Analytical and numerical study for a fractional boundary value problem with a conformable fractional derivative of Caputo and its fractional integral

• Autorzy: Bekkouche M. Moumen , Guebbai H.

Identyfikatory

DOI

10.17512/jamcm.2020.2.03

• Języki publikacji: EN

Abstrakty

EN

We study the existence and uniqueness of the solution of a fractional boundary value problem with conformable fractional derivation of the Caputo type, which increases the interest of this study. In order to study this problem we have introduced a new definition of fractional integral as an inverse of the conformable fractional derivative of Caputo, therefore, the proofs are based upon the reduction of the problem to a equivalent linear Volterra-Fredholm integral equations of the second kind, and we have built the minimum conditions to obtain the existence and uniqueness of this solution. The analytical study is followed by a complete numerical study.

Słowa kluczowe

EN

fractional boundary value problem; Volterra-Fredholm integral equation; fractional derivative; fractional integral

PL

równanie całkowe Volterry-Fredholma; ułamkowa wartość brzegowa; pochodna ułamkowa Caputo

• Wydawca: Wydawnictwo Politechniki Częstochowskiej
• Czasopismo: Journal of Applied Mathematics and Computational Mechanics
• Rocznik: 2020
• Tom: Vol. 19, nr 2
• Strony: 31--42
• Opis fizyczny: Bibliogr. 22 poz., rys.

Twórcy

autor

Bekkouche M. Moumen

• Faculty of Exact Sciences, Department of Mathematics El Oued University El Oued 39000, Algeria
• Laboratoire des Math´ematiques Appliqu´ees et Mod´elisation, Universit´e 8 Mai 1945 Guelma 24000, Algeria

autor

Guebbai H.

• Laboratoire des Math´ematiques Appliqu´ees et Mod´elisation, Universit´e 8 Mai 1945 Guelma 24000, Algeria

Bibliografia

• [1] Machado, J.T., Kiryakova, V., & Mainardi, F. (2011). Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation, 16(3), 1140-1153.
• [2] Kiryakova, V.S. (1993). Generalized Fractional Calculus and Applications. CRC Press.
• [3] Miller, K.S., & Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley.
• [4] Wang, Y., &Wang, H. (2020). Triple positive solutions for fractional differential equation boundary value problems at resonance. Applied Mathematics Letters, 106376.
• [5] Esmaili, S., Nasresfahani, F., & Eslahchi, M.R. (2020). Solving a fractional parabolic-hyperbolic free boundary problem which models the growth of tumor with drug application using finite difference-spectral method. Chaos, Solitons & Fractals, 132, 109538.
• [6] Munkhammar, J. (2005). Fractional calculus and the Taylor-Riemann series. Rose-Hulman Undergraduate Mathematics Journal, 6(1), 6.
• [7] Samko, S.G., Kilbas, A.A., & Marichev, O.I. (1993). Fractional Integrals and Derivatives (Vol. 1). Yverdon-les-Bains, Switzerland: Gordon and Breach Science Publishers.
• [8] Diethelm, K. (2010). The Analysis of Fractional Differential Equations: An Application-oriented Exposition using Differential Operators of Caputo Type. Springer Science & Business Media.
• [9] Losada, J., & Nieto, J.J. (2015). Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl., 1(2), 87-92.
• [10] Hilfer, R. (Ed.). (2000). Applications of Fractional Calculus in Physics (Vol. 35, No. 12, pp. 87-130). Singapore: World Scientific.
• [11] Linz, P. (1985). Analytical and Numerical Methods for Volterra Equations (Vol. 7). Siam.
• [12] Baleanu, D., G¨uvenc¸, Z.B., & Machado, J.T. (Eds.). (2010). New Trends in Nanotechnology and Fractional Calculus Applications (pp. xii+-531). New York: Springer.
• [13] Kilbas, A.A., Srivastava, H.M., & Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations (Vol. 204). Elsevier.
• [14] Iskenderoglu, G., & Kaya, D. (2020). Symmetry analysis of initial and boundary value problems for fractional differential equations in Caputo sense. Chaos, Solitons & Fractals, 134, 109684.
• [15] Mehandiratta, V., Mehra, M., & Leugering, G. (2019). Existence and uniqueness results for a nonlinear Caputo fractional boundary value problem on a star graph. Journal of Mathematical Analysis and Applications, 477(2), 1243-1264.
• [16] Podlubny, I. (1998). Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Elsevier.
• [17] Ye, H., & Huang, R. (2015). Initial value problem for nonlinear fractional differential equations with sequential fractional derivative. Advances in Difference Equations, 2015, 291, 1-13.
• [18] Błasik, M., & Klimek, M. (2013). Exact Solution of Two-term Nonlinear Fractional Differential Equation with Sequential Riemann-Liouville Derivatives. In Advances in the Theory and Applications of Non-integer Order Systems (pp. 161-170). Heidelberg: Springer.
• [19] Bai, Z., & L¨u, H. (2005). Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications, 311(2), 495-505.
• [20] Caputo, M., & Fabrizio, M. (2015). A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl., 1(2), 1-13.
• [21] Wazwaz, A.M. (2015). A First Course in Integral Equations. World Scientific Publishing Company.
• [22] Wazwaz, A.M. (2011). Linear and Nonlinear Integral Equations (Vol. 639). Berlin: Springer.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).