Investigation the Existence of a Solution for a Multi-Singular Fractional Differential Equation with Multi-Points Boundary Conditions


Download PDF

Authors: M. TALAEE, M. SHABIBI, A. GILANI AND S. REZAPOUR

DOI: 10.46793/KgJMat2204.549T

Abstract:

We should try to increase our abilities in solving of complicate differential equations. One type of complicate equations are multi-singular pointwise defined fractional differential equations. We investigate the existence of solutions for a multi-singular pointwise defined fractional differential equation with multi-points boundary conditions. We provide an example to illustrate our main result.



Keywords:

Caputo derivative, fixed point, multi-singular equation, multi-points boundary conditions.



References:

[1]   E. Akbari Kojabad and S. Rezapour, Approximate solutions of a sum-type fractional integro-differential equation by using Chebyshev and Legendre polynomials, Adv. Difference Equ. 2017 (2017), Article ID 351, 18 pages.

[2]   S. Alizadeh, D. Baleanu and S. Rezapour, Analyzing transient response of the parallel RCL circuit by using the Caputo-Fabrizio fractional derivative, Adv. Difference Equ. 2020 (2020), Paper ID 55, 19 pages.

[3]   A. Alsaedi, D. Baleanu, S. Etemad and S. Rezapour, On coupled systems of time-fractional differential problems by using a new fractional derivative, J. Funct. Spaces 2016 (2015), Article ID 4626940, 8 pages.

[4]   M. S. Aydogan, D. Baleanu, A. Mousalou and S. Rezapour, On high order fractional integro-differential equations including the Caputo-Fabrizio derivative, Bound. Value Probl. 2018 (2018), Article ID 90, 15 pages.

[5]   S. M. Aydogan, D. Baleanu, A. Mousalou and S. Rezapour, On approximate solutions for two higher-order Caputo-Fabrizio fractional integro-differential equations, Adv. Difference Equ. 2017 (2017), Paper ID 221, 11 pages.

[6]   D. Baleanu, R. Agarwal, H. Mohammadi and S. Rezapour, Some existence results for a nonlinear fractional differential equation on partially ordered Banach spaces, Bound. Value Probl. 2013 (2013), Paper ID 112, 8 pages.

[7]   D. Baleanu, S. Etemad, S. Pourrazi and S. Rezapour, On the new fractional hybrid boundary value problems with three-point integral hybrid conditions, Adv. Difference Equ. 2019 (2019), Paper ID 473, 21 pages.

[8]   D. Baleanu, K. Ghafarnezhad and S. Rezapour, On a three steps crisis integro-differential equation, Adv. Difference Equ. 2018 (2018), Paper ID 153, 19 pages.

[9]   D. Baleanu, K. Ghafarnezhad, S. Rezapour and M. Shabibi, On the existence of solutions of a three steps crisis integro-differential equation, Adv. Difference Equ. 2018 (2018), Paper ID 135, 20 pages.

[10]   D. Baleanu, V. Hedayati, S. Rezapour and M. M. Al-Qurashi, On two fractional differential inclusions, Springer Plus 5 (2016), Paper ID 882, 15 pages.

[11]   D. Baleanu, H. Khan, H. Jafari, R. A. Khan and M. Alipour, On existence results for solutions of a coupled system of hybrid boundary value problems with hybrid conditions, Adv. Difference Equ. 2015 (2015), Paper ID 318, 20 pages.

[12]   D. Baleanu, H. Mohammadi and S. Rezapour, The existence of solutions for a nonlinear mixed problem of singular fractional differential equations, Adv. Difference Equ. 2013 (2013), Paper ID 359, 14 pages.

[13]   D. Baleanu, H. Mohammadi and S. Rezapour, On a nonlinear fractional differential equation on partially ordered metric spaces, Adv. Difference Equ. 2013 (2013), Paper ID 83, 12 pages.

[14]   D. Baleanu, H. Mohammadi and S. Rezapour, Analysis of the model of hiv-1 infection of CD4+ T-cell with a new approach of fractional derivative, Adv. Difference Equ.2020 (2020), Paper ID 71, 10 pages.

[15]   D. Baleanu, A. Mousalou and S. Rezapour, A new method for investigating approximate solutions of some fractional integro-differential equations involving the Caputo-Fabrizio derivative, Adv. Difference Equ. 2017 (2017), Paper ID 51, 12 pages.

[16]   D. Baleanu, A. Mousalou and S. Rezapour, On the existence of solutions for some infinite coefficient-symmetric Caputo-Fabrizio fractional integro-differential equations, Bound. Value Probl. 2017 (2017), Paper ID 145, 11 pages.

[17]   D. Baleanu, A. Mousalou and S. Rezapour, The extended fractional Caputo-Fabrizio derivative of order 0 σ < 1 on C[0, 1] and the existence of solutions for two higher-order series-type differential equations, Adv. Difference Equ. 2018 (2018), Paper ID 255, 11 pages.

[18]   D. Baleanu, S. Rezapour and H. Mohammadi, Some existence results on nonlinear fractional differential equations, Philos. Trans. Roy. Soc. A 371 (2013), DOI 10.1098/rsta.2012.0144.

[19]   D. Baleanu, S. Rezapour and Z. Saberpour, On fractional integro-differential inclusions via the extended fractional Caputo-Fabrizio derivation, Bound. Value Probl. 2019 (2019), Paper ID 79, 17 pages.

[20]   N. Balkani, S. Rezapour and R. H. Haghi, Approximate solutions for a fractional q-integro-difference equation, Journal of Mathematical Extension 13(3) (2019), 201–214.

[21]   M. De La Sena, V. Hedayati, Y. Gholizade Atani and S. Rezapour, The existence and numerical solution for a k-dimensional system of multi-term fractional integro-differential equations, Nonlinear Anal. Model. Control 22 (2017), 188–209.

[22]    V. Hedayati and S. Rezapour, On a Caputo fractional differential inclusion with integral boundary condition for convex-compact and nonconvex-compact valued multifunctions, Kragujevac J. Math. 41 (1) (2017), 143–158.

[23]   V. Hedayati and M. E. Samei, Positive solutions of fractional differential equation with two pieces in chain interval and simultaneous dirichlet boundary conditions, Bound. Value Probl. 2019 (2019), Paper ID 141, 23 pages.

[24]   S. Hristova and C. Tunc, Stability of nonlinear Volterra integro-differential equations with Caputo fractional derivative and bounded delays, Electron. J. Differential Equations 2019 (2019), 1–11.

[25]   M. Jleli, E. Karapinar and B. Samet, Positive solutions for multipoints boundary value problems for singular fractional differential equations, J. Appl. Math. 2014 (2014), Article ID 596123, 7 pages.

[26]   V. Kalvandi and M. E. Samei, New stability results for a sum-type fractional q-integro-differential equation, J. Adv. Math. Stud. 12 (2019), 201–209.

[27]   H. Khan, C. Tunc, W. Chen and A. Khan, Existence theorems and Hyers-Ulam stability for a class of hybrid fractional differential equations with P-Laplacian operator, J. Appl. Anal. Comput. 8 (2018), 1211–1226.

[28]   S. Liang and M. E. Samei, New approach to solutions of a class of singular fractional q-differential problem via quantum calculus, Adv. Difference Equ. 2020 (2020), Paper ID 14, 22 pages.

[29]   S. K. Ntouyas and M. E. Samei, Existence and uniqueness of solutions for multi-term fractional q-integro-differential equations via quantum calculus, Adv. Difference Equ. 2019 (2019), Paper ID 475, 20 pages.

[30]   I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[31]   M. E. Samei, Existence of solutions for a system of singular sum fractional q-differential equations via quantum calculus, Adv. Difference Equ. 2020 (2020), Paper ID 23, 23 pages.

[32]   M. E. Samei, V. Hedayati and G. K. Ranjbar, The existence of solution for k-dimensional system of Langevin Hadamard-type fractional differential inclusions with 2k different fractional orders, Mediterr. J. Math. 17 (2020), Paper ID 37, 23 pages.

[33]   M. E. Samei, V. Hedayati and S. Rezapour, Existence results for a fraction hybrid differential inclusion with Caputo-Hadamard type fractional derivative, Adv. Difference Equ. 2019 (2019), Paper ID 163, 15 pages.

[34]   M. E. Samei, G. Khalilzadeh Ranjbar and V. Hedayati, Existence of solutions for a class of Caputo fractional q-difference inclusion on multifunctions by computational results, Kragujevac J. Math. 45 (2021), 543–570.

[35]   B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154–2165.

[36]   M. Shabibi, M. Postolache, S. Rezapour and S. M. Vaezpour, Investigation of a multisingular pointwise defined fractional integro-differential equation, J. Math. Anal. 7 (2016), 61–77.

[37]   M. Shabibi, S. Rezapour and S. M. Vaezpour, A singular fractional integro-differential equation, Sci. Bull. Univ. Politec. Bush. Series A 79 (2017), 109–118.

[38]   M. Talaee, M. Shabibi, A. Gilani and S. Rezapour, On the existence of solutions for a pointwise defined multi-singular integro-differential equation with integral boundary condition, Adv. Difference Equ. 2020 (2020), Paper ID 41, 16 pages.

[39]   S. W. Vong, Positive solutions of singular fractional differential equations with integral boundary conditions, Mathematical and Computer Modelling 57 (2013), 1053–1059.