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Journals / Konuralp Journal of Mathematics / 2021 / Volume: 9 Issue: 2

Uniform Convergence of Generalized Fourier Series of Hahn-Sturm-Liouville Problem

Bilender Paşaoğlu(SÜLEYMAN DEMİREL ÜNİVERSİTESİ)
Hüseyin Tuna(MEHMET AKİF ERSOY ÜNİVERSİTESİ)0000-0001-7240-8687
Pages
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Publication Date
2021-10-15
DOI
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In this work, we consider the Hahn-Sturm-Liouville boundary value problem defined by $$ (Ly)\left( x\right) :=\frac{1}{r\left( x\right) }\left[ -q^{-1} D_{-\omega q^{-1},q^{-1}}(p\left( x\right) D_{\omega,q}y\left( x\right) )+v\left( x\right) y\left( x\right) \right] =\lambda y\left( x\right) ,\ x\in J_{\omega_{0},a}^{0}=\{x:x=\omega _{0}+(a-\omega_{0})q^{n}, n=1,2,...\} $$ with the boundary conditions $$ y\left( \omega_{0}\right) -h_{1}p\left( \omega_{0}\right) D_{-\omega q^{-1},q^{-1}}y\left( \omega_{0}\right) =0, y\left( a\right) +h_{2}p\left( h^{-1}\left( a\right) \right) D_{-\omega q^{-1},q^{-1}}y\left( a\right) =0,$$ where $q\in\left( 0,1\right) ,\ \omega>0,\ h_{1},h_{2}>0,\ \lambda$ is a complex eigenvalue parameter, $p,v,r$ are real-valued continuous functions at $\omega_{0},$ defined on $J_{\omega_{0},h^{-1}(a)}$ and $p(x)>0,$ $r\left( x\right) >0,\ v\left( x\right) >0,\ x\in J_{\omega_{0},h^{-1}(a)},$ $h^{-1}\left( a\right) =q^{-1}(a-\omega)>a,$ $h^{-1}\left( \omega _{0}\right) =\omega_{0},$ $J_{\omega_{0},a}=\{x:x=\omega_{0}+(a-\omega _{0})q^{n},$ $n=0,1,2...\}\cup\{\omega_{0}\}.$ The existence of a countably infinite set of eigenvalues and eigenfunctions is proved and a uniformly convergent expansion formula in the eigenfunctions is established.

📌 3 Key Takeaways

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Objective & Hypothesis:In this work, we consider the Hahn-Sturm-Liouville boundary value problem defined by $$ (Ly)\left( x\right) :=\frac{1}{r\left( x\right) }\left[ -q^{-1} D_{-\omega q^{-1},q^{-1}}(p\left( x\right) D_{\omega,q}y\left( x\right) )+v\left( x\right) y\left( x\right) \right] =\lambda y\left( x\right) ,\ x\in J_{\omega_{0},a}^{0}=\{x:x=\omega _{0}+(a-\omega_{0})q^{n}, n=1,2,...\} $$ with the boundary conditions $$ y\left( \omega_{0}\right) -h_{1}p\left( \omega_{0}\right) D_{-\omega q^{-1},q^{-1}}y\left( \omega_{0}\right) =0, y\left( a\right) +h_{2}p\left( h^{-1}\left( a\right) \right) D_{-\omega q^{-1},q^{-1}}y\left( a\right) =0,$$ where $q\in\left( 0,1\right) ,\ \omega>0,\ h_{1},h_{2}>0,\ \lambda$ is a complex eigenvalue parameter, $p,v,r$ are real-valued continuous functions at $\omega_{0},$ defined on $J_{\omega_{0},h^{-1}(a)}$ and $p(x)>0,$ $r\left( x\right) >0,\ v\left( x\right) >0,\ x\in J_{\omega_{0},h^{-1}(a)},$ $h^{-1}\left( a\right) =q^{-1}(a-\omega)>a,$ $h^{-1}\left( \omega _{0}\right) =\omega_{0},$ $J_{\omega_{0},a}=\{x:x=\omega_{0}+(a-\omega _{0})q^{n},$ $n=0,1,2...\}\cup\{\omega_{0}\}.$ The existence of a countably infinite set of eigenvalues and eigenfunctions is proved and a uniformly convergent expansion formula in the eigenfunctions is established.
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Key Findings & Data:In this work, we consider the Hahn-Sturm-Liouville boundary value problem defined by $$ (Ly)\left( x\right) :=\frac{1}{r\left( x\right) }\left[ -q^{-1} D_{-\omega q^{-1},q^{-1}}(p\left( x\right) D_{\omega,q}y\left( x\right) )+v\left( x\right) y\left( x\right) \right] =\lambda y\left( x\right) ,\ x\in J_{\omega_{0},a}^{0}=\{x:x=\omega _{0}+(a-\omega_{0})q^{n}, n=1,2,...\} $$ with the boundary conditions $$ y\left( \omega_{0}\right) -h_{1}p\left( \omega_{0}\right) D_{-\omega q^{-1},q^{-1}}y\left( \omega_{0}\right) =0, y\left( a\right) +h_{2}p\left( h^{-1}\left( a\right) \right) D_{-\omega q^{-1},q^{-1}}y\left( a\right) =0,$$ where $q\in\left( 0,1\right) ,\ \omega>0,\ h_{1},h_{2}>0,\ \lambda$ is a complex eigenvalue parameter, $p,v,r$ are real-valued continuous functions at $\omega_{0},$ defined on $J_{\omega_{0},h^{-1}(a)}$ and $p(x)>0,$ $r\left( x\right) >0,\ v\left( x\right) >0,\ x\in J_{\omega_{0},h^{-1}(a)},$ $h^{-1}\left( a\right) =q^{-1}(a-\omega)>a,$ $h^{-1}\left( \omega _{0}\right) =\omega_{0},$ $J_{\omega_{0},a}=\{x:x=\omega_{0}+(a-\omega _{0})q^{n},$ $n=0,1,2...\}\cup\{\omega_{0}\}.$ The existence of a countably infinite set of eigenvalues and eigenfunctions is proved and a uniformly convergent expansion formula in the eigenfunctions is established.
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Conclusion & Impact:In this work, we consider the Hahn-Sturm-Liouville boundary value problem defined by $$ (Ly)\left( x\right) :=\frac{1}{r\left( x\right) }\left[ -q^{-1} D_{-\omega q^{-1},q^{-1}}(p\left( x\right) D_{\omega,q}y\left( x\right) )+v\left( x\right) y\left( x\right) \right] =\lambda y\left( x\right) ,\ x\in J_{\omega_{0},a}^{0}=\{x:x=\omega _{0}+(a-\omega_{0})q^{n}, n=1,2,...\} $$ with the boundary conditions $$ y\left( \omega_{0}\right) -h_{1}p\left( \omega_{0}\right) D_{-\omega q^{-1},q^{-1}}y\left( \omega_{0}\right) =0, y\left( a\right) +h_{2}p\left( h^{-1}\left( a\right) \right) D_{-\omega q^{-1},q^{-1}}y\left( a\right) =0,$$ where $q\in\left( 0,1\right) ,\ \omega>0,\ h_{1},h_{2}>0,\ \lambda$ is a complex eigenvalue parameter, $p,v,r$ are real-valued continuous functions at $\omega_{0},$ defined on $J_{\omega_{0},h^{-1}(a)}$ and $p(x)>0,$ $r\left( x\right) >0,\ v\left( x\right) >0,\ x\in J_{\omega_{0},h^{-1}(a)},$ $h^{-1}\left( a\right) =q^{-1}(a-\omega)>a,$ $h^{-1}\left( \omega _{0}\right) =\omega_{0},$ $J_{\omega_{0},a}=\{x:x=\omega_{0}+(a-\omega _{0})q^{n},$ $n=0,1,2...\}\cup\{\omega_{0}\}.$ The existence of a countably infinite set of eigenvalues and eigenfunctions is proved and a uniformly convergent expansion formula in the eigenfunctions is established.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Hahn's Sturm-Liouville equation · Green's function · Parseval equality

Abstract

In this work, we consider the Hahn-Sturm-Liouville boundary value problem defined by $$ (Ly)\left( x\right) :=\frac{1}{r\left( x\right) }\left[ -q^{-1} D_{-\omega q^{-1},q^{-1}}(p\left( x\right) D_{\omega,q}y\left( x\right) )+v\left( x\right) y\left( x\right) \right] =\lambda y\left( x\right) ,\ x\in J_{\omega_{0},a}^{0}=\{x:x=\omega _{0}+(a-\omega_{0})q^{n}, n=1,2,...\} $$ with the boundary conditions $$ y\left( \omega_{0}\right) -h_{1}p\left( \omega_{0}\right) D_{-\omega q^{-1},q^{-1}}y\left( \omega_{0}\right) =0, y\left( a\right) +h_{2}p\left( h^{-1}\left( a\right) \right) D_{-\omega q^{-1},q^{-1}}y\left( a\right) =0,$$ where $q\in\left( 0,1\right) ,\ \omega>0,\ h_{1},h_{2}>0,\ \lambda$ is a complex eigenvalue parameter, $p,v,r$ are real-valued continuous functions at $\omega_{0},$ defined on $J_{\omega_{0},h^{-1}(a)}$ and $p(x)>0,$ $r\left( x\right) >0,\ v\left( x\right) >0,\ x\in J_{\omega_{0},h^{-1}(a)},$ $h^{-1}\left( a\right) =q^{-1}(a-\omega)>a,$ $h^{-1}\left( \omega _{0}\right) =\omega_{0},$ $J_{\omega_{0},a}=\{x:x=\omega_{0}+(a-\omega _{0})q^{n},$ $n=0,1,2...\}\cup\{\omega_{0}\}.$ The existence of a countably infinite set of eigenvalues and eigenfunctions is proved and a uniformly convergent expansion formula in the eigenfunctions is established.

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Cite This Article

Paşaoğlu, B., Tuna, H. (2021). Uniform Convergence of Generalized Fourier Series of Hahn-Sturm-Liouville Problem. Konuralp Journal of Mathematics, Volume: 9 Issue: 2. https://ajindex.com/articles/1521591

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