	0.9996994842	1.00000000	-0.670316746
	$ \beta_m $	-0.999999761	-1.0000000	0.0694451202	-0.363172581	0.4710444283
NLIVP6	$ \alpha_m $	9.8817e-08	0.499999967	-0.128335779	-0.225730262	0.4574883022
	$ \beta_m $	1.00000000	-0.334983343	0.0548712929	0.1919886974	0.4900184664
NLBVP1	$ \alpha_m $	6.0988e-07	-1.0000000	-0.168001861	0.1680024128	1.00000000
	$ \beta_m $	0.3182717396	0.5556535208	0.2683350684	0.1086186654	-0.242687286
NLBVP2	$ \alpha_m $	1.00000000	0.00000000	0.00000000	0.00000000	0.00000000
	$ \beta_m $	0.00000000	0.00000000	0.00000000	0.0069444444	0.00000000
NLBVP3	$ \alpha_m $	-2.979e-11	-1.512033065	0.5287945264	-0.376563712	1.359802251
	$ \beta_m $	1.512152004	0.2390046775	1.385502167	0.2230003398	0.4953153062
NLBVP4	$ \alpha_m $	2.6966e-06	-3.398796238	-0.916386583	-3.997792287	-2.079546692
	$ \beta_m $	-4.00000000	-1.420908971	-2.843010022	-3.999816103	1.8712207261

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Table 5. Comparison of expansions: PEA, TEA and FEA. The capital letters P, T and F denotes when PEA, TEA and FEA is the best results respectively.

Equation	PEA			TEA			FEA
	No. It	RMSE	Time (s)	No. It	RMSE	Time (s)	No. It	RMSE	Time (s)
LIVP1	047	6.98e-04	26.79	113	9.70e-04	37.70	349	3.28e-02	090.10	P
LIVP2	017	3.35e-04	10.37	062	8.50e-04	21.39	500	4.68e-02	130.30	P
LBVP1	019	6.49e-04	09.25	081	9.87e-04	21.64	128	9.69e-04	027.38	P
NLIVP1	049	9.53e-04	25.92	122	9.90e-04	39.58	500	1.50e-02	093.13	P
NLIVP2	006	4.95e-04	03.61	003	0.00e-00	02.06	500	2.90e-02	193.67	T
NLIVP3	006	6.83e-04	09.67	500	1.36e-03	247.7	500	9.90e-03	233.29	P
NLIVP4	028	9.80e-04	28.07	500	8.60e-03	189.1	500	6.00e-02	181.89	P
NLIVP5	008	6.70e-04	7.122	500	2.60e-03	238.3	500	2.30e-02	201.20	P
NLIVP6	014	7.95e-04	18.14	040	9.90e-04	36.58	500	9.10e-03	339.87	P
NLBVP1	014	9.01e-04	08.89	500	2.10e-02	151.0	500	4.80e-03	152.02	P
NLBVP2	013	8.69e-04	12.08	500	2.14e-01	143.2	500	6.30e-01	126.00	P
NLBVP3	014	8.59e-04	16.40	500	5.80e-03	158.5	500	1.49e-03	226.66	P
NLBVP4	013	4.58e-04	10.62	500	2.30e-03	200.7	500	2.15e-03	121.30	P

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Table 6. Comparison of fitness function: WRF, LSWF and DLSWF. W, L and D symbols indicates that method PEA-PSO-WRF, PEA-PSO-LSWF and PEA-PSO-DLSWF is the best results respectively.

Equation	WRF			LSWF			DLSWF
Equation	No. It	RMSE	Time (s)	No. It	RMSE	Time (s)	No. It	RMSE	Time (s)
LIVP1	012	8.03e-04	14.13	014	5.91e-03	22.88	039	1.24e-03	58.56	W
LIVP2	012	8.98e-04	15.68	016	9.57e-04	29.54	030	6.09e-04	22.33	W
LBVP1	031	6.76e-04	23.92	027	9.14e-04	25.76	018	9.81e-04	11.42	D
NLIVP1	212	8.38e-04	270.9	150	9.94e-04	182.0	057	9.51e-04	77.09	D
NLIVP2	004	1.42e-17	07.64	003	1.44e-17	04.26	003	0.00e-00	07.48	L
NLIVP3	003	3.79e-17	08.43	003	3.79e-17	08.77	003	3.79e-17	07.71	D
NLIVP4	077	9.81e-04	172.9	246	8.19e-04	486.1	016	8.96e-04	39.77	D
NLIVP5	013	7.60e-04	40.40	018	8.52e-04	36.23	009	6.52e-04	25.20	D
NLIVP6	027	7.14e-04	85.23	018	7.86e-04	39.10	012	9.79e-04	23.62	D
NLBVP1	034	6.13e-04	54.26	074	9.14e-04	114.2	018	8.97e-04	33.88	D
NLBVP2	019	1.86e-04	35.08	027	7.55e-05	48.71	012	2.52e-04	17.02	D
NLBVP3	016	3.75e-04	27.01	011	8.91e-04	18.40	018	9.11e-04	28.41	L
NLBVP4	015	8.11e-04	25.80	016	7.19e-04	22.62	021	8.84e-04	24.08	L

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