Year: 2013
Author: S. D. Purohit
Advances in Applied Mathematics and Mechanics, Vol. 5 (2013), Iss. 5 : pp. 639–651
Abstract
The aim of this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators, occurring in quantum mechanics. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms, in terms of the Fox's $H$-function. Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented. The results given earlier by Saxena et al. [Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177-190] and Purohit and Kalla [J. Phys. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.12-m1298
Advances in Applied Mathematics and Mechanics, Vol. 5 (2013), Iss. 5 : pp. 639–651
Published online: 2013-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 13
Keywords: Fractional Schrödinger equation Laplace transform Fourier transform Hilfer fractional derivative Fox's $H$-function and Quantum mechanics.
Author Details
-
Generalized fractional integrals of product of twoH-functions and a general class of polynomials
Baleanu, D. | Kumar, Dinesh | Purohit, S.D.International Journal of Computer Mathematics, Vol. 93 (2016), Iss. 8 P.1320
https://doi.org/10.1080/00207160.2015.1045886 [Citations: 20] -
The Analytical Analysis of Time-Fractional Fornberg–Whitham Equations
Alderremy, A. A. | Khan, Hassan | Shah, Rasool | Aly, Shaban | Baleanu, DumitruMathematics, Vol. 8 (2020), Iss. 6 P.987
https://doi.org/10.3390/math8060987 [Citations: 23] -
On the solutions for generalised multiorder fractional partial differential equations arising in physics
Purohit, Sunil Dutt | Baleanu, Dumitru | Jangid, KamleshMathematical Methods in the Applied Sciences, Vol. 46 (2023), Iss. 7 P.8139
https://doi.org/10.1002/mma.7431 [Citations: 3] -
Successive iterations and positive extremal solutions for a Hadamard type fractional integro-differential equations on infinite domain
Pei, Ke | Wang, Guotao | Sun, YanyanApplied Mathematics and Computation, Vol. 312 (2017), Iss. P.158
https://doi.org/10.1016/j.amc.2017.05.056 [Citations: 45] -
Solving fractional partial differential equations by using Bernstein polynomials with artificial neural networks
Mohammad, Susan H. | Al-Rawi, Ekhlass S.PHYSICAL MESOMECHANICS OF CONDENSED MATTER: Physical Principles of Multiscale Structure Formation and the Mechanisms of Nonlinear Behavior: MESO2022, (2023), P.060020
https://doi.org/10.1063/5.0157008 [Citations: 1] -
On Euler type integrals involving extended Mittag-Leffler functions
Mittal, Ekta | Joshi, Sunil | Pandey, Rupakshi MishraBoletim da Sociedade Paranaense de Matemática, Vol. 38 (2018), Iss. 2 P.123
https://doi.org/10.5269/bspm.v38i2.34829 [Citations: 1] -
A second-order parareal algorithm for fractional PDEs
Wu, Shu-Lin
Journal of Computational Physics, Vol. 307 (2016), Iss. P.280
https://doi.org/10.1016/j.jcp.2015.12.007 [Citations: 3] -
On fractional kinetic equationsk-Struve functions based solutions
Nisar, Kottakkaran Sooppy | Mondal, Saiful Rahman | Belgacem, Fethi Bin MuhammadAlexandria Engineering Journal, Vol. 57 (2018), Iss. 4 P.3249
https://doi.org/10.1016/j.aej.2018.01.010 [Citations: 3] -
Fractional calculus pertaining to multivariable Aleph-function
Kumar, Dinesh | Ayant, FredericBoletim da Sociedade Paranaense de Matemática, Vol. 40 (2022), Iss. P.1
https://doi.org/10.5269/bspm.42941 [Citations: 0] -
Numerical Study of Schrödinger Equation Using Differential Quadrature Method
Bhatia, Rachna | Mittal, R. C.International Journal of Applied and Computational Mathematics, Vol. 4 (2018), Iss. 1
https://doi.org/10.1007/s40819-017-0470-x [Citations: 1] -
Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel
Yépez-Martínez, H. | Gómez-Aguilar, J.F. | Atangana, Abdon | Mophou, Gisèle | Hristov, Jordan | Hammouch, ZakiaMathematical Modelling of Natural Phenomena, Vol. 13 (2018), Iss. 1 P.13
https://doi.org/10.1051/mmnp/2018002 [Citations: 17] -
The composition of extended Mittag-Leffler functions with pathway integral operator
Rahman, G | Ghaffar, A | Mubeen, S | Arshad, M | Khan, SUAdvances in Difference Equations, Vol. 2017 (2017), Iss. 1
https://doi.org/10.1186/s13662-017-1237-8 [Citations: 5] -
Some generating functions and properties of extended second Appell function
Parmar, Rakesh K. | Purohit, Sunil DuttBoletim da Sociedade Paranaense de Matemática, Vol. 37 (2017), Iss. 1 P.169
https://doi.org/10.5269/bspm.v37i1.30725 [Citations: 0] -
COMPARISON PRINCIPLES FOR HADAMARD-TYPE FRACTIONAL DIFFERENTIAL EQUATIONS
YIN, CHUNTAO | MA, LI | LI, CHANGPINFractals, Vol. 26 (2018), Iss. 04 P.1850056
https://doi.org/10.1142/S0218348X18500561 [Citations: 9] -
Certain Fractional Integral Formulas Involving the Product of Generalized Bessel Functions
Baleanu, D. | Agarwal, P. | Purohit, S. D. | Atangana, A. | Noutchie, S. C. O. | Ray, S. S. | Secer, A.The Scientific World Journal, Vol. 2013 (2013), Iss. 1
https://doi.org/10.1155/2013/567132 [Citations: 6] -
On the solutions of certain fractional kinetic equations involving k-Mittag-Leffler function
Agarwal, P. | Chand, M. | Baleanu, D. | O’Regan, D. | Jain, ShilpiAdvances in Difference Equations, Vol. 2018 (2018), Iss. 1
https://doi.org/10.1186/s13662-018-1694-8 [Citations: 34] -
Wavelet methods for fractional electrical circuit equations
Tural-Polat, Sadiye Nergis | Dincel, Arzu TuranPhysica Scripta, Vol. 98 (2023), Iss. 11 P.115203
https://doi.org/10.1088/1402-4896/acfacc [Citations: 1] -
Fractional operators with generalized Mittag-Leffler k-function
Mubeen, Shahid | Safdar Ali, RanaAdvances in Difference Equations, Vol. 2019 (2019), Iss. 1
https://doi.org/10.1186/s13662-019-2458-9 [Citations: 6] -
Analysis of semi-analytical method for solving fuzzy fractional differential equations with strongly nonlinearity under caputo derivative sense
Hashim, Dulfikar Jawad | Jameel, Ali Fareed | Ying, Teh YuanTHE SECOND INTERNATIONAL SCIENTIFIC CONFERENCE (SISC2021): College of Science, Al-Nahrain University, (2023), P.020010
https://doi.org/10.1063/5.0118685 [Citations: 0] -
Numerical Solutions for Space Fractional Schrödinger Equation Through Semiclassical Approximation
Gao, Yijin | Sacks, Paul | Luo, SongtingCommunications on Applied Mathematics and Computation, Vol. (2024), Iss.
https://doi.org/10.1007/s42967-024-00384-z [Citations: 0] -
RETRACTED ARTICLE: Research on statistical algorithm optimization of fractional differential equations of quantum mechanics in ecological compensation
Zhao, Wei | Leng, Kaijun | Chen, Jinbo | Jiao, Yuanze | Zhao, QiongThe European Physical Journal Plus, Vol. 134 (2019), Iss. 7
https://doi.org/10.1140/epjp/i2019-12700-5 [Citations: 3] -
A basic study of a fractional integral operator with extended Mittag-Leffler kernel
Rahman, Gauhar | Suwan, Iyad | Nisar, Kottakkaran Sooppy | Abdeljawad, Thabet | Samraiz, Muhammad | Ali, AsadAIMS Mathematics, Vol. 6 (2021), Iss. 11 P.12757
https://doi.org/10.3934/math.2021736 [Citations: 1] -
Mathematical analysis and numerical simulation of chaotic noninteger order differential systems with Riemann‐Liouville derivative
Owolabi, Kolade M.
Numerical Methods for Partial Differential Equations, Vol. 34 (2018), Iss. 1 P.274
https://doi.org/10.1002/num.22197 [Citations: 40] -
On flow of electric current in RL circuit using Hilfer type composite fractional derivative
Kachhia, Krunal B. | Prajapati, J. C. | Pandya, K. S. | Jadea, R.Proyecciones (Antofagasta), Vol. 38 (2019), Iss. 4 P.625
https://doi.org/10.22199/issn.0717-6279-2019-04-0040 [Citations: 1] -
A numerical method for fractional Schrödinger equation of Lennard-Jones potential
Al-Raeei, Marwan | Sayem El-Daher, MoustafaPhysics Letters A, Vol. 383 (2019), Iss. 26 P.125831
https://doi.org/10.1016/j.physleta.2019.07.019 [Citations: 22] -
RETRACTED ARTICLE: Fractional-order scheme for bovine babesiosis disease and tick populations
Zafar, Zain Ul Abadin | Rehan, Kashif | Mushtaq, MAdvances in Difference Equations, Vol. 2017 (2017), Iss. 1
https://doi.org/10.1186/s13662-017-1133-2 [Citations: 23] -
An approximate analytical view of physical and biological models in the setting of Caputo operator via Elzaki transform decomposition method
Rashid, Saima | Kubra, Khadija Tul | Sultana, Sobia | Agarwal, Praveen | Osman, M.S.Journal of Computational and Applied Mathematics, Vol. 413 (2022), Iss. P.114378
https://doi.org/10.1016/j.cam.2022.114378 [Citations: 34] -
Mathematical Analysis and a Second-Order Compact Scheme for Nonlinear Caputo–Hadamard Fractional Sub-diffusion Equations
Guan, Kaijing | Ou, Caixia | Wang, ZhiboMediterranean Journal of Mathematics, Vol. 21 (2024), Iss. 3
https://doi.org/10.1007/s00009-024-02617-0 [Citations: 5] -
Exact solutions of (1+2)-dimensional non-linear time-space fractional PDEs
Kumar, Manoj
Arab Journal of Mathematical Sciences, Vol. 30 (2024), Iss. 1 P.30
https://doi.org/10.1108/AJMS-11-2021-0282 [Citations: 0] -
Integro-differential fractional boundary value problem on an unbounded domain
Wang, Dong | Wang, GuotaoAdvances in Difference Equations, Vol. 2016 (2016), Iss. 1
https://doi.org/10.1186/s13662-016-1051-8 [Citations: 8] -
A new nonlinear triadic model of predator–prey based on derivative with non-local and non-singular kernel
Saad T Alkahtani, Badr | Atangana, Abdon | Koca, IlknurAdvances in Mechanical Engineering, Vol. 8 (2016), Iss. 11
https://doi.org/10.1177/1687814016681906 [Citations: 9] -
Marichev-Saigo-Maeda fractional calculus operators, Srivastava polynomials and generalized Mittag-Leffler function
Mishra, Vishnu Narayan | Suthar, D.L. | Purohit, S.D. | Srivastava, Hari M.Cogent Mathematics, Vol. 4 (2017), Iss. 1 P.1320830
https://doi.org/10.1080/23311835.2017.1320830 [Citations: 17] -
Existence and Uniqueness Results for Hadamard-Type Fractional Differential Equations with Nonlocal Fractional Integral Boundary Conditions
Thiramanus, Phollakrit | Ntouyas, Sotiris K. | Tariboon, JessadaAbstract and Applied Analysis, Vol. 2014 (2014), Iss. P.1
https://doi.org/10.1155/2014/902054 [Citations: 20] -
Analysis of non-homogeneous heat model with new trend of derivative with fractional order
Alkahtani, Badr Saad T. | Atangana, AbdonChaos, Solitons & Fractals, Vol. 89 (2016), Iss. P.566
https://doi.org/10.1016/j.chaos.2016.03.027 [Citations: 94] -
Multigrid Waveform Relaxation for the Time-Fractional Heat Equation
Gaspar, Francisco J. | Rodrigo, CarmenSIAM Journal on Scientific Computing, Vol. 39 (2017), Iss. 4 P.A1201
https://doi.org/10.1137/16M1090193 [Citations: 21] -
A Hybrid Method to Solve Time-Space Fractional PDEs with Proportional Delay
Kumar, Manoj
International Journal of Applied and Computational Mathematics, Vol. 8 (2022), Iss. 2
https://doi.org/10.1007/s40819-022-01277-6 [Citations: 0] -
Certain integral transforms concerning the product of family of polynomials and generalized incomplete functions
Meena, Sapna | Bhatter, Sanjay | Jangid, Kamlesh | Purohit, Sunil DuttMoroccan Journal of Pure and Applied Analysis, Vol. 6 (2020), Iss. 2 P.243
https://doi.org/10.2478/mjpaa-2020-0019 [Citations: 4] -
A non-integer order dengue internal transmission model
Zafar, Zain Ul Abadin | Mushtaq, Muhammad | Rehan, KashifAdvances in Difference Equations, Vol. 2018 (2018), Iss. 1
https://doi.org/10.1186/s13662-018-1472-7 [Citations: 19] -
COMPARISON THEOREMS FOR CAPUTO–HADAMARD FRACTIONAL DIFFERENTIAL EQUATIONS
MA, LI
Fractals, Vol. 27 (2019), Iss. 03 P.1950036
https://doi.org/10.1142/S0218348X19500361 [Citations: 21] -
Using hierarchical matrices in the solution of the time-fractional heat equation by multigrid waveform relaxation
Hu, Xiaozhe | Rodrigo, Carmen | Gaspar, Francisco J.Journal of Computational Physics, Vol. 416 (2020), Iss. P.109540
https://doi.org/10.1016/j.jcp.2020.109540 [Citations: 6] -
Analysis of reaction–diffusion system via a new fractional derivative with non-singular kernel
Saad, Khaled M. | Gómez-Aguilar, J.F.Physica A: Statistical Mechanics and its Applications, Vol. 509 (2018), Iss. P.703
https://doi.org/10.1016/j.physa.2018.05.137 [Citations: 105]