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Existence of mild solutions for fractional Schrodinger equations in extended Colombeau algebras | ||
| International Journal of Nonlinear Analysis and Applications | ||
| مقاله 4، دوره 14، شماره 2، اردیبهشت 2023، صفحه 31-44 اصل مقاله (419.78 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22075/ijnaa.2022.27296.3553 | ||
| نویسندگان | ||
| Elomari M'hamed* ؛ Said Melliani؛ Fatima Ezzahra Bourhim؛ Ali El Mfadel | ||
| Laboratory of Applied Mathematics Scientific Calculus, Sultan Moulay Slimane University, BP 523, 23000, Beni Mellal, Morocco | ||
| تاریخ دریافت: 05 خرداد 1401، تاریخ بازنگری: 10 مهر 1401، تاریخ پذیرش: 13 مهر 1401 | ||
| چکیده | ||
| The main crux of this research manuscript is to study the existence and uniqueness of generalized mild solutions for nonlinear Schrodinger equations with singular initial conditions in the extended algebras of generalized functions. The proofs are based on generalized semigroups theory and Gronwall's inequality. As an application, our theoretical results have been illustrated by providing a suitable example. | ||
| کلیدواژهها | ||
| Extended Colombeau algebra؛ generalized mild solution؛ generalized function؛ Schr\"odinger equation | ||
| مراجع | ||
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[1] J.F. Colombeau, Elementary Introduction to New Generalized Function, North Holland, Amsterdam, 1985.
[2] J.F. Colombeau, New Generalized Function and Multiplication of Distribution, North Holland, Amsterdam, NewYork, Oxford, 1984. [3] M. Grosser, M. Kunzinger and M. Oberguggenberger and R. Steinbauer, Geometric Theory of Generalized Functions with Applications to General Relativity, Mathematics and its Applications, Kluwer Acad. Publ, Dordrecht, 2001. [4] R. Hermann and M. Oberguggenberger, Ordinary differential equations and generalized functions, in: Non-linear Theory of Generalized Functions, Chapman & Hall, 1999. [5] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in: Jan van Mill (Ed.), North-Holland Mathematics Studies, vol. 204, Amsterdam, Netherlands, 2006. [6] F. Mainardi, Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, World Scientific, 2010. [7] F. Mainardi, P. Paolo and G. Rudolf, Probability distributions generated by fractional diffusion equations, arXiv preprint arXiv:0704.0320. (2007). [8] S. Mirjana, Nonlinear Schrodinger equation with singular potential and initial data, Commun. Contemp. Math. 64 (2006), no. 7, 1460–1474. [9] M. Nedeljkov, S. Pilipovic and D. Rajter, Heat equation with singular potential and singular data, Proc. Sec. A, Math. Royal Soc. Edin. 135 (2005), no.. 4, 863–886. [10] M. Oberguggenberger, Multiplication of Distributions and Applications to Partial Differential Equations, Pitman Research Notes in Mathematics, 1992. [11] K. Oldham and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, Elsevier, 1974. [12] I. Podlubny, Fractional Differential Equations, An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998. [13] M. Stojanovic, Extension of Colombeau algebra to derivatives of arbitrary order Dα, α ∈ R+ ∪ {0}: Application to ODEs and PDEs with entire and fractional derivatives, Nonlinear Anal. 71 (2009), 5458–5475. [14] Y. Zhou and F. Jiao, Existence of mild solutions for fractional neutral evolution equations, Comput. Math. Appl.59 (2010), 1063–1077. | ||
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