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Comparison Criteria For Two-İnterval Sturm-Liouville Equations Coupled With Transmission Conditions

Yıl 2023, Cilt: 12 Sayı: 3, 61 - 70, 31.12.2023

Öz

The purpose of this work is to establish some comparison properties of the Sturm-Liouville equations defined on two non-intersecting intervals with a common end, on which two conditions for the interaction of the left and right solutions are given, the so-called transmission conditions. We will call equations of this type as two-interval Sturm-Liouville equations, coupled with transmission conditions. It is not clear how to apply the classical methods of Sturm’s comparison theory to two-interval differential equations under given transmission conditions. We have derived some new criteria for comparison properties and have developed new approaches to obtain these criteria. The results obtained are an extension and generalization of the corresponding classical comparison properties for the Sturm-Liouville equations.

Kaynakça

  • Akbarfam, I., Jodayree, A., 2014. Resolvent Operator and Self-Adjointness of Sturm-Liouville Operators with a Finite Number of Transmission Conditions, Mediterranean Journal Of Mathematics, 11(2), 447-462.
  • Allahverdiev, B. P., Bairamov, E., Ugurlu, E., 2013. Eigenparameter dependent Sturm-Liouville problems in boundary conditions with transmission conditions, Journal of Mathematical Analysis and Applications, 401, No. 1, 388-396 .
  • Allahverdiev, B. P., Tuna, H., 2018. Titchmarsh-Weyl Theory for Dirac Systems with Transmission Conditions, Mediterranean Journal of Mathematics 15.4 (2018): 1-12.
  • Allahverdiev, B. P., Tuna, H., 2019. Eigenfunction Expansion for Singular Sturm-Liouville Problems with Transmission Conditions, Electronic Journal of Differential Equations, 2019(03), 1-10.
  • Allegretto, W. 2001. Sturm theorems for degenerate elliptic equations. Proc. Am. Math. Soc. 129, 3031-3035.
  • Ao, J., Sun, J., 2014. Matrix representations of Sturm-Liouville problems with coupled eigenparameter- dependent boundary conditions, Applied Mathematics and Computation 244 (2014) 142-148
  • Aydemir, K., Mukhtarov, O. Sh., 2016. Qualitative analysis of eigenvalues and eigenfunctions of one boundary value-transmission problem, Boundary Value Problems, 1-16.
  • Aydemir, K., Mukhtarov, O. Sh., 2017. Generalized fourier series as Green’s function expansion for multi- interval Sturm-Liouville systems, Mediterr. J. Math. (2017) 14:100.
  • Aydemir, K., Olğar, H., Mukhtarov, O. Sh., Muhtarov, F., 2018. Differential Operator Equations with Interface Conditions in Modified Direct Sum Spaces, Filomat 32(3), 921-931.
  • Bairamov, E., Ugurlu, E., 2012. On the characteristic values of the real component of a dissipative boundary value transmission problem, Appl. Math. and Comp. 218(2012), 9657-9663.
  • Binding, P.A., Langer, H., Möller, M., 2004. Oscillation results for Sturm–Liouville problems with an indefinite weight function. Journal of computational and applied mathematics, 171(1):93–101.
  • Binding, P.A., Rynne, P. B., 2004. Half-eigenvalues of periodic Sturm-Liouville problems, Journal of Differential Equations 206.2 (2004): 280-305.
  • Cannon, J. R., Meyer, G.H., 1971. On a Diffusion in a Fractured Medium, SIAM J. Appl. Math., 3 (1971), pp. 434-448.
  • Dzurina, J., 2018. Oscillation of the second order advanced differential equations, Electron. J. Qual. Theory Differ. Equ. 2018, No. 20, 1-9.
  • Grace, S. R., El-Morshedy, H. A., 2000. Oscillation criteria of comparison type for second order difference equations. Journal of Applied Analysis, 6(1):87–102, DOI: 10.1515/JAA.2000.87.
  • Graef, J. R., Jadlovsk, I., Tun, E., 2022. Oscillation of odd-order differential equations with a nonpositive sublinear neutral term and distributed deviating arguments, Applicable Analysis and Discrete Mathematics, 16(2), 350-364.
  • Kreith, K.,1973. Oscillation Theory. Lecture Notes in Mathematics, vol. 324. Springer, Berlin (1973).
  • Mukhtarov, O. Sh., Aydemir, K., 2021. Two-linked periodic Sturm-Liouville problems with transmission conditions, Mathematical Methods In The Applied Sciences 44 (18),14664-14676.
  • Mukhtarov, O. Sh., Aydemir, K., 2021. Oscillation Properties for Non-Classical Sturm-Liouville Problems with Additional Transmission Conditions, Mathematical Modelling and Analysis 26 (3), 432-443.
  • Mukhtarov, O. Sh., Olğar, H., Aydemir, K., Jabbarov I. S., 2018. Operator-pencil realization of one Sturm- Liouville problem with transmission conditions, Applied and Computational Mathematics 17.2 (2018): 284-294.
  • Mukhtarov, O. S., Yücel, M., 2020. A study of the eigenfunctions of the singular Sturm–Liouville problem using the analytical method and the decomposition technique. Mathematics, 8(3), 415.
  • Mukhtarov, O. S., Yücel, M., Aydemir, K., 2020. Treatment a new approximation method and its justification for Sturm–Liouville problems. Complexity, 2020, 1-8.
  • Picone, M., 1909. Sui valori eccezionali di un parametro da cui dipende un’equazione differenziale lineare ordinaria del second’ordine, Ann. Scuola Norm. Sup. Pisa 11 (1909) 1-141.
  • Qiu, J., 2012. Positive solutions for a nonlinear periodic boundary-value problem with a parameter, Electronic Journal of Differential Equations 2012.133 (2012): 1- 10.
  • Simon, B., 2005. Sturm oscillation and comparison theorems, [in:] W.O. Amrein, A.M. Hintz, D.B. Hinton (eds), Sturm-Liouville Theory: Past and Present, Birkhuser Verlag, Basel, 2005; pp. 29-43.
  • Swanson, C. A., 1968. Comparison and oscillation theory of linear differential equations, Vol. 48, Academic Prees, New York and London.
  • Şen, E., 2021. Spectrum, Trace and Nodal Points of a Sturm-Liouville Type Delayed Differential Operator with Interface Conditions Rocky Mountain Journal of Mathematics (2021) 51 (1), 283-294.
  • Tiryaki, A., Sahiner, S., Mısırlı, E., 2016. Sturm comparison theorems for some elliptic type equations via Picone-tpye inequalities, Electron. J. Qual. Theory Differ. Equ., Proc. 10’th Coll. Qualitative Theory of Diff. Equ. 2016, No. 23, 1-20.
  • Ugurlu, E., 2020. On the characteristic values of the real component of a dissipative boundary value transmission problem,Quaestiones Mathematicae 43.4 (2020): 507-521. Yoshida, N., 2008. Oscillation theory of partial differential equations, World Scientific.
  • Yücel, M., Muhtarov, F., 2023. Parameterized Differential Transform Method and Its Application to Boundary Value Transmission Problems. Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 28(2), 431- 442.
  • Yücel, M., Mukhtarov, O. S., Aydemir, K., 2023. Computation of eigenfunctions of nonlinear boundary-value- transmission problems by developing some approximate techniques. Boletim da Sociedade Paranaense de Matemática, 41, 1-12.

Geçiş Şartları İle Birleştirilmiş İki Aralıklı Sturm-Liouville Denklemleri İçin Karşılaştırma Kriterleri

Yıl 2023, Cilt: 12 Sayı: 3, 61 - 70, 31.12.2023

Öz

Bu çalışmanın amacı ortak uç noktasına sahip olan iki ayrık aralıkta tanımlı olan Sturm-Liouville denklemleri için bazı karşılaştırma özelliklerini elde etmektir. Ortak uç noktada sağ ve sol çözümleri birbirine birleştiren ve geçiş şartları olarak adlandırılan iki şart verilmiştir. Bu çeşit denklemleri geçiş şartları ile birleştirilmiş iki aralıklı Sturm-Liouville denklemleri olarak adlandıracağız. Geçiş şartları ile birbirine birleştirilmiş iki aralıklı diferansiyel denklemlere Sturm karşılaştırma teorisinin klasik yöntemlerinin nasıl uygulanacağı açık değildir. Bu çalışmada karşılaştırma özellikleri için bazı yeni kriterler elde ettik ve bu kriterleri elde etmek için yeni yaklaşımlar geliştirdik. Elde edilen sonuçlar Sturm-Liouville denklemleri için klasik karşılaştırma özelliklerinin genişlemesi ve genelleştirilmesidir.

Kaynakça

  • Akbarfam, I., Jodayree, A., 2014. Resolvent Operator and Self-Adjointness of Sturm-Liouville Operators with a Finite Number of Transmission Conditions, Mediterranean Journal Of Mathematics, 11(2), 447-462.
  • Allahverdiev, B. P., Bairamov, E., Ugurlu, E., 2013. Eigenparameter dependent Sturm-Liouville problems in boundary conditions with transmission conditions, Journal of Mathematical Analysis and Applications, 401, No. 1, 388-396 .
  • Allahverdiev, B. P., Tuna, H., 2018. Titchmarsh-Weyl Theory for Dirac Systems with Transmission Conditions, Mediterranean Journal of Mathematics 15.4 (2018): 1-12.
  • Allahverdiev, B. P., Tuna, H., 2019. Eigenfunction Expansion for Singular Sturm-Liouville Problems with Transmission Conditions, Electronic Journal of Differential Equations, 2019(03), 1-10.
  • Allegretto, W. 2001. Sturm theorems for degenerate elliptic equations. Proc. Am. Math. Soc. 129, 3031-3035.
  • Ao, J., Sun, J., 2014. Matrix representations of Sturm-Liouville problems with coupled eigenparameter- dependent boundary conditions, Applied Mathematics and Computation 244 (2014) 142-148
  • Aydemir, K., Mukhtarov, O. Sh., 2016. Qualitative analysis of eigenvalues and eigenfunctions of one boundary value-transmission problem, Boundary Value Problems, 1-16.
  • Aydemir, K., Mukhtarov, O. Sh., 2017. Generalized fourier series as Green’s function expansion for multi- interval Sturm-Liouville systems, Mediterr. J. Math. (2017) 14:100.
  • Aydemir, K., Olğar, H., Mukhtarov, O. Sh., Muhtarov, F., 2018. Differential Operator Equations with Interface Conditions in Modified Direct Sum Spaces, Filomat 32(3), 921-931.
  • Bairamov, E., Ugurlu, E., 2012. On the characteristic values of the real component of a dissipative boundary value transmission problem, Appl. Math. and Comp. 218(2012), 9657-9663.
  • Binding, P.A., Langer, H., Möller, M., 2004. Oscillation results for Sturm–Liouville problems with an indefinite weight function. Journal of computational and applied mathematics, 171(1):93–101.
  • Binding, P.A., Rynne, P. B., 2004. Half-eigenvalues of periodic Sturm-Liouville problems, Journal of Differential Equations 206.2 (2004): 280-305.
  • Cannon, J. R., Meyer, G.H., 1971. On a Diffusion in a Fractured Medium, SIAM J. Appl. Math., 3 (1971), pp. 434-448.
  • Dzurina, J., 2018. Oscillation of the second order advanced differential equations, Electron. J. Qual. Theory Differ. Equ. 2018, No. 20, 1-9.
  • Grace, S. R., El-Morshedy, H. A., 2000. Oscillation criteria of comparison type for second order difference equations. Journal of Applied Analysis, 6(1):87–102, DOI: 10.1515/JAA.2000.87.
  • Graef, J. R., Jadlovsk, I., Tun, E., 2022. Oscillation of odd-order differential equations with a nonpositive sublinear neutral term and distributed deviating arguments, Applicable Analysis and Discrete Mathematics, 16(2), 350-364.
  • Kreith, K.,1973. Oscillation Theory. Lecture Notes in Mathematics, vol. 324. Springer, Berlin (1973).
  • Mukhtarov, O. Sh., Aydemir, K., 2021. Two-linked periodic Sturm-Liouville problems with transmission conditions, Mathematical Methods In The Applied Sciences 44 (18),14664-14676.
  • Mukhtarov, O. Sh., Aydemir, K., 2021. Oscillation Properties for Non-Classical Sturm-Liouville Problems with Additional Transmission Conditions, Mathematical Modelling and Analysis 26 (3), 432-443.
  • Mukhtarov, O. Sh., Olğar, H., Aydemir, K., Jabbarov I. S., 2018. Operator-pencil realization of one Sturm- Liouville problem with transmission conditions, Applied and Computational Mathematics 17.2 (2018): 284-294.
  • Mukhtarov, O. S., Yücel, M., 2020. A study of the eigenfunctions of the singular Sturm–Liouville problem using the analytical method and the decomposition technique. Mathematics, 8(3), 415.
  • Mukhtarov, O. S., Yücel, M., Aydemir, K., 2020. Treatment a new approximation method and its justification for Sturm–Liouville problems. Complexity, 2020, 1-8.
  • Picone, M., 1909. Sui valori eccezionali di un parametro da cui dipende un’equazione differenziale lineare ordinaria del second’ordine, Ann. Scuola Norm. Sup. Pisa 11 (1909) 1-141.
  • Qiu, J., 2012. Positive solutions for a nonlinear periodic boundary-value problem with a parameter, Electronic Journal of Differential Equations 2012.133 (2012): 1- 10.
  • Simon, B., 2005. Sturm oscillation and comparison theorems, [in:] W.O. Amrein, A.M. Hintz, D.B. Hinton (eds), Sturm-Liouville Theory: Past and Present, Birkhuser Verlag, Basel, 2005; pp. 29-43.
  • Swanson, C. A., 1968. Comparison and oscillation theory of linear differential equations, Vol. 48, Academic Prees, New York and London.
  • Şen, E., 2021. Spectrum, Trace and Nodal Points of a Sturm-Liouville Type Delayed Differential Operator with Interface Conditions Rocky Mountain Journal of Mathematics (2021) 51 (1), 283-294.
  • Tiryaki, A., Sahiner, S., Mısırlı, E., 2016. Sturm comparison theorems for some elliptic type equations via Picone-tpye inequalities, Electron. J. Qual. Theory Differ. Equ., Proc. 10’th Coll. Qualitative Theory of Diff. Equ. 2016, No. 23, 1-20.
  • Ugurlu, E., 2020. On the characteristic values of the real component of a dissipative boundary value transmission problem,Quaestiones Mathematicae 43.4 (2020): 507-521. Yoshida, N., 2008. Oscillation theory of partial differential equations, World Scientific.
  • Yücel, M., Muhtarov, F., 2023. Parameterized Differential Transform Method and Its Application to Boundary Value Transmission Problems. Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 28(2), 431- 442.
  • Yücel, M., Mukhtarov, O. S., Aydemir, K., 2023. Computation of eigenfunctions of nonlinear boundary-value- transmission problems by developing some approximate techniques. Boletim da Sociedade Paranaense de Matemática, 41, 1-12.
Toplam 31 adet kaynakça vardır.

Ayrıntılar

Birincil Dil Türkçe
Konular Bilgi Sistemleri Geliştirme Metodolojileri ve Uygulamaları
Bölüm Araştırma Makaleleri
Yazarlar

Sevda Nur Öztürk

Oktay Mukhtarov

Kadriye Aydemir 0000-0002-8378-3949

Erken Görünüm Tarihi 28 Aralık 2023
Yayımlanma Tarihi 31 Aralık 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 12 Sayı: 3

Kaynak Göster

APA Öztürk, S. N., Mukhtarov, O., & Aydemir, K. (2023). Geçiş Şartları İle Birleştirilmiş İki Aralıklı Sturm-Liouville Denklemleri İçin Karşılaştırma Kriterleri. Gaziosmanpaşa Bilimsel Araştırma Dergisi, 12(3), 61-70.
AMA Öztürk SN, Mukhtarov O, Aydemir K. Geçiş Şartları İle Birleştirilmiş İki Aralıklı Sturm-Liouville Denklemleri İçin Karşılaştırma Kriterleri. GBAD. Aralık 2023;12(3):61-70.
Chicago Öztürk, Sevda Nur, Oktay Mukhtarov, ve Kadriye Aydemir. “Geçiş Şartları İle Birleştirilmiş İki Aralıklı Sturm-Liouville Denklemleri İçin Karşılaştırma Kriterleri”. Gaziosmanpaşa Bilimsel Araştırma Dergisi 12, sy. 3 (Aralık 2023): 61-70.
EndNote Öztürk SN, Mukhtarov O, Aydemir K (01 Aralık 2023) Geçiş Şartları İle Birleştirilmiş İki Aralıklı Sturm-Liouville Denklemleri İçin Karşılaştırma Kriterleri. Gaziosmanpaşa Bilimsel Araştırma Dergisi 12 3 61–70.
IEEE S. N. Öztürk, O. Mukhtarov, ve K. Aydemir, “Geçiş Şartları İle Birleştirilmiş İki Aralıklı Sturm-Liouville Denklemleri İçin Karşılaştırma Kriterleri”, GBAD, c. 12, sy. 3, ss. 61–70, 2023.
ISNAD Öztürk, Sevda Nur vd. “Geçiş Şartları İle Birleştirilmiş İki Aralıklı Sturm-Liouville Denklemleri İçin Karşılaştırma Kriterleri”. Gaziosmanpaşa Bilimsel Araştırma Dergisi 12/3 (Aralık 2023), 61-70.
JAMA Öztürk SN, Mukhtarov O, Aydemir K. Geçiş Şartları İle Birleştirilmiş İki Aralıklı Sturm-Liouville Denklemleri İçin Karşılaştırma Kriterleri. GBAD. 2023;12:61–70.
MLA Öztürk, Sevda Nur vd. “Geçiş Şartları İle Birleştirilmiş İki Aralıklı Sturm-Liouville Denklemleri İçin Karşılaştırma Kriterleri”. Gaziosmanpaşa Bilimsel Araştırma Dergisi, c. 12, sy. 3, 2023, ss. 61-70.
Vancouver Öztürk SN, Mukhtarov O, Aydemir K. Geçiş Şartları İle Birleştirilmiş İki Aralıklı Sturm-Liouville Denklemleri İçin Karşılaştırma Kriterleri. GBAD. 2023;12(3):61-70.