June  2021, 16(2): 155-185. doi: 10.3934/nhm.2021003

Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph

1. 

Department of Mathematics, Indian Institute of Technology Delhi, 110016, Delhi, India

2. 

Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Lehrstuhl Angewandte Mathematik II, Cauerstr. 11, 91058 Erlangen, Germany

* Corresponding author: Mani Mehra

Received  July 2020 Revised  November 2020 Published  June 2021 Early access  January 2021

In this paper, we study a nonlinear fractional boundary value problem on a particular metric graph, namely, a circular ring with an attached edge. First, we prove existence and uniqueness of solutions using the Banach contraction principle and Krasnoselskii's fixed point theorem. Further, we investigate different kinds of Ulam-type stability for the proposed problem. Finally, an example is given in order to demonstrate the application of the obtained theoretical results.

Citation: Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks and Heterogeneous Media, 2021, 16 (2) : 155-185. doi: 10.3934/nhm.2021003
References:
[1]

B. Ahmad, Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Applied Mathematics Letters, 23 (2010), 390-394.  doi: 10.1016/j.aml.2009.11.004.

[2]

Z. AliA. Zada and K. Shah, On Ulam's stability for a coupled systems of nonlinear implicit fractional differential equations, Bulletin of the Malaysian Mathematical Sciences Society, 42 (2019), 2681-2699.  doi: 10.1007/s40840-018-0625-x.

[3]

R. AlmeidaN. R. O. BastosM. Teresa and T. Monteiro, Modelling some real phenomena by fractional differential equations, Math. Meth. Appl. Sci., 39 (2016), 4846-4855.  doi: 10.1002/mma.3818.

[4]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, preprint, arXiv: 1602.03408. doi: 10.1063/1.5026284.

[5]

D. Baleanu, S. Rezapour and H. Mohammadi, Some existence results on nonlinear fractional differential equations, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371 (2013), 20120144. doi: 10.1098/rsta.2012.0144.

[6]

H. M. Baskonus and J. F. G. Aguilar, New singular soliton solutions to the longitudinal wave equation in a magneto-electro-elastic circular rod with M-derivative, Modern Physics Letters B, 33 (2019), 1950251. doi: 10.1142/S0217984919502518.

[7]

J. V. Below, Sturm-Liouville eigenvalue problems on networks, Math. Meth. Appl. Sci., 10 (1988), 383-395.  doi: 10.1002/mma.1670100404.

[8]

U. Brauer and G. Leugering, On boundary observability estimates for semi-discretizations of a dynamic network of elastic strings, Control and Cybernetics, 28 (1999), 421-447. 

[9]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1-13. 

[10]

V. F. M. DelgadoJ. F. G. Aguilar and M. A. T. Hernandez, Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense, AEU-International Journal of Electronics and Communications, 85 (2018), 108-117. 

[11]

P. Exner, P. Kuchment and B. Winn, On the location of spectral edges in-periodic media, Journal of Physics A: Mathematical and Theoretical, 43 (2010), 474022. doi: 10.1088/1751-8113/43/47/474022.

[12]

H. Fazli and J. J. Nieto, Fractional Langevin equation with anti-periodic boundary conditions, Chaos, Solitons & Fractals, 114 (2018), 332-337.  doi: 10.1016/j.chaos.2018.07.009.

[13]

B. Ghanbari and J. F. G. Aguilar, New exact optical soliton solutions for nonlinear Schrödinger equation with second-order spatio-temporal dispersion involving M-derivative, Modern Physics Letters B, 33 (2019), 1950235. doi: 10.1142/S021798491950235X.

[14]

B. Ghanbari and J. F. G. Aguilar, Optical soliton solutions of the Ginzburg-Landau equation with conformable derivative and Kerr law nonlinearity, Revista Mexicana de Física, 65 (2019), 73–81.

[15]

S. Gnutzmann and D. Waltner, Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory, Phys. Rev. E, 93 (2016). doi: 10.1103/physreve.93.032204.

[16]

C. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23 (2010), 1050-1055.  doi: 10.1016/j.aml.2010.04.035.

[17]

D. G. GordezianiM. KupreishvliH. V. Meladze and T. D. Davitashvili, On the solution of boundary value problem for differential equations given in graphs, Appl. Math. Lett., 13 (2008), 80-91. 

[18]

J. R. GraefL. Kong and M. Wang, Existence and uniqueness of solutions for a fractional boundary value problem on a graph, Fractional Calculus and Applied Analysis, 17 (2014), 499-510.  doi: 10.2478/s13540-014-0182-4.

[19]

A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, 2003. doi: 10.1007/978-0-387-21593-8.

[20]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747.

[21]

D. H. Hyers, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27 (1941), 222-224.  doi: 10.1073/pnas.27.4.222.

[22]

A. KhanH. KhanJ. F. G. Aguilar and T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos, Solitons & Fractals, 127 (2019), 422-427.  doi: 10.1016/j.chaos.2019.07.026.

[23]

H. KhanW. Chen and H. Sun, Analysis of positive solution and Hyers-Ulam stability for a class of singular fractional differential equations with p-Laplacian in Banach space, Mathematical Methods in the Applied Sciences, 41 (2018), 3430-3440.  doi: 10.1002/mma.4835.

[24]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.

[25]

C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140364. doi: 10.1098/rspa.2014.0364.

[26]

P. Kuchment, Quantum graphs: An introduction and a brief survey, preprint, arXiv: 0802.3442. doi: 10.1090/pspum/077/2459876.

[27]

P. Kuchment and L. Kunyansky, Differential operators on graphs and photonic crystals, Advances in Computational Mathematics, 16 (2002), 263-290.  doi: 10.1023/A:1014481629504.

[28]

N. Kumar and M. Mehra, Collocation method for solving nonlinear fractional optimal control problems by using Hermite scaling function with error estimates, Optimal Control Applications and Methods, (2020). doi: 10.1002/oca.2681.

[29]

N. Kumar and M. Mehra, Legendre wavelet collocation method for fractional optimal control problems with fractional Bolza cost, Numerical Methods for Partial Differential Equations, (2020). doi: 10.1002/num.22604.

[30]

J. E. LagneseG. Leugering and E. J. P. G. Schmidt, Control of planar networks of Timoshenko beams, SIAM J. Control Optim., 31 (1993), 780-811.  doi: 10.1137/0331035.

[31]

G. Leugering, Domain decomposition of an optimal control problem for semi-linear elliptic equations on metric graphs with application to gas networks, Applied Mathematics, 8 (2017), 1074-1099.  doi: 10.4236/am.2017.88082.

[32]

G. Lumer, Connecting of local operators and evolution equtaions on a network, Lect. Notes Math., 787 (1980), 219-234. 

[33]

R. L. Magin and M. Ovadia, Modeling the cardiac tissue electrode interface using fractional calculus, Journal of Vibration and Control, 14 (2008), 1431-1442. 

[34]

H. M. Martínez and J. F. G. Aguilar, Local M-derivative of order $\alpha$ and the modified expansion function method applied to the longitudinal wave equation in a magneto electro-elastic circular rod, Optical and Quantum Electronics, 50 (2018), 375.

[35]

H. Y. Martínez and J. F. G. Aguilar, Fractional sub-equation method for Hirota–Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana's conformable derivative, Waves in Random and Complex Media, 29 (2019), 678-693.  doi: 10.1080/17455030.2018.1464233.

[36]

H. Y. Martínez and J. F. G. Aguilar, M-derivative applied to the dispersive optical solitons for the Schrödinger-Hirota equation, The European Physical Journal Plus, 134 (2019), 1-11. 

[37]

H. Y. Martínez and J. F. G. Aguilar, M-derivative applied to the soliton solutions for the Lakshmanan–Porsezian–Daniel equation with dual-dispersion for optical fibers, Optical and Quantum Electronics, 51 (2019), 31.

[38]

H. Y. Martínez and J. F. G. Aguilar, Optical solitons solution of resonance nonlinear Schrödinger type equation with Atangana's-conformable derivative using sub-equation method, Waves in Random and Complex Media, (2019), 1–24.

[39]

H. Y. Martínez, J. F. G. Aguilar and A. Atangana, First integral method for non-linear differential equations with conformable derivative, Mathematical Modelling of Natural Phenomena, 13 (2018), 14. doi: 10.1051/mmnp/2018012.

[40]

V. Mehandiratta and M. Mehra, A difference scheme for the time-fractional diffusion equation on a metric star graph, Applied Numerical Mathematics, 158 (2020), 152-163.  doi: 10.1016/j.apnum.2020.07.022.

[41]

V. MehandirattaM. Mehra and G. Leugering, Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph, Journal of Mathematical Analysis and Applications, 477 (2019), 1243-1264.  doi: 10.1016/j.jmaa.2019.05.011.

[42]

V. MehandirattaM. Mehra and G. Leugering, Fractional optimal control problems on a star graph: Optimality system and numerical solution, Mathematical Control and Related Fields, 11 (2021), 189-209.  doi: 10.3934/mcrf.2020033.

[43]

V. Mehandiratta, M. Mehra and G. Leugering, An approach based on Haar wavelet for the approximation of fractional calculus with application to initial and boundary value problems, Math. Meth. Appl. Sci., (2020). doi: 10.1002/mma.6800.

[44]

M. Mehra and R. K. Malik, Solutions of differential–difference equations arising from mathematical models of granulocytopoiesis, Differential Equations and Dynamical Systems, 22 (2014), 33-49.  doi: 10.1007/s12591-013-0159-5.

[45]

G. Mophou, G. Leugering and P. S. Fotsing, Optimal control of a fractional Sturm-Liouville problem on a star graph, Optimization, (2020), 1–29. doi: 10.1080/02331934.2020.1730371.

[46]

S. Nicaise, Some results on spectral theory over networks, applied to nerve impulses transmission, Lect. Notes Math., 1771 (1985), 532-541.  doi: 10.1007/BFb0076584.

[47]

K. S. Patel and M. Mehra, Fourth order compact scheme for space fractional advection-diffusion reaction equations with variable coefficients, Journal of Computational and Applied Mathematics, 380 (2020), 112963. doi: 10.1016/j.cam.2020.112963.

[48]

B. S. Pavlov and M. Faddeev, Model of free electrons and the scattering problem, Teor. Mat. Fiz., 55 (1983), 257-269. 

[49] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999. 
[50]

Y. V. Pokornyi and A. V. Borovskikh, Differential equations on networks (geometric graphs), Journal of Mathematical Sciences, 119 (2004), 691-718.  doi: 10.1023/B:JOTH.0000012752.77290.fa.

[51]

T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72 (1978), 297-300.  doi: 10.1090/S0002-9939-1978-0507327-1.

[52]

A. Shukla, M. Mehra and G. Leugering, A fast adaptive spectral graph wavelet method for the viscous Burgers' equation on a star-shaped connected graph, Mathematical Methods in the Applied Sciences, 43, (2020), 7595–7614. doi: 10.1002/mma.5907.

[53]

A. K. Singh and M. Mehra, Uncertainty quantification in fractional stochastic integro-differential equations using Legendre wavelet collocation method, Lect. Notes Comput. Sci., 12138, (2020), 58–71. doi: 10.1007/978-3-030-50417-5_5.

[54]

X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22, (2009), 64–69. doi: 10.1016/j.aml.2008.03.001.

[55]

S. M. Ulam, A Collection of Mathematical Problems, New York, 1960.

[56]

C. Urs, Coupled fixed point theorems and applications to periodic boundary value problems, Miskolc Mathematical Notes, 14, (2013), 323–333. doi: 10.18514/MMN.2013.598.

[57]

J. R. Wang, A. Zada and H. Waheed, Stability analysis of a coupled system of nonlinear implicit fractional anti-periodic boundary value problem, Mathematical Methods in the Applied Sciences, 42, (2019), 6706–6732. doi: 10.1002/mma.5773.

[58]

L. Xiping, J. Mei and G. Weiago, The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator, Applied Mathematics Letters, 65, (2017), 56–62. doi: 10.1016/j.aml.2016.10.001.

[59]

W. Zhang and W. Liu, Existence and Ulam's type stability results for a class of fractional boundary value problems on a star graph, Mathematical Methods in the Applied Sciences, (2020). doi: 10.1002/mma.6516.

show all references

References:
[1]

B. Ahmad, Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations, Applied Mathematics Letters, 23 (2010), 390-394.  doi: 10.1016/j.aml.2009.11.004.

[2]

Z. AliA. Zada and K. Shah, On Ulam's stability for a coupled systems of nonlinear implicit fractional differential equations, Bulletin of the Malaysian Mathematical Sciences Society, 42 (2019), 2681-2699.  doi: 10.1007/s40840-018-0625-x.

[3]

R. AlmeidaN. R. O. BastosM. Teresa and T. Monteiro, Modelling some real phenomena by fractional differential equations, Math. Meth. Appl. Sci., 39 (2016), 4846-4855.  doi: 10.1002/mma.3818.

[4]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, preprint, arXiv: 1602.03408. doi: 10.1063/1.5026284.

[5]

D. Baleanu, S. Rezapour and H. Mohammadi, Some existence results on nonlinear fractional differential equations, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371 (2013), 20120144. doi: 10.1098/rsta.2012.0144.

[6]

H. M. Baskonus and J. F. G. Aguilar, New singular soliton solutions to the longitudinal wave equation in a magneto-electro-elastic circular rod with M-derivative, Modern Physics Letters B, 33 (2019), 1950251. doi: 10.1142/S0217984919502518.

[7]

J. V. Below, Sturm-Liouville eigenvalue problems on networks, Math. Meth. Appl. Sci., 10 (1988), 383-395.  doi: 10.1002/mma.1670100404.

[8]

U. Brauer and G. Leugering, On boundary observability estimates for semi-discretizations of a dynamic network of elastic strings, Control and Cybernetics, 28 (1999), 421-447. 

[9]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1-13. 

[10]

V. F. M. DelgadoJ. F. G. Aguilar and M. A. T. Hernandez, Analytical solutions of electrical circuits described by fractional conformable derivatives in Liouville-Caputo sense, AEU-International Journal of Electronics and Communications, 85 (2018), 108-117. 

[11]

P. Exner, P. Kuchment and B. Winn, On the location of spectral edges in-periodic media, Journal of Physics A: Mathematical and Theoretical, 43 (2010), 474022. doi: 10.1088/1751-8113/43/47/474022.

[12]

H. Fazli and J. J. Nieto, Fractional Langevin equation with anti-periodic boundary conditions, Chaos, Solitons & Fractals, 114 (2018), 332-337.  doi: 10.1016/j.chaos.2018.07.009.

[13]

B. Ghanbari and J. F. G. Aguilar, New exact optical soliton solutions for nonlinear Schrödinger equation with second-order spatio-temporal dispersion involving M-derivative, Modern Physics Letters B, 33 (2019), 1950235. doi: 10.1142/S021798491950235X.

[14]

B. Ghanbari and J. F. G. Aguilar, Optical soliton solutions of the Ginzburg-Landau equation with conformable derivative and Kerr law nonlinearity, Revista Mexicana de Física, 65 (2019), 73–81.

[15]

S. Gnutzmann and D. Waltner, Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory, Phys. Rev. E, 93 (2016). doi: 10.1103/physreve.93.032204.

[16]

C. Goodrich, Existence of a positive solution to a class of fractional differential equations, Appl. Math. Lett., 23 (2010), 1050-1055.  doi: 10.1016/j.aml.2010.04.035.

[17]

D. G. GordezianiM. KupreishvliH. V. Meladze and T. D. Davitashvili, On the solution of boundary value problem for differential equations given in graphs, Appl. Math. Lett., 13 (2008), 80-91. 

[18]

J. R. GraefL. Kong and M. Wang, Existence and uniqueness of solutions for a fractional boundary value problem on a graph, Fractional Calculus and Applied Analysis, 17 (2014), 499-510.  doi: 10.2478/s13540-014-0182-4.

[19]

A. Granas and J. Dugundji, Fixed Point Theory, Springer-Verlag, 2003. doi: 10.1007/978-0-387-21593-8.

[20]

R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. doi: 10.1142/9789812817747.

[21]

D. H. Hyers, On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27 (1941), 222-224.  doi: 10.1073/pnas.27.4.222.

[22]

A. KhanH. KhanJ. F. G. Aguilar and T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos, Solitons & Fractals, 127 (2019), 422-427.  doi: 10.1016/j.chaos.2019.07.026.

[23]

H. KhanW. Chen and H. Sun, Analysis of positive solution and Hyers-Ulam stability for a class of singular fractional differential equations with p-Laplacian in Banach space, Mathematical Methods in the Applied Sciences, 41 (2018), 3430-3440.  doi: 10.1002/mma.4835.

[24]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, 2006.

[25]

C. Klein, C. Sparber and P. Markowich, Numerical study of fractional nonlinear Schrödinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470 (2014), 20140364. doi: 10.1098/rspa.2014.0364.

[26]

P. Kuchment, Quantum graphs: An introduction and a brief survey, preprint, arXiv: 0802.3442. doi: 10.1090/pspum/077/2459876.

[27]

P. Kuchment and L. Kunyansky, Differential operators on graphs and photonic crystals, Advances in Computational Mathematics, 16 (2002), 263-290.  doi: 10.1023/A:1014481629504.

[28]

N. Kumar and M. Mehra, Collocation method for solving nonlinear fractional optimal control problems by using Hermite scaling function with error estimates, Optimal Control Applications and Methods, (2020). doi: 10.1002/oca.2681.

[29]

N. Kumar and M. Mehra, Legendre wavelet collocation method for fractional optimal control problems with fractional Bolza cost, Numerical Methods for Partial Differential Equations, (2020). doi: 10.1002/num.22604.

[30]

J. E. LagneseG. Leugering and E. J. P. G. Schmidt, Control of planar networks of Timoshenko beams, SIAM J. Control Optim., 31 (1993), 780-811.  doi: 10.1137/0331035.

[31]

G. Leugering, Domain decomposition of an optimal control problem for semi-linear elliptic equations on metric graphs with application to gas networks, Applied Mathematics, 8 (2017), 1074-1099.  doi: 10.4236/am.2017.88082.

[32]

G. Lumer, Connecting of local operators and evolution equtaions on a network, Lect. Notes Math., 787 (1980), 219-234. 

[33]

R. L. Magin and M. Ovadia, Modeling the cardiac tissue electrode interface using fractional calculus, Journal of Vibration and Control, 14 (2008), 1431-1442. 

[34]

H. M. Martínez and J. F. G. Aguilar, Local M-derivative of order $\alpha$ and the modified expansion function method applied to the longitudinal wave equation in a magneto electro-elastic circular rod, Optical and Quantum Electronics, 50 (2018), 375.

[35]

H. Y. Martínez and J. F. G. Aguilar, Fractional sub-equation method for Hirota–Satsuma-coupled KdV equation and coupled mKdV equation using the Atangana's conformable derivative, Waves in Random and Complex Media, 29 (2019), 678-693.  doi: 10.1080/17455030.2018.1464233.

[36]

H. Y. Martínez and J. F. G. Aguilar, M-derivative applied to the dispersive optical solitons for the Schrödinger-Hirota equation, The European Physical Journal Plus, 134 (2019), 1-11. 

[37]

H. Y. Martínez and J. F. G. Aguilar, M-derivative applied to the soliton solutions for the Lakshmanan–Porsezian–Daniel equation with dual-dispersion for optical fibers, Optical and Quantum Electronics, 51 (2019), 31.

[38]

H. Y. Martínez and J. F. G. Aguilar, Optical solitons solution of resonance nonlinear Schrödinger type equation with Atangana's-conformable derivative using sub-equation method, Waves in Random and Complex Media, (2019), 1–24.

[39]

H. Y. Martínez, J. F. G. Aguilar and A. Atangana, First integral method for non-linear differential equations with conformable derivative, Mathematical Modelling of Natural Phenomena, 13 (2018), 14. doi: 10.1051/mmnp/2018012.

[40]

V. Mehandiratta and M. Mehra, A difference scheme for the time-fractional diffusion equation on a metric star graph, Applied Numerical Mathematics, 158 (2020), 152-163.  doi: 10.1016/j.apnum.2020.07.022.

[41]

V. MehandirattaM. Mehra and G. Leugering, Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph, Journal of Mathematical Analysis and Applications, 477 (2019), 1243-1264.  doi: 10.1016/j.jmaa.2019.05.011.

[42]

V. MehandirattaM. Mehra and G. Leugering, Fractional optimal control problems on a star graph: Optimality system and numerical solution, Mathematical Control and Related Fields, 11 (2021), 189-209.  doi: 10.3934/mcrf.2020033.

[43]

V. Mehandiratta, M. Mehra and G. Leugering, An approach based on Haar wavelet for the approximation of fractional calculus with application to initial and boundary value problems, Math. Meth. Appl. Sci., (2020). doi: 10.1002/mma.6800.

[44]

M. Mehra and R. K. Malik, Solutions of differential–difference equations arising from mathematical models of granulocytopoiesis, Differential Equations and Dynamical Systems, 22 (2014), 33-49.  doi: 10.1007/s12591-013-0159-5.

[45]

G. Mophou, G. Leugering and P. S. Fotsing, Optimal control of a fractional Sturm-Liouville problem on a star graph, Optimization, (2020), 1–29. doi: 10.1080/02331934.2020.1730371.

[46]

S. Nicaise, Some results on spectral theory over networks, applied to nerve impulses transmission, Lect. Notes Math., 1771 (1985), 532-541.  doi: 10.1007/BFb0076584.

[47]

K. S. Patel and M. Mehra, Fourth order compact scheme for space fractional advection-diffusion reaction equations with variable coefficients, Journal of Computational and Applied Mathematics, 380 (2020), 112963. doi: 10.1016/j.cam.2020.112963.

[48]

B. S. Pavlov and M. Faddeev, Model of free electrons and the scattering problem, Teor. Mat. Fiz., 55 (1983), 257-269. 

[49] I. Podlubny, Fractional Differential Equations, Academic Press, an Diego, 1999. 
[50]

Y. V. Pokornyi and A. V. Borovskikh, Differential equations on networks (geometric graphs), Journal of Mathematical Sciences, 119 (2004), 691-718.  doi: 10.1023/B:JOTH.0000012752.77290.fa.

[51]

T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72 (1978), 297-300.  doi: 10.1090/S0002-9939-1978-0507327-1.

[52]

A. Shukla, M. Mehra and G. Leugering, A fast adaptive spectral graph wavelet method for the viscous Burgers' equation on a star-shaped connected graph, Mathematical Methods in the Applied Sciences, 43, (2020), 7595–7614. doi: 10.1002/mma.5907.

[53]

A. K. Singh and M. Mehra, Uncertainty quantification in fractional stochastic integro-differential equations using Legendre wavelet collocation method, Lect. Notes Comput. Sci., 12138, (2020), 58–71. doi: 10.1007/978-3-030-50417-5_5.

[54]

X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22, (2009), 64–69. doi: 10.1016/j.aml.2008.03.001.

[55]

S. M. Ulam, A Collection of Mathematical Problems, New York, 1960.

[56]

C. Urs, Coupled fixed point theorems and applications to periodic boundary value problems, Miskolc Mathematical Notes, 14, (2013), 323–333. doi: 10.18514/MMN.2013.598.

[57]

J. R. Wang, A. Zada and H. Waheed, Stability analysis of a coupled system of nonlinear implicit fractional anti-periodic boundary value problem, Mathematical Methods in the Applied Sciences, 42, (2019), 6706–6732. doi: 10.1002/mma.5773.

[58]

L. Xiping, J. Mei and G. Weiago, The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator, Applied Mathematics Letters, 65, (2017), 56–62. doi: 10.1016/j.aml.2016.10.001.

[59]

W. Zhang and W. Liu, Existence and Ulam's type stability results for a class of fractional boundary value problems on a star graph, Mathematical Methods in the Applied Sciences, (2020). doi: 10.1002/mma.6516.

Figure 1.  A general star graph with k edges and k+1 vertices
Figure 2.  A circular ring with an attached edge
[1]

Jeffrey W. Lyons. An application of an avery type fixed point theorem to a second order antiperiodic boundary value problem. Conference Publications, 2015, 2015 (special) : 775-782. doi: 10.3934/proc.2015.0775

[2]

Parin Chaipunya, Poom Kumam. Fixed point theorems for cyclic operators with application in Fractional integral inclusions with delays. Conference Publications, 2015, 2015 (special) : 248-257. doi: 10.3934/proc.2015.0248

[3]

Marta García-Huidobro, Raul Manásevich. A three point boundary value problem containing the operator. Conference Publications, 2003, 2003 (Special) : 313-319. doi: 10.3934/proc.2003.2003.313

[4]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615

[5]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834

[6]

John R. Graef, Bo Yang. Multiple positive solutions to a three point third order boundary value problem. Conference Publications, 2005, 2005 (Special) : 337-344. doi: 10.3934/proc.2005.2005.337

[7]

Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485

[8]

John R. Graef, Johnny Henderson, Bo Yang. Positive solutions to a fourth order three point boundary value problem. Conference Publications, 2009, 2009 (Special) : 269-275. doi: 10.3934/proc.2009.2009.269

[9]

Chunqing Wu, Patricia J.Y. Wong. Global asymptotical stability of the coexistence fixed point of a Ricker-type competitive model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3255-3266. doi: 10.3934/dcdsb.2015.20.3255

[10]

Nguyen Huy Tuan. On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5465-5494. doi: 10.3934/dcdsb.2020354

[11]

Yu-Feng Sun, Zheng Zeng, Jie Song. Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 157-164. doi: 10.3934/naco.2019045

[12]

Eric R. Kaufmann. Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Conference Publications, 2009, 2009 (Special) : 416-423. doi: 10.3934/proc.2009.2009.416

[13]

Changming Song, Hong Li, Jina Li. Initial boundary value problem for the singularly perturbed Boussinesq-type equation. Conference Publications, 2013, 2013 (special) : 709-717. doi: 10.3934/proc.2013.2013.709

[14]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems and Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355

[15]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569

[16]

Kateryna Marynets. Stability analysis of the boundary value problem modelling a two-layer ocean. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2433-2445. doi: 10.3934/cpaa.2022083

[17]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839

[18]

K. Q. Lan, G. C. Yang. Optimal constants for two point boundary value problems. Conference Publications, 2007, 2007 (Special) : 624-633. doi: 10.3934/proc.2007.2007.624

[19]

Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial and Management Optimization, 2021, 17 (4) : 2161-2180. doi: 10.3934/jimo.2020063

[20]

Tiziana Cardinali, Paola Rubbioni. Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1947-1955. doi: 10.3934/dcdss.2020152

2020 Impact Factor: 1.213

Metrics

  • PDF downloads (359)
  • HTML views (385)
  • Cited by (0)

[Back to Top]