MJ Journal on Differential Equations 1 (1) (2016) 15-21 Website: www.math-journals.com/index.php/JDE doi: 10.14419/jde.v1i1.81 Research paper
Semi-analytic technique to solve Emden_fowler type differential equations Luma. N. M. Tawfiq *, Reem. W. Hussein Department of Mathematics, College of Education for Pure Science / Ibn Al-Haitham, Baghdad University, Iraq *Corresponding author E-mail: dr.lumanaji@yahoo.com
Abstract The aim of this paper is to solve second order Emden–Fowler equations as a type of singular ordinary differential equations. We suggest the semi-analytic technique by using two point osculator interpolation method to overcome the difficulty at the singular point of non-homogeneous, non-linear Emden–Fowler equations; especially when the singularity appears on the right-hand side of this type of equations. Also, various examples are introduced to illustrate the suggested method. The results of the examples are compared with exact solutions when available, also, compared with other methods to assign the effectiveness and accuracy of the suggested method. Keywords: Differential Equations, Singular Initial Value Problems, Emden–Fowler Equations, Semi-Analytic Technique. AMS Subject Classification: 34L05, 35A05, 35A08, 35A20, 35A21.
1. Introduction In the recent years the studies of singular initial value problems (SIVP′s) in the second order ordinary differential equations have attracted the attention of many mathematicians and physicists. The search for efficient numerical methods to solve SIVP is strongly motivated by numerous applications from physics, chemistry, mechanics, ecology, or economics [1]. [2]. Also, research activities in related fields, like the computation of connecting orbits in dynamical systems [3], [4], or singular Sturm-Liouville problems [5] . Hoog et al [6] applied a theory for linear multi-step schemes to the IVP for a nonlinear first order system of differential equations with a singularity of the first kind, also, de Hoog et al [7] used explicit R-K schemes to solve SIVP with singularity of first kind. Tawfiq et al [8] solved some classes of Lane-Emden type equations using semi - analytic technique. K1ymaz et al [9] used differential transformation method (DTM) to solve second order SIVP but Ramos [10] used linearisation methods to solve non-linear Lane – Emden equations as a SIVP in second order ODE. Hasan [11] solved non-linear, non-homogeneous SIVP by using Taylor series method. Demir [12] used DTM to solve the Emden-Fowler type equations in the second order singular ordinary differential equations. Mohyud-Din et al [13] used He’s polynomials for solving second order SIVP, the suggested algorithm is tested on Emden-Fowler equation. Tripathi [14] gave solutions of a subclass of singular second order differential equations of Lane-Emden and Emden – Fowler type. In this paper a semi-analytic technique used for constructing polynomial solutions of Emden-Fowler type differential equations using two-points osculatory interpolation which is based on Taylor series expansion. In fact this method takes computationally short time of the necessary derivatives of the data functions rather than Taylor series method. Thus an important feature of the paper is the use of the symbolic computational package MATLAB in the process of implementation together with its IVP library codes as a checking device.
2. Emden-fowler equation In this paper, we study the second order nonlinear Emden–Fowler equations of the form: y" + (a/x) y' + b xm-1 yn = g(x) ; 0 ≤ x ≤ 1, a 0, b ϵ R
(1)
Under the following initial conditions: y (0) = A, y'(0) = 0 Where A, a, b, m are constants, n ≠ 0, n ≠ 1. When m = 1, a = 2 and b = 1, Eq. (1) reduces to the Lane – Emden equation [15].
Copyright © 2016 Luma. N. M. Tawfiq, Reem. W. Hussein. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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3. Osculator interpolation polynomial In this paper we use two-point osculatory interpolation polynomial; essentially this is a generalisation of interpolation using Taylor polynomials. The idea is to approximate a function y by a polynomial P in which values of y and any number of its derivatives at a given points are fitted by the corresponding function values and derivatives of P [16]. We are particularly concerned with fitting function values and derivatives at the two end points of a finite interval, say [0, 1], i.e., P (j)(xi) = f(j)(xi), j = 0, . . . , n, xi = 0, 1, where a useful and succinct way of writing osculatory interpolant P 2n+1 of degree 2n + 1 was given for example by Phillips [16] as: n
P2n+1(x) =
{y
( j)
(0) q j (x) + (-1)
j
y
( j)
(1) q j (1-x) },
(2)
xs = Q j (x) / j!,
(3)
j 0
j
q j (x) = ( x / j!)(1-x)
n 1
n s s
n j
s 0
So that (2) with (3) satisfies:
y
( j)
( j)
(0) =
( j)
P2 n1 (0) , y ( j ) (1) = P2 n1 (1) , j = 0, 1, 2,…, n .
Implying that P2n+1 agrees with the appropriately truncated Taylor series for y about x = 0 and x = 1. We observe that Eq. (2) can be written directly in terms of the Taylor coefficients
and
about x = 0 and x = 1 respectively, as:
n
P2n+1(x) =
{ aj Q j (x) + (-1)
j
bj Q j (1-x) },
(4)
j 0
4. Solution of the second order emden-fowler equations In this section, we suggest a semi-analytic technique which is based on osculatory interpolating polynomials P2n+1 and Taylor series expansion to solve second order Emden-Fowler equations. A general form of second order Emden-Fowler equations is : y"(x) = f( x, y, y' ), 0 ≤ x ≤ 1
(5a)
Subject to the initial condition(IC) Y (0) = A, y (1) = B,
(5b)
Where f, in general, nonlinear function of their arguments. Now, we solve the problem by the suggested method by doing the following steps: Step one:Evaluate Taylor series of y(x) about x = 0:
yy= i 0 ai x i
= a 0 + a1 x +
i 2
ai x
i
(6)
Where y (0) = a0, y' (0) = a1, y"(0) / 2! = a2… y (i) (0) / i! = ai, i= 3, 4… And evaluate Taylor series of y(x) about x = 1:
yy= i 0 bi ( x 1) i = b 0 + b 1 (x-1) +
i 2
b i (x-1)
i
(7)
where, y(1) = b0, y'(1) = b1, y"(1) / 2! = b2 , … , y(i)(1) / i! = bi , i = 3,4,… Step two: Insert the series form Eq. (6) into Eq. (5a) and put x = 0, then equate the coefficients of powers of x to obtain a 2. Insert the series form Eq. (7) into Eq. (5a) and put x = 1, then equate the coefficients of powers of (x-1) to obtain b2. Step three: Derive Eq. (5a) with respect to x, to get new form of equation say (8) as follows:
y ' ' ' ( x)
df ( x, y, y ' ) dx
(8)
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Then, insert the series form Eq. (6) into Eq. (8) and put x = 0 and equate the coefficients of powers of x to obtain a 3, again insert the series form Eq. (7) into Eq. (8) and put x = 1, then equate the coefficients of powers of (x-1) to obtain b3. Step four:Iterate the above process many times to obtain a4, b4 then a5, b5 and so on, that is, to get ai and bi for all i ≥ 2, the resulting equations can be solved using MATLAB version 7.10, to obtain ai and bi for all i 2. Step five:The notation implies that the coefficients depend only on the indicated unknowns a0, a1, b0, b1, , use the IC to get two coefficients from these, i.e., a0, a1, therefore, we have only two unknown coefficients: b 0, b1. Now, we can construct two point osculatory interpolating polynomial P2n+1(x) by insert these coefficients ( aiۥs and biۥs ) into Eq. (4). Step six:To find the unknowns coefficients integrate equation (5a) on [0, x] to obtain: x
y' (x) – y'(0) –
f(s, y, y') ds = 0
(9a)
0
And again integrate Eq. (9a) on [0, x] to obtain: x
Y (x) – y (0) – y'(0) x –
(1– s) f(s, y, y') ds = 0
(9b)
0
Step seven:Putting x = 1 in equations (9) to get: 1
b1 – a1 –
f(s, y, y') ds = 0
(10a)
0
And 1
b0 – a0 – a1 – (1– s) f(s, y, y') ds = 0
(10b)
0
Step eight:Use P2n+1(x) which constructed in step five as a replacement of y(x), we see that Eq. (10) have only two unknown coefficients from b0, b1. So, we can find these coefficients by solving the system of algebraic Eq. (10) using MATLAB, so insert the value of the unknown coefficients into Eq. (4), thus Eq. (4) represent the solution of the problem.
5. Example In this section, we investigate the theory using examples of Emden_Fowler equation. The algorithm was implemented in MATLAB 7.10. The bvp4c solver of MATLAB has been modified accordingly so that it can solve Some class of Emden_Fowler equation Also, we report a more conventional measure of the error, namely the error relative to the larger of the magnitude of the solution component and taking advantage of having a continuous approximate solution, we report the largest error found at 10 equally spaced points in [0, 1]. Example 1 Consider the following nonlinear Emden _ Fowler equation: y" + (8/x) y' + x y2 = x5 + x4 ; 0 ≤ x ≤ 1 With IC : y(0) = 1, y'(0) = 0 It is clear that x= 0 singular point of the first kind, we solve this example using semi-analytic technique, from Eq. (4), we have: P9 = – 0.00078826882 x9 + 0.0012952791 x8 + 0.00879760172 x7 + 0.014414828109 x6– 0.000150192x5 – 0.0333333333337578 x3 + 1.0 Higher accuracy can be obtained by evaluating higher n, now, take n= 5, i.e., P11= – 0.00021822695x11+ 0.00086427359x10- 0.00189361335x9+ 0.0014538812 x + .009581210401 x7 + 0.0137816833 x6 0.033333333333758 x3 + 1.0 Now, increase n, to get higher accuracy, let n = 6, i.e. P13= - 0.0000996316x13 + 0.0004488478x12 - 0.000849659317x11 + 0.0007848859x10 - 0.0007538654 x9 + 0.0001852901x8 + 0.01017813169x7 + 0.01367521368x6 - 0.03333333333375776x3 + 1.0
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Demir et al [12] solved this example by DTM and gave the following solution:
Also, this example is solved using numerical method with the Maple11 given in [12] and the results compared with the suggested method and given in Table 1a and 1b. Fig. 1 gave a comparison between the numerical solution in [12] and P 13. b0 b1 xi 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Table 1a: Result Of Suggested Method for N = 2, 3, 4and Method in [12] For Example1 0.990235914910346 0.990235874852203 0.0505890348752984 0.050589341476757 Numerical solution (y)in[12] P9 P11 1 1 1 0.99996668000000 0.999966680471501 ..999966681419201 0.999734330000000 0.999734323342549 0.999734340836996 0.999112190000000 0.999112136946651 0.999112205091858 0.997939330000000 0.997939228062444 0.997939352185325 0.996126220000000 0.996126122885822 0.996126243561995 0.993720970000000 0.993720947623996 0.993720981359821 0.9910046300000000 0.991004695132483 0.991004619559273 0.988619280000000 0.988619382084590 0.988619263799973 0.987723192000000 0.987731993184585 0.987731909996439 0.990235880000000 0.990235914910346 0.990235874852203
0.990235879509409 0.050589305215627 P13 1 .0.999966681360863 0.999734338980186 0.999112195858900 0.997939333529043 0.996126225634366 0.993720978536793 0.991004633063186 0.988619280474356 0.987731920033536 0.990235879509409
the soulution at n=6 1 Numerical solution p13
y
0.995
0.99
0.985
0
0.1
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
0.9
1
Fig. 1: Comparison between the Numerical Solution in [12] and P13 for Example 1.
xi 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 SSE
Table 1b: Comparison between Suggested Method & Method in [12] For Example 1 Error |y-P9| Error |y-P11| Error |y-P13| 0 0 0 4.715006163279999e-010 1.419200645891300e-009 1.360863421950850e-009 6.65745103487000e-009 1.083699618931180e-008 8.980186372298251e-009 5.305334893623799e-008 1.509185754144940e-008 .5.858900142818871e-009 1.019375558408700e-007 2.218532491049530e-008 3.529043057781680e-009 9.711417825464299e-008 2.356199502173690e-008 5.634365973783420e-009 2.237600371657300e-008 1.135982052868910e-008 8.536792939217721e-009 6.513248329564900e-008 1.044072739375680e-008 3.063186371221600e-009 1.020845900034930e-007 1.620002665347900e-008 4.743559989250900e-010 7.318458528082800e-008 1.000356053548760e-008 3.353606281564000e-011 3.491034628044800e-008 5.147797188520500e-009 4.905914563480000e-010 4.442053594825927-014 2.021641245529962e-015 2.437493949383256e-016
Example 2: Consider the following nonlinear Emden_Fowler equation: y" + (3/x) y' + 2 x2 y2 = 0 ; 0 ≤ x ≤ 1 With IC : y(0) = 1, y'(0) = 0 It is clear that x= 0 is nonlinear singular point of the first kind, we solve this example using semi-analytic technique, from Eq. (4), we have:
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P9 = – 0.0042317087 x9 + 0.015796436085 x8 – 0.01293372442 x7 + 0.00670634 x6 – 0.00134516637 x5 – 0.0833333333339 x4 + 1.0 higher accuracy can be obtained by evaluating higher n, now, take n= 5, i.e., P11 = 0.00025504311 x11 – 0.0027057371983 x10 + 0.00541846451 x9 – 0.0008254728 x8 + 0.002254816286 x7 – 0.00040522198 x6 – 0.0833333333339 x4 + 1.0 Now, increase n, to get higher accuracy, let n = 6, i.e. P13 = 0.0002435144x13–0.0010236556x12+ 0.00130252139x11– 0.00111969692x10 + 0.0005528258x9 + 0.0040204432x8 + 0.000015911476x7 – 0.0833333333339x4 + 1 Demir et al. [12] solved this example by DTM (n =20) and gave the following solution:
Also, this example solved using numerical method with the Maple11 [12] and the results compared with the suggested method which illustrated in Table 2. Fig. 2 gave a comparison between the numerical solution in [12] and P 13. b0 b1 xi 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Table2: Comparison between Suggested Method and Method in [12] for Example 2 0.920658530505436 -0.302070388178564 Numerical solution(y)in[12] P13 Error |y-P13| 1 1 0 0.9999916600 0.999991666708916 000000006708915645248226 0.999866670000000 0.999866677353741 00000007353740838489387 0.999325270000000 0.999325273332881 0.00000000333288063814052 0.997869390000000 0.997869394176759 00000000417675916075666. 0.994807890000000 0.994807897922098 00000007922098421708768 0.989269580000000 0.989269589539375 0.00000000953937473457955 0.980229370000000 0.980229373226163 00000003226163003411386 0.966553400000000 0.966553422481230 00000002248123009973568 0.947068570000000 0.947068573806556 00000003806555626262309 0.920658530000000 0.920658530505436 0.00000000050543647045486 SSE e-0168.119589439030920 the solution at n=6 1 Numerical solution p13
0.99 0.98
y
0.97 0.96 0.95 0.94 0.93 0.92
0
0.1
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
0.9
1
Fig. 2: Comparison between the Numerical Solution and P13 of Example 2.
6. Error / defect weights Every known IVP software package reports an estimate of either the relative error or the maximum relative defect. The weights used to scale either the error or the maximum defect differ among IVP software. Therefore, the IVP component of pythODE allows users to select the weights they wish to use. The default weights depend on whether an estimate of the error or maximum defect is being used. If the error is being estimated, then the IVP component of pythODE used, in this paper is modified this package to consist Emden – Fowler and is named "pythSODE", which is defined as: ║y(x) − P(x) ║∞ / (1 + ║P(x) ║∞), 0 ≤ x ≤ 1,
(11)
Where y (x) is exact solution and P (x) is suggested solution of Emden _ Fowler equation. If the maximum defect is being estimated, then the Emden – Fowler component of pythSODE uses: ║P(x) −f(x, P(x), P(x)) ║∞ / (1 +║f(x, P(x), P(x)) ║∞); 0 ≤ x ≤ 1, The relative estimate of both the error and the maximum defect are slightly modified from the one used in SIVP SOLVER. We apply this package for example 1, to Eq. (12) as the following:
(12)
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║P13(x) - f(x, P13(x), P13(x)) ║∞ / (1 + ║f(x, P13(x), P13(x)) ║∞) = E-007 / 1.999966681360863 = 8.419941391601780e-008. 1.683960224221478 For more details see Table 3. Table 3: Max. Error of Example 1 b0 b1 xi 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.990235879509409 0.050589305215627 P''13 -0.019954693976617 -0.039206687602793 -0.055639331969225 - 0.065139098987678 -0.061118434331738 -0.034078685565394 0.028767057993287 0.143788913036560 0.331125863858524 0.614718461206487 Max. Error
Error | P''13-f(x, P13 (x), P'13(x) | 9.826181241272956e-009 9.851733395474671e-008 1.564345449579437e-007 6.350270132559042e-009 e-0071.683960224221478 1.159391418267258e-007 4.083249415301991e-008 6.01923548237699e-008 7.582132859063062e-009 4.761522773288895e-012 e-0071.683960224221478
Again apply Eq. (12) for example 2 as the following: ║P13(x) - f(x, P13(x), P13(x)) ║∞ / (1 + ║f(x, P13(x), P13(x)) ║∞) = = 3.198141909012086e-008 1.999991666708916e-008 / 6.396257166976717 For more details see Table 4.
7. Conclusion In this paper, the semi-analytic technique is proposed to find the solution of second order nonlinear Emden–Fowler equations as a class of nonlinear SIVP, which plays very important role because of the singularity behavior at the origin and its considering generally form from Lane-Emden equation as illustrated in section 2. It is shown that suggested method is a very fast convergent, very powerful, precise and efficient tool for solving the Emden-Fowler equations, which achieves to the approximate solution of the SIVP, which is easy to implement with less computational work comparing with other numerical methods. Table 4: Max. Error of Example 2 b0 b1 xi 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.92065853050543645 -0.3020703881785643 P''13 - 0.009999765069596 - 0.039985065075665 - 0.089830062644003 -0.159046808013498 -0.246377475354887 -0.349257338733297 - 0.463214947120430 -0.581335265708468 -0.693980297050578 -0.789013095048386 Max. Error
Error | P''13-f(x, P13(x), P'13 (x) ) | 0.000000002628238047217215 0.00000001054073230455181 0.00000002292241687095314 0.0000000527894570136308 0.000000007779590550099647 0.00000006396257166976717 0.00000001524427631328061 0.00000002756204082023442 0.000000005845518177144404 0.000000000009595692296304748 e-0086.396257166976717
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[11] Hasan, Y. Q., and Zhu, L. M., Solving Singular Initial Value Problems in the Second - Order Ordinary Differential Equations, Journal of Applied Sciences, Vol. 7, No. 17, pp: 2505-2508, 2007. http://dx.doi.org/10.3923/jas.2007.2505.2508. [12] Demir, H., and Süngü, İ. Ç., Numerical Solution of a Class of Nonlinear Emden-Fowler Equations by Using Differential Transform Method, Journal of Arts and Sciences, Vol.12, pp: 75-82, 2009. [13] Mohyud-Din, S. T., Noor, M. A., and Noor, K. I., Solving Second order Singular Problems Using He’s Polynomials, World Applied Sciences Journal, Vol. 6, No. 6, pp: 769-775, 2009. [14] Tripathi, B. K., Solutions of a Subclass of Singular Second Order Differential Equations of Lane-Emden and Emden – Fowler Type , World Applied Sciences Journal , Vol. 14, No. 4, pp: 607- 610, 2011. [15] Rachůnková, I., Staněk, S., and Tvrdý, M., Solvability of Nonlinear Singular Problems for Ordinary Differential Equations, New York, USA, 2008. [16] Tawfiq, L. N. M., and Hussein, R. W., Singular Initial Value Problems, LAP LAMBERT Academic Publishing, 2013.