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  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 & Heterogeneous Media, doi: 10.3934/nhm.2021003
References:
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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.   Google Scholar

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

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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

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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

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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.   Google Scholar

[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. Google Scholar

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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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.   Google Scholar

[9]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[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.  Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

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

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[55]

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