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Abstract
In the present work, an initial value problem involving the Atangana-Baleanu derivative is considered. An explicit solution of the given problem in integral form is obtained by using the Laplace transform. The use of the given initial value problem is illustrated by considering a boundary value problem in which the solution is expressed in the form of a series expansion using an orthogonal basis obtained by separation of variables. Some examples are also given to illustrate the obtained results.
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References
- Caputo, M. and Fabrizio, M. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 2015,2, 73-85.
- Atangana, A. and Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Thermal Science, 2016,20, 763-769.
- Algahtani, R.T. Atangana-Baleanu derivative with fractional order applied to the model of groundwater within an unconfined aquifer. Journal of Nonlinear Sciences and Applications, 2016,9, 3647-3654.
- Alkahtani, B.S.T. Chuas circuit model with Atangana-Baleanu derivative with fractional order. Chaos, Solitons and Fractals. 2016,89, 547-551.
- Alkahtani, B.S.T. and Atangana, A. Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order. Chaos, Solitons and Fractals, 2016,89, 539-546.
- Al-Salti, N., Karimov, E. and Sadarangani, K. On a differential equation with Caputo-Fabrizio fractional derivative of order 1 < β ≤ 2 and application to mass-spring-damper system. Progress in Fractional Differentiation and Applications, 2016, 2, 257-263.
- G´omez-Aguilar, J.F., Atangana, A. and Morales-Delgado, V.F. Elecrtical circuits RC, LC, and RL described by Atangana-Baleanu fractional derivatives. International Journal of Circuit Theory and Applications, 2017, 1514–1533.
- Al-Salti, N., Karimov, E. and Kerbal, S. Boundary-value problem for fractional heat equation involving Caputo-Fabrizio derivative. New Trend in Mathematical Sciences, 2016,4, 79-89.
- Atangana, A. Non-validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A, 2018,505, 688-706.
- Atangana, A. and G´omez -Aguilar, J.F. Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 2018,133, 1-22.
- Atangana, A. and Koca, I. Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos, Solitons and Fractals, 2016, 89, 446-454.
- Caputo, M. and Fabrizio, M. Applications of new time and spatial fractional derivatives with exponential kernels. Progress in Fractional Differentiation and Applications, 2016,1, 1-11.
- Djida, J.D., Atangana, A. and Area, I. Numerical computation of a fractional derivative with non-local and non-singular kernel. Mathematical Modelling of Natural Phenomena, 2017,12, 4-13.
- Jefain, O. and Algahtani, J. Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model. Chaos, Solitons and Fractals, 2016,89, 552-559.
- Losada, J. and Nieto, J. Properties of a new fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 2015,2, 87-92.
- Sheikh, N.A., Ali, F., Saqib, M., Khan, I., Jan, S.A.A., Alshomrani, A.S. and Alghamdi, M.S. Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction. Results in Physics, 2017,7, 789-800.
- Tateishi, A.A., Haroldo V. Ribeiro, H.V. and Lenzi, E. K. The Role of Fractional Time-Derivative Operators on Anomalous Diffusion, Frontiers in Physics, 2017,5, 1-9.
- Podlubny, I. Fractional differential equations. Academic Press Inc., San Diego, CA, 1999.
- Al-Musalhi, F., Al-Salti, N. and Karimov, E., Karimov, Initial and boundary value problems for fractional differential equations involving Atangana-Baleanu derivative, 2017, arxiv: 1706.00740.
- Al-Refai, M. Comparison principles for differential equations involving Caputo fractional derivative with Mittag-Leffler non-singular kernel, Electronic Journal of Differential Equations, 2018,2018, 1-10.
- Baleanu, D. and Fernandez, A. On some new properties of fractional derivatives with Mittag-Leffler kernel. Communications in Nonlinear Science and Numerical Simulation, 2018,59, 444-462.
- Djida, J.D., Mophou, G.M and Area, I. Optimal control of diffusion equation with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel, Journal of Optimization Theory and Applications (2018). https://doi.org/10.1007/s10957-018-1305-6.
- Moiseev, E.I. The basis property of systems of sines and cosines. Doklady Akademii Nauk SSSR, 1984,275, 794–798.
References
Caputo, M. and Fabrizio, M. A new definition of fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 2015,2, 73-85.
Atangana, A. and Baleanu, D. New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model. Thermal Science, 2016,20, 763-769.
Algahtani, R.T. Atangana-Baleanu derivative with fractional order applied to the model of groundwater within an unconfined aquifer. Journal of Nonlinear Sciences and Applications, 2016,9, 3647-3654.
Alkahtani, B.S.T. Chuas circuit model with Atangana-Baleanu derivative with fractional order. Chaos, Solitons and Fractals. 2016,89, 547-551.
Alkahtani, B.S.T. and Atangana, A. Controlling the wave movement on the surface of shallow water with the Caputo-Fabrizio derivative with fractional order. Chaos, Solitons and Fractals, 2016,89, 539-546.
Al-Salti, N., Karimov, E. and Sadarangani, K. On a differential equation with Caputo-Fabrizio fractional derivative of order 1 < β ≤ 2 and application to mass-spring-damper system. Progress in Fractional Differentiation and Applications, 2016, 2, 257-263.
G´omez-Aguilar, J.F., Atangana, A. and Morales-Delgado, V.F. Elecrtical circuits RC, LC, and RL described by Atangana-Baleanu fractional derivatives. International Journal of Circuit Theory and Applications, 2017, 1514–1533.
Al-Salti, N., Karimov, E. and Kerbal, S. Boundary-value problem for fractional heat equation involving Caputo-Fabrizio derivative. New Trend in Mathematical Sciences, 2016,4, 79-89.
Atangana, A. Non-validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A, 2018,505, 688-706.
Atangana, A. and G´omez -Aguilar, J.F. Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, The European Physical Journal Plus, 2018,133, 1-22.
Atangana, A. and Koca, I. Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos, Solitons and Fractals, 2016, 89, 446-454.
Caputo, M. and Fabrizio, M. Applications of new time and spatial fractional derivatives with exponential kernels. Progress in Fractional Differentiation and Applications, 2016,1, 1-11.
Djida, J.D., Atangana, A. and Area, I. Numerical computation of a fractional derivative with non-local and non-singular kernel. Mathematical Modelling of Natural Phenomena, 2017,12, 4-13.
Jefain, O. and Algahtani, J. Comparing the Atangana-Baleanu and Caputo-Fabrizio derivative with fractional order: Allen Cahn model. Chaos, Solitons and Fractals, 2016,89, 552-559.
Losada, J. and Nieto, J. Properties of a new fractional derivative without singular kernel. Progress in Fractional Differentiation and Applications, 2015,2, 87-92.
Sheikh, N.A., Ali, F., Saqib, M., Khan, I., Jan, S.A.A., Alshomrani, A.S. and Alghamdi, M.S. Comparison and analysis of the Atangana-Baleanu and Caputo-Fabrizio fractional derivatives for generalized Casson fluid model with heat generation and chemical reaction. Results in Physics, 2017,7, 789-800.
Tateishi, A.A., Haroldo V. Ribeiro, H.V. and Lenzi, E. K. The Role of Fractional Time-Derivative Operators on Anomalous Diffusion, Frontiers in Physics, 2017,5, 1-9.
Podlubny, I. Fractional differential equations. Academic Press Inc., San Diego, CA, 1999.
Al-Musalhi, F., Al-Salti, N. and Karimov, E., Karimov, Initial and boundary value problems for fractional differential equations involving Atangana-Baleanu derivative, 2017, arxiv: 1706.00740.
Al-Refai, M. Comparison principles for differential equations involving Caputo fractional derivative with Mittag-Leffler non-singular kernel, Electronic Journal of Differential Equations, 2018,2018, 1-10.
Baleanu, D. and Fernandez, A. On some new properties of fractional derivatives with Mittag-Leffler kernel. Communications in Nonlinear Science and Numerical Simulation, 2018,59, 444-462.
Djida, J.D., Mophou, G.M and Area, I. Optimal control of diffusion equation with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel, Journal of Optimization Theory and Applications (2018). https://doi.org/10.1007/s10957-018-1305-6.
Moiseev, E.I. The basis property of systems of sines and cosines. Doklady Akademii Nauk SSSR, 1984,275, 794–798.