On the nonlinear Volterra equation with conformable derivative

Yıl 2023, Cilt: 7 Sayı: 2, 292 - 302, 23.07.2023

Tuan Nguyen Hoang Hai Nguyen Minh Nguyen Duc Phuong

https://doi.org/10.31197/atnaa.1281575

Öz

In this paper, we are interested to study a nonlinear Volterra equation with conformable derivative. This kind of such equation has various applications, for example physics, mechanical engineering, heat conduction theory.
First, we show that our problem have a mild soltution which exists locally in time. Then we prove that the convergence of the mild solution when the parameter tends to zero.

Anahtar Kelimeler

conformable differential equation, memory term, local existence, Banach fixed point theorem

Proje Numarası

This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under grant number C2022-44-12.

Kaynakça

• [1] T. Abdeljawad, On conformable fractional calculus J. Comput. Appl. Math. 279 (2015), 5766
• [2] K. Balachandran, J.J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces Nonlinear Anal. 72 (2010), no. 12, 45874593
• [3] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative J. Comput. Appl. Math. 264 (2014), 6570
• [4] A.A. Abdelhakim, J.A. Tenreiro Machado, A critical analysis of the conformable derivative Nonlinear Dynamics, 2019, Volume 95, Issue 4, pp 30633073.
• [5] A. Jaiswal, D. Bahuguna, Semilinear Conformable Fractional Differential Equations in Banach Spaces Dier. Equ. Dyn. Syst. 27 (2019), no. 1-3, 313325
• [6] M. Conti, M.E. Marchini, A remark on nonclassical diffusion equations with memory Appl. Math. Optim. 73 (2016), no. 1, 121
• [7] X. Wang, C. Zhong, Attractors for the non-autonomous nonclassical diffusion equations with fading memory Nonlinear Anal. 71 (2009), no. 11, 57335746.
• [8] E.C. Aifantis, On the problem of diffusion in solids, Acta Mech. 37 (1980) 265296 [9] T.W. Ting, Certain non-steady flows of second-order fluids Arch. Rational Mech. Anal., 1963, 14: 126 [10] D. Baleanu, M. Jleli, S. Kumar, B. Samet, A fractional derivative with two singular kernels and application to a heat conduction problem Adv. Difference Equ. 2020, Paper No. 252, 19 pp [11] M. Hajipour, A. Jajarmi, A. Malek, D. Baleanu, Positivity-preserving sixth-order implicit nite difference weighted essentially non-oscillatory scheme for the nonlinear heat equation Appl. Math. Comput. 325 (2018), 146158
• [12] E. Karapinar, H.D. Binh, N.H. Luc, N.H. Can, On the continuity of the fractional derivative of the time- fractional semilinear pseudo-parabolic systems Adv. Difference Equ. 2021, Paper No. 70, 24 pp.
• [13] R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On The Solutions Of Fractional Differential Equations Via Geraghty Type Hybrid Contractions, Appl. Comput. Math., V.20, N.2, 2021,313-333 [14] N.H. Tuan, N.V. Tien, D. O'regan, N.H. Can, V.T. Nguyen, New results on continuity by order of derivative for conformable parabolic equations, FRACTALS, to appear, https://doi.org/10.1142/S0218348X23400145. [15] N.H. Tuan, N.V. Tien, C. Yang, On an initial boundary value problem for fractional pseudo-parabolic equation with conformable derivative, Math. Biosci. Eng. 19 (2022), no. 11, 1123211259 [16] N.H. Tuan, T.B. Ngoc, D. Baleanu, D. O'Regan, On well-posedness of the sub-diffusion equation with a conformable derivative model, Communications in Nonlinear Science and Numerical Simulation, 89 (2020), 105332, 26 pp.
• [17] N.A. Tuan, Z. Hammouch, E. Karapinar, N.H. Tuan, On a nonlocal problem for a Caputo time-fractional pseudoparabolic equation Math. Methods Appl. Sci. 44 (2021), no. 18, 1479114806.
• [18] N.A. Triet, N.A. Tuan, An iterative method for inverse source parabolic equation Lett. Nonlinear Anal. Appl. Volume 1, Issue 2, Pages:7281, Year: 2023
• [19] N.A. Triet, N.H. Tuan, Global existence for nonlinear bi-parabolic equation under global Lipschitz condition Lett. Nonlinear Anal. Appl. Volume 1, Issue 3, Pages: 89-95, Year: 2023
• [20] M.L. Heard, S. M.RankinIII, A semilinear parabolic Volterra integro-dierential equation J. Dierential Equations 71 (1988), no. 2, 201233.
• [21] J.V. C. Sousa, F. G. Rodrigues, E.C. Oliveira, Stability of the fractional Volterra integral-differential equation by means of ψ-Hilfer operator Math. Methods Appl. Sci. 42 (2019), no. 9, 30333043.
• [22] H.T.K. Van, Non-classical heat equation with singular memory term, Thermal Science, Volume 25, Special issue 2, 2021