Spectral parameter power series method for discontinuous coefficients
Let (a,b) be a finite interval and 1/p, q, r∈L1[a,b]. We show that a general solution (in the weak sense) of the equation (pu′)′+qu = λru on (a,b) can be constructed in terms of power series of the spectral parameter λ. The series converge uniformly on [a,b] and the corresponding coefficients are co...
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| Published in | Mathematical methods in the applied sciences Vol. 38; no. 10; pp. 2000 - 2011 |
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| Format | Journal Article |
| Language | English |
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Blackwell Publishing Ltd
15.07.2015
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| Abstract | Let (a,b) be a finite interval and 1/p, q, r∈L1[a,b]. We show that a general solution (in the weak sense) of the equation (pu′)′+qu = λru on (a,b) can be constructed in terms of power series of the spectral parameter λ. The series converge uniformly on [a,b] and the corresponding coefficients are constructed by means of a simple recursive procedure. We use this representation to solve different types of eigenvalue problems. Several numerical tests are discussed. Copyright © 2014 John Wiley & Sons, Ltd. |
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| AbstractList | Let (
a
,
b
) be a finite interval and 1/
p
,
q
,
r
∈
L
1
[
a
,
b
]. We show that a general solution (in the weak sense) of the equation (
p
u
′
)
′
+
q
u
=
λ
r
u
on (
a
,
b
) can be constructed in terms of power series of the spectral parameter
λ
. The series converge uniformly on [
a
,
b
] and the corresponding coefficients are constructed by means of a simple recursive procedure. We use this representation to solve different types of eigenvalue problems. Several numerical tests are discussed. Copyright © 2014 John Wiley & Sons, Ltd. Let (a,b) be a finite interval and 1/p, q, r∈L1[a,b]. We show that a general solution (in the weak sense) of the equation (pu′)′+qu = λru on (a,b) can be constructed in terms of power series of the spectral parameter λ. The series converge uniformly on [a,b] and the corresponding coefficients are constructed by means of a simple recursive procedure. We use this representation to solve different types of eigenvalue problems. Several numerical tests are discussed. Copyright © 2014 John Wiley & Sons, Ltd. Let (a,b) be a finite interval and 1/p, q, rL super(1)[a,b]. We show that a general solution (in the weak sense) of the equation (pu super(')) super(')+q u = lambda ru on (a,b) can be constructed in terms of power series of the spectral parameter lambda . The series converge uniformly on [a,b] and the corresponding coefficients are constructed by means of a simple recursive procedure. We use this representation to solve different types of eigenvalue problems. Several numerical tests are discussed. |
| Author | Campos, Hugo M. Khmelnytskaya, Kira V. Blancarte, Herminio |
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| Cites_doi | 10.1063/1.3579991 10.1080/00036811.2013.794940 10.1002/mma.2732 10.1002/mma.3213 10.1080/17476930802102894 10.1016/S0362-546X(96)00287-8 10.1007/978-0-387-70914-7 10.1016/j.amc.2013.07.035 10.1063/1.3602275 10.1007/978-0-8176-4656-1 10.1007/s10665-013-9644-7 10.1016/j.jmaa.2012.01.004 10.1007/s11075-012-9609-3 10.1002/mma.1205 10.1016/j.amc.2012.09.055 |
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| References_xml | – year: 2011 – article-title: Eigenvalue problems, spectral parameter power series, and modern applications publication-title: Mathematical Methods in the Applied Sciences – year: 2009 – year: 1964 – volume: 85 start-page: 179 issue: 1 year: 2014 end-page: 209 article-title: The phase retrieval problem: a spectral parameter power series approach publication-title: Journal of Engineering Mathematics – volume: 11 start-page: 065707 (6pp) issue: 6 year: 2009 article-title: Galdeano Efficient calculation of the reflectance and transmittance of finite inhomogeneous layers publication-title: Journal of Optics A: Pure and Applied Optics – volume: 219 start-page: 3610 issue: 8 year: 2012 end-page: 3624 article-title: Spectral parameter power series for fourth‐order Sturm‐Liouville problems publication-title: Applied Mathematics and Computation – year: 2005 – volume: 389 start-page: 1222 issue: 2 year: 2012 end-page: 1238 article-title: Transmutations, L‐bases and complete families of solutions of the stationary Schrödinger equation in the plane publication-title: Journal of Mathematical Analysis and Applications – volume: 30 start-page: 719 issue: 2 year: 1997 end-page: 726 article-title: Lipschitzian composition operators in some function spaces publication-title: Nonlinear Analysis: Theory, Methods & Applications – volume: 93 start-page: 729 issue: 4 year: 2014 end-page: 755 article-title: Spectral parameter power series analysis of isotropic planarly layered waveguides publication-title: Applicable Analysis – volume: 53 start-page: 775 issue: 8 year: 2008 end-page: 789 article-title: A representation for solutions of the Sturm‐Liouville equation publication-title: Complex Variables and Elliptic Equations – volume: 220 start-page: 676 issue: 1 year: 2013 end-page: 694 article-title: Spectral parameter power series for perturbed Bessel equations publication-title: Applied Mathematics and Computation – volume: 63 start-page: 27 issue: 1 year: 2013 end-page: 48 article-title: Yildirim Ahmet, Numerical computation of eigenvalues of discontinuous Stirm‐Liouville problems with parameter dependent boundary conditions using sinc method publication-title: In Numerical Algorithms – year: 1974 – volume: 52 start-page: 043522 issue: 4 year: 2011 article-title: Dispersion equation and eigenvalues for quantum wells using spectral parameter power series publication-title: Journal of Mathematical Physics – volume: 33 start-page: 459 year: 2010 end-page: 468 article-title: Spectral parameter power series for Sturm‐Liouville problems publication-title: Mathematical Methods in the Applied Sciences – year: 1993 – volume: 36 start-page: 1878 issue: 14 year: 2013 end-page: 1891 article-title: The heat transfer problem for inhomogeneous materials in photoacoustic applications and spectral parameter power series publication-title: Mathematical Methods in the Applied Sciences – volume: 52 issue: 6 year: 2011 article-title: Dispersion equation and eigenvalues for the Zakharov‐Shabat system using spectral parameter power series publication-title: Journal of Mathematical Physics – year: 1999 – ident: e_1_2_6_6_1 doi: 10.1063/1.3579991 – ident: e_1_2_6_5_1 doi: 10.1080/00036811.2013.794940 – ident: e_1_2_6_13_1 doi: 10.1002/mma.2732 – ident: e_1_2_6_7_1 doi: 10.1002/mma.3213 – ident: e_1_2_6_2_1 doi: 10.1080/17476930802102894 – ident: e_1_2_6_19_1 doi: 10.1016/S0362-546X(96)00287-8 – volume-title: Theory of Functions of a Real Variable year: 1964 ident: e_1_2_6_16_1 contributor: fullname: Natanson I – ident: e_1_2_6_17_1 doi: 10.1007/978-0-387-70914-7 – ident: e_1_2_6_21_1 doi: 10.1016/j.amc.2013.07.035 – ident: e_1_2_6_9_1 doi: 10.1063/1.3602275 – volume-title: Numerical Solution of Sturm‐Liouville Problems. Monographs on Numerical Analysis year: 1993 ident: e_1_2_6_23_1 contributor: fullname: Pryce JD – volume: 11 start-page: 065707 (6pp) issue: 6 year: 2009 ident: e_1_2_6_10_1 article-title: Galdeano Efficient calculation of the reflectance and transmittance of finite inhomogeneous layers publication-title: Journal of Optics A: Pure and Applied Optics contributor: fullname: Castillo Pérez R – ident: e_1_2_6_14_1 doi: 10.1007/978-0-8176-4656-1 – ident: e_1_2_6_4_1 doi: 10.1007/s10665-013-9644-7 – ident: e_1_2_6_8_1 – ident: e_1_2_6_12_1 doi: 10.1016/j.jmaa.2012.01.004 – ident: e_1_2_6_22_1 doi: 10.1007/s11075-012-9609-3 – ident: e_1_2_6_3_1 doi: 10.1002/mma.1205 – volume-title: Elements of the Theory of Functions and Functional Analysis year: 1999 ident: e_1_2_6_15_1 contributor: fullname: Kolmogorov AN – volume-title: Mathematical Surveys and Monographs year: 2005 ident: e_1_2_6_20_1 contributor: fullname: Zettl A – volume-title: Mathematical Analysis year: 1974 ident: e_1_2_6_18_1 contributor: fullname: Apostol TM – ident: e_1_2_6_11_1 doi: 10.1016/j.amc.2012.09.055 |
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| Snippet | Let (a,b) be a finite interval and 1/p, q, r∈L1[a,b]. We show that a general solution (in the weak sense) of the equation (pu′)′+qu = λru on (a,b) can be... Let ( a , b ) be a finite interval and 1/ p , q , r ∈ L 1 [ a , b ]. We show that a general solution (in the weak sense) of the equation ( p u ′ ) ′ + q u = λ... Let (a,b) be a finite interval and 1/p, q, rL super(1)[a,b]. We show that a general solution (in the weak sense) of the equation (pu super(')) super(')+q u =... |
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| Title | Spectral parameter power series method for discontinuous coefficients |
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