MJ Journal on Applied Mathematics 1 (1) (2016) 9-15 Website: www.math-journals.com/index.php/JAM doi: 10.14419/jam.v1i1.38 Research paper
Solving the singular multi-parameter eigenvalue problem using repeated interpolation technique Luma N. M. Tawfiq * Department of Mathematics, College of Education Ibn Al-Haitham, Baghdad University, Iraq *Corresponding author E-mail:dr.lumanaji@yahoo.com
Abstract This paper is concerned with a repeated interpolation technique to solve the singular multi-parameter eigenvalue problems for ordinary differential equations. The method finds the multi parameter eigenvalues and the corresponding nonzero eigenvectors can be decoupled using new technique which represent the solution of the problem in a certain domain. Illustration example is presented, which confirm the theoretical predictions with a comparison between suggested technique and other methods. Keywords: Eigenvalue, Eigenvector, Ordinary Differential Equation, Multi-Parameter Eigenvalue Problem, Singular Differential Equation. AMS Subject Classification: 65L10, 65D30, 26D10, 41A05, 30E25, 45Exx, 65F15, 65F50, 15A18, 15A69, 41A05, 45C05, 141A80.
1. Introduction The study of multi-parameter eigenvalue problems for singular ordinary differential equations with boundary conditions is a topic of great interest. In many cases singular multi-parameter eigenvalue problems (SMPEVP′s) model important physical processes [1] The multi-parameter eigenvalue problems for ordinary differential equations have been studied by a number of authors. The first was studied by Shimasaki in (1995) he used continuation method for a special class of MPEVP. Hochstenbach, in (2002) used Jacobi-Davidson type method for solving MPEVP. Hochstenbach, and Košir, in (2005) used Two-sided Jacobi-Davidson type method to solve MPEVP (see, [4]). MICHIEL, and PLESTENJAK, in (2006) study harmonic and refined Rayleigh–Ritz techniques for the multi-parameter eigenvalue problem [7]. AUZINGER et al used Collocation Methods to solve of the eigenvalue problems for singular ODEs [8].
2. Osculator interpolation polynomial In this paper we use two-point osculatory interpolation polynomial, essentially this is a generalization 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 [1]. 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 [see 2] as: n
P2n+1(x) =
{ y ( j ) (0) q j (x) + (-1) j y ( j ) (1) q j (1-x) },
(1)
j 0
Copyright © 2016 Authors. 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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q j (x) = (x j / j!)(1-x) n 1
n j
s 0
n s s x = Q j (x) / j!, s
(2)
So that (1) with (2) satisfies: (j)
(j)
y ( j ) (0) = P2 n 1 (0) , y ( j ) (1) = P2 n 1 (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 (1) can be written directly in terms of the Taylor coefficients a i and bi about x = 0 and x = 1 respectively, as: n
P2n+1(x) =
{ a j Q j (x) + (-1)
j
bj Q j (1-x) },
(3)
j 0
3. Solution of the singular multi - parameter eigenvalue problems In this section, we suggest a semi-analytic technique which is based on osculatory interpolating polynomials P2n+1 and Taylor series expansion to solve 2nd order singular multi eigenvalue Problems (SMPEVP′s). A general form of 2nd order SMPEVP's is: xm y"(x) = f(x, y, y', λ, μ, δ),
0≤x≤1
(4a)
Where f are in general nonlinear functions of their arguments and m is integer. Subject to the boundary condition (BC): In the case Dirichlet BC: y (0) = A, y (1) = B, where A, B ϵ R
(4b)
In the case Neumann BC: y′ (0) = A, y′ (1) = B, where A, B ϵ R
(4c)
In the case Cauchy or mixed BC: y (0) = A, y′ (1) = B, where A, B ϵ R
(4d)
Or y′ (0) = A, y (1) = B, where A, B ϵ R
(4e)
Now, to solve these problems by suggested method doing the following steps: Step one Evaluate Taylor series of y(x) about x = 0, i.e. y = y i 0 ai x i = a 0 + a 1 x +
i 2
ai xi
(5)
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, i.e. y = y i 0 bi (x 1)i = b 0 + b 1 (x-1) +
i 2
b i (x-1) i
(6)
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 (5) into equation (4a) and put x = 0, then equate the coefficients of powers of x to obtain a2. Insert the series form (6) into equation (4a) and put x = 1, then equate the coefficients of powers of (x-1) to obtain b2. Step three Derive equation (4a) with respect to x, to get new form of equation say (7) as follows: m xm-1 y"(x) + xm y '''(x )
df (x , y , y ', , , ) , dx
(7)
Then, insert the series form (5) into equation (7) and put x = 0 and equate the coefficients of powers of x to obtain a3, again insert the series form (6) into equation (7) 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.
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Step five The notation implies that the coefficients depend only on the indicated unknowns a0, a1, b0, b1, and λ, μ, δ use the BC′s to get two coefficients from these, therefore, we have only two unknown coefficients and λ,μ,δ Now, we can construct two point osculatory interpolating polynomial P 2n+1(x) by insert these coefficients ( aiۥs and biۥs ) as the following: n
P2n+1(x) =
{ a j Q j (x) + (-1)
j
bj Q j (1-x) },
j 0
n j
q j (x) = ( x j / j!)(1-x) n 1
s 0
n s s x = Q j (x) / j!, s
Step six To find the unknowns coefficients integrate equation (4a) on [0, x] to obtain: x
x
0
0
xm y'(x) – m xm-1y(x) + m(m–1) sm-2 y(s) ds – f(s, y, y', λ, μ, δ) ds = 0,
(8a)
And again integrate equation (8a) on [0, x] to obtain: x
xmy(x)–2m sm-1y(s) ds +m(m-1) 0
x
x
(1-s)sm-2y(s)ds +
0
(1-s)f(s, y, y',λ,μ ,δ) = 0,
(8b)
0
Step seven Putting x = 1 in equations (8) to get: 1
b1 – mb0 + m (m-1)
1
sm-2 y(s) ds +
0
f(s, y, y', λ ,μ, δ) ds = 0,
(9a)
0
And 1
1
0
0
B0–2m sm-1y(s) ds +m (m–1) (1–s)sm-2 y(s) ds +
1
(1–s)f(s, y, y', λ, μ, δ)ds =0,
0
(9b) Step eight Use P2n+1(x) which constructed in step five as a replacement of y(x), we see that equations(9) have only two unknown coefficients from a0, a1, b0, b1 and λ,μ , δ. If the BC is Dirichlet boundary condition, that is, we have a0 and b0, then equations (9) have two unknown coefficients a1, b1 and λ, μ ,δ. If the BC is Neumann, that is, we have a1 and b1, then equations (9) have two unknown coefficients a0, b0 and λ , μ ,δ. Finally, if the BC is Cauchy or mixed condition, i.e., we have a0 and b1 or a1 and b0, then equations (9) have two unknown coefficients a1, b0 or a0, b1 and λ ,μ, δ. Step nine In the case Dirichlet BC, we have: 1
F(a1, b1, λ,μ,δ) = b1 – mb0 + m (m–1)
1
sm-2 y(s) ds +
0
f(s, y, y', λ,μ,δ) ds = 0,
(10a)
0
1
1
0
0
G(a1, b1, λ,μ,δ) = b0 –2m sm-1y(s)ds + m(m–1) (1–s)sm-2y(s)ds +
1
(1–s)f(s, y, y', λ,μ,δ)ds = 0,
(10b)
0
(∂F/∂a1) – (∂G/∂a1) = 0,
(10c)
(∂F/∂b1) – (∂G/∂b1) = 0,
(10d)
(∂F/∂a1) (∂G/∂b1) – (∂F/∂b1) (∂G/∂a1) = 0,
(10e)
In the case Neumann BC, we have:
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F(a0, b0, λ,μ,δ ) = b1 – mb0 + m (m–1)
1
sm-2 y(s) ds +
0
f(s, y, y', λ,μ,δ) ds = 0,
(11a)
0
1
1
0
0
G(a0, b0, λ,μ,δ) = b0 –2m sm-1y(s)ds + m(m–1) (1–s)sm-2y(s)ds +
1
(1–s)f(s, y, y', λ,μ,δ)ds = 0
(11b)
0
(∂F/∂a0) - (∂G/∂a0) = 0,
(11c)
(∂F/∂b0) - (∂G/∂b0) = 0,
(11d)
(∂F/∂a0)(∂G/∂b0) – (∂F/∂b0)(∂G/∂a0) = 0,
(11e)
In the case mixed BC, we have: 1
F (a1, b0, λ,μ,δ) = b1 – mb0 + m (m–1)
0
1
sm-2 y(s) ds +
1
0
f(s, y, y', λ,μ,δ) ds = 0,
1
(12a)
1
G (a1, b0, λ,μ,δ) = b0 –2m 0 sm-1y(s)ds + m(m–1) 0 (1–s)sm-2y(s)ds +
0
(1–s)f(s, y, y', λ,μ,δ)ds = 0
(12b)
(∂F/∂a1) - (∂G/∂a1) = 0
(12b)
(∂F/∂b0) - (∂G/∂b0) = 0
(12c)
(∂F/∂a1)(∂G/∂b0) – (∂F/∂b0)(∂G/∂a1) = 0,
(12e)
Or 1
F (a0, b1, λ ,μ δ) = b1 – mb0 + m (m–1)
0
1
1
sm-2 y(s) ds +
0
f(s, y, y', λ ,μ ,δ) ds = 0,
1
G(a0,b1,λ,μ,δ) = b0 –2m 0 sm-1y(s)ds + m(m–1) 0 (1–s)sm-2y(s)ds +
(13a)
1
0
(1–s)f(s,y,y',λ ,μ,δ)ds = 0
(13b)
(∂F/∂a0) - (∂G/∂a0) = 0
(13c)
(∂F/∂b1) - (∂G/∂b1) = 0
(13d)
(∂F/∂a0) (∂G/∂b1) – (∂F/∂b1) (∂G/∂a0) = 0,
(13e)
So, we can find these coefficients by solving the system of algebraic equations (10) or (11) or (12) or (13) using MATLAB, so insert the value of the unknown coefficients into equation (3), thus equation (3) represent the solution of the problem. IV. Examples In this section, we investigate the theory using example of singular multi_ parameter eigenvalue problem. The algorithm was implemented in MATLAB version 7.11. The bvp4c solver of MATLAB has been modified accordingly so that it can solve Some class of singular multi_ parameter eigenvalue problems as effectively as it previously solved eigenvalue problems. Example 4.1 The following problem arises in a study of heat and mass transfer in a porous spherical catalyst with a first order reaction. There is a singular coefficient arising from the reduction of a partial differential equation to an ODE by symmetry [3]. The SMPEVP is: (1 y )
xy " 2 y ' x 2 ye 1 (1 y )
, x ϵ [0, 1]
The BC (mixed case) are y (1) = 1 and the symmetry condition y'(0) = 0. Now, we solve this problem by suggested method. Here equations (13) become: F (a0, b1, λ,α,β) = b1 – 1 + λ2 s ds = 0,
(13a)
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G (a0, b1, λ,α,β) = 1 – 2 y(s)ds + λ2 s (1– s) f(s, y, y') ds = 0,
(13a)
(∂F/∂a0) – (∂G/∂a0) = 0
(13c)
(∂F/∂b1) – (∂G/∂b1) = 0
(13d)
(∂F/∂a0) (∂G/∂b1) - (∂F/∂b1) (∂G/∂a0) = 0
(13e)
Now, we have to solve equations (13) for the unknowns a0, b1, λ, α, β using MATLAB, then we have a0 = 0.64, b1 = 0, λ = 0.6 = 40 and = 0.2. Then from equation (3) we have (where n = 4): P9 = 18. 63018391074087x9 – 81.86548669822514x8 + 136.0606573968419x7 – 101.7163034006953x6 + 29.5919367027597 x5 – 1.076125341642182 x4 + 0.7351374287263839 x2 + 0.64. Higher accuracy can be obtained by evaluating higher n, now take n=10 P21=52535.82045238554x21-549779.0547394637 x20+ 2594544.959251828x19-272910.909707223x18 +13413618.49119748x17-17012485.96003969x16+ 15031675.07293067x15 - 9139740.80568491x14 + 3661565.477401016x13- 73218.8815826758x12+ 94190.70616882079x11+7.614357331206804x103.760089006497641x8+1.931071345092823x6-1.076125341636627x4+ 0.7351374287263839x2 + 0.64 If n= 11 P23=2307273.5847575x22-201189.35259898x23-12048976.340348x21+37826952.9633766x2079339407.916265x19+116759503.0267302x18-123051968.6595243x17+92896106.64761737x1649247501.14161112x15+17467140.105114x14-3732006.240472856x13+364068.2388733x12+ 7.614357331206804x10 - 3.7600890064976x8 + 1.9310713450928x6 - 0.0761253416366x4 + 0.735137428726x2 + 0.64. Table 4.1: Comparison between P23 an P21d for Example 4.1
P23 0.640000000000000 0.647246012889855 0.668049729702792 0.705306859754149 0.778311965383488 0.889394245754096 0.981440965886602 1.011922794591708 1.008022846066691 1.001928697887349 0.999999994519209
P21 0.640000000000000 0.647245780444789 0.667954814870555 0.704121734347959 0.776224985001824 0.890921071212835 0.985476906381475 1.013556553221331 1.008140292157062 1.001928959494405 0.999999960351312
|p23-p21| 0 0.000000232445066 0.000094914832236 0.001185125406190 0.002086980381664 0.001526825458740 0.004035940494873 0.001633758629623 0.000117446090371 0.000000261607056 0.000000034167897
the solution at n=4 1.05
p9
1 0.95 0.9 0.85
y
X 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.8 0.75 0.7 0.65
0
0.1
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
Fig. 4.1: Illustration Solution of P9 for Example 4.1.
0.9
1
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MJ Journal on Applied Mathematics the solution at n=11 1.05 p23 1 0.95 0.9
y
0.85 0.8 0.75 0.7 0.65
0
0.1
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
0.9
1
Fig. 4.2: Illustration Solution of P23 for Example 4.1.
Kubiček et al (see [3], [4]) solved this problem by the collection method (there are three solutions) assembled by consider a range of parameter values: λ, β, α, such that, the values λ = 0.6, α = 0.1, β = 0.2 used in ex6bvp.m lead to three solutions that are displayed in figure (3.2). Table (3.2) compares the solutions in [3] at the origin to values reported in [5], [6]. Table 4.2: Computed Y (0) for Three Solutions of the Example 4.1
Kubiček et alia [3] 0.9070 0.3639 0.0001
Method in [5] 0.9071 0.3637 0.0001
Fig. 4.3: Multiple Solutions of Example 4. 1, Gave In [5]
Another solution of this problem gave in [7] using collocation method with different code in MATLAB and FORTRAN given in Table (4.2) and Figure (4.2). Table 4.3: Comparisons for Different Code of Solution in [5] with TOL = 10−7
N fcount
bvp4c 205 6856
CW-4 41 1080
CW-6 21 840
sbvp4 57 6051
sbvp4g 57 3699
sbvp6 22 3741
sbvp6g 15 1431
Where: bvp4c (MATLAB 6.0 routine): which is based on collocation at three Lobatto points (see [7]). This is a method of order 4 for regular problems. COLNEW (Fortran 90 code): The basic method here is collocation at Gaussian points, We chose the polynomial degrees m = 4 (CW-4) and m = 6 (CW-6), which results in (super convergent) methods of orders 8 and 12 respectively (for regular problems). – sbvp is used with equidistant (sbvp4 and sbvp6) and Gaussian (sbvp4g and sbvp6g) collocation points and polynomials of degrees 4 and 6 respectively. Although for reasons mentioned here, the comparison of all three codes is difficult. Table (3.3) show the number of mesh points (N) and the number of function evaluations ( fcount ) that the different solvers required to reach tolerance (TOL).
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Fig. 4.4: Solution of Example 4.1 Gave in [7].
4. Conclusions In the present paper, we have proposed repeated interpolation technique to solve eigenvalue problems using osculator interpolation. It may be concluded that the suggested technique is a very powerful, efficient and can be used successfully for finding the solution of nonlinear eigenvalue problems with boundary condition. The bvp4c solver of MATLAB has been modified accordingly so that it can solve some class of eigenvalue problems as effectively as it previously solved BVP.
References [1] Tretter, c., boundary eigenvalue problems for differential equations nη = λpη with λ-polynomial boundary conditions, journal of differential equations, 170 (2001), 408-471. http://dx.doi.org/10.1006/jdeq.2000.3829. [2] C. tretter, on buckling problems, z. angew. Math. mech. 80 (2000), 633-639. http://dx.doi.org/10.1002/15214001(200009)80:9<633::AID-ZAMM633>3.0.CO;2-1. [3] Arbenz, p., lecture notes on solving large scale eigenvalue problems, spring, 2012. [4] B., plestenjak, multiparameter eigenvalue problems, the 2nd international conference on structured matrices, Hong Kong Baptist University, 2006. [5] Michiel, e. h., and plestenjak, b., harmonic rayleigh–ritz for the multiparameter eigenvalue problem, journal of mathematics and computing science, 30 (2006):1-16. [6] W. auzinger, e. karner, o. koch, and e. weinmüller, collocation methods for the solution of eigenvalue problems for singular ordinary differential equations, vienna university of technology, institute for analysis and scientific computing, [7] Wiedner hauptstrasse 8{10/101, a-1040 wien. [8] Burden, l. r., and faires, j. d., numerical analysis, seventh edition, 2001. [9] Phillips, g .m, explicit forms for certain hermite approximations, bit, 13(1973): 177-180. [10] Shampine, l. f., singular boundary value problems for odes, southern methodist university j.,1(2002): 1-14. [11] Tawfiq, l. n. m. and rasheed, h. w., on solving singular boundary value problems and its applications, lap lambert academic publishing, 2013. [12] Shampine, l. f., reichelt, m. w., and kierzenka, j., solving boundary value problems for ordinary differential equations in matlab with bvp4c, 2000. [13] Tawfiq, l. n. m. and mjthap, h. z., solving singular eigenvalue problem using semi-analytic technique, international journal of modern mathematical sciences, 7 (1) (2013): 121-131. [14] Morgado, l., and Lima, p., (2009), numerical methods for a singular boundary value problem with application to a heat conduction model in the human head, proceedings of the international conference on computational and mathematical methods in science and engineering (cmmse).