Can Fractional Calculus Be Generalized: Problems and Efforts
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 3, 2018, 1058-1099 ISSN 1307-5543 { www.ejpam.com Published by New York Business Global Can Fractional Calculus be Generalized? Problems and Efforts Syamal K.Sen1;2, J. Vasundhara Devi1;3,∗, R.V.G. Ravi Kumar3 1 GVP-Prof.V.Lakshmikantham Institute for Advanced Studies, GVP College of Engineering Campus, Visakhapatnam 530048, India 2 Department of Computer Science and Engineering, GVP College of Engineering, Visakhapatnam, AP, India 3 Department of Mathematics, GVP College of Engineering(A), Visakhapatnam, AP, India Abstract. Fractional order calculus always includes integer-order too. The question that crops up is: Can it be a widely accepted generalized version of classical calculus? We attempt to highlight the current problems that come in the way to define the fractional calculus that will be universally accepted as a perfect generalized version of integer-order calculus and to point out the efforts in this direction. Also, we discuss the question: Given a non-integer fractional order differential equation as a mathematical model can we readily write the corresponding physical model and vice versa in the same way as we traditionally do for classical differential equations? We demonstrate numerically computationally the pros and cons while addressing the questions keeping in the background the generalization of the inverse of a matrix. 2010 Mathematics Subject Classifications: 26A33, 26A36, 34A08, 44A45 Key Words and Phrases: Differintegrals, fractional calculus, fractional differential equations, generalization of calculus, integer order calculus 1. Introduction Fractional calculus is the calculus investigating the properties of derivatives and inte- grals of both integer order and non-integer fractional order called fractional derivatives and fractional integrals, in short differintegrals.
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