On the solution of a nonlinear fractional integro-differential equation with non-local boundary condition

[1] J.A. Adam. A simplified mathematical model of tumor growth, Math. Biosci. 81 (1986), no. 2, 229–244.

[2] J.A. Adam, A mathematical model of tumor growth. ii. effects of geometry and spatial nonuniformity on stability, Math. Biosci. 86 (1987), no. 2, 183–211.

[3] J.A. Adam and S.A. Maggelakis, Mathematical models of tumor growth. iv. effects of a necrotic core, Math. Biosci. 7 (1989), no. 1, 121—136.

[4] N.S. Asaithambi and J.B. Garner, Pointwise solution bounds for a class of singular diffusion problems in physiology, Appl.Math. Comput. 30 (1989), no. 3, 215–222.

[5] M. Bhakta, A. Biswas, D. Ganguly, and L. Montoro, Integral representation of solutions using green function for fractional hardy equations, J. Differ. Equ. 269 (2020), no. 7, 5573–5594.

[6] A.C. Burton, Rate of growth of solid tumours as a problem of diffusion, Growth 30 (1966), no. 2, 157–176.

[7] A. Cabada, P. Habets, and R.L. Pouso, Optimal existence conditions for ϕ-laplacian equations with upper and lower solutions in the reversed order, J. Diff. Equ. 166 (2000), 385–401.

[8] A. Cabada and Z. Hamdi, Nonlinear fractional differential equations with integral boundary value conditions, Appl. Math. Comput. 228 (2014), 251-257.

[9] A. Cabada and G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 389 (2012), no. 1, 403-411.

[10] K. Diethelm, The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type, Springer Science & Business Media, 2010.

[11] A. Dinmohammadi, A. Razani, and E. Shivanian, Analytical solution to the nonlinear singular boundary value problem arising in biology, Boundary Value Problems 2017 (2017), no. 1, 1-9.

[12] A. Dinmohammadi, E. Shivanian, and A. Razani, Existence and uniqueness of solutions for a class of singular nonlinear two-point boundary value problems with sign-changing nonlinear terms, Numer. Funct. Anal. Optim. 38 (2017), no. 3, 344–359.

[13] R.C. Duggan and A.M. Goodman. Pointwise bounds for a nonlinear heat conduction model of the human head, Bull. Math. Bio. 48 (1986), no. 2, 229—236.

[14] U. Flesch, The distribution of heat sources in the human head: a theoretical consideration, J. Theore. Bio. 54 (1975), no. 2, 285—287.

[15] A. Granas and J. Dugundji, Elementary fixed point theorems, Fixed Point Theory, Springer, 2003, pp. 9—84.

[16] B.F Gray, The distribution of heat sources in the human head—theoretical considerations, J. Theore. Bio. 82 (1980), no. 3, 473—476.

[17] H.P. Greenspan. Models for the growth of a solid tumor by diffusion, Stud. Appl. Math. 51 (1972), no. 4, 317—340.

[18] L.N. Huynh, N.H. Luc, D. Baleanu, and L.D. Long, Recovering the space source term for the fractional-diffusion equation with Caputo–Fabrizio derivative, J. Inequal. Appl. 2021 (2021), no. 1, 1–20.

[19] A. Kilbas, H.M. Srivastava, and J.J Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier, 2006.

[20] R. Klages, G. Radons, and I.M. Sokolov. Anomalous transport, Wiley Online Library, 2008. [21] R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29 (1966), no. 1, 255.

[22] R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, volume 31, Springer Science & Business Media, 2012.

[23] Y. Li and Y. Wang, The existence and asymptotic behavior of solutions to fractional stochastic evolution equations with infinite delay, J. Differ. Equ. 266 (2019), no. 6, 3514–3558.

[24] R.L. Magin, Fractional calculus in bioengineering, volume 2. Begell House Redding, 2006.

[25] N.I. Mahmudov, M. Awadalla, and K. Abuassba, Nonlinear sequential fractional differential equations with nonlocal boundary conditions, Adv. Difference Equ. 2017 (2017), no. 1, 1-–15.

[26] D.L.S. McElwain, A re-examination of oxygen diffusion in a spherical cell with michaelis-menten oxygen uptake kinetics, J. Theore. Bio. 71 (1978), no. 2, 255—263.

[27] H. Mohammadi, D. Baleanu, S. Etemad, and S. Rezapour, Criteria for existence of solutions for a Liouville–Caputo boundary value problem via generalized Gronwall’s inequality, J. Inequal. Appl. 2021 (2021), no. 1, 1—19.

[28] S. Pashayi, S. Shahmorad, M.S. Hashemi, and M. Inc, Lie symmetry analysis of two dimensional weakly singular integral equations, J. Geom. Phys. 170 (2021), 04385.

[29] E. Shivanian, Existence of weak solutions to a kind of system of fractional semi-linear Fredholm-Volterra boundary value problem, Int. J. Ind. Math. 14 (2022), no. 2, 153–163.

[30] E. Shivanian, Existence of at least three distinct weak solutions for a class of nonlinear system of fractional differential equations, Numer. Funct. Anal. Optim. 41 (2020), no. 10, 1228-1245.

[31] E. Shivanian, To study existence of at least three weak solutions to a system of over-determined Fredholm fractional integro-differential equations, Commun. Nonlinear Sci. Numer. Simul. 101 (2021), 105892.

[32] E. Shivanian, Error estimate and stability analysis on the study of a high-order nonlinear fractional differential equation with Caputo-derivative and integral boundary condition, Comput. Appl. Math. 41 (2022), no. 8, 395.

[33] E. Shivanian, S.J. Hosseini Ghoncheh, and H. Goudarzi, A unique weak solution for the fractional integrodifferential Schrodinger equations, Math. Sci. 17 (2023), 15–19.