The fractional derivative was introduced in 1695 by leibnitz as a generalization of the integer order derivative and was reconsidered also by euler, abel, riemann liouville, grunwald and letnikov. In this work, we considered the qdifferential equations of order 0 fractional calculus. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. The existence and uniqueness theorems are derived using successive approximations, leading to systems of equations with retarded arguments. Fractional calculus fc started with the ideas of gottfried leibniz by the end of the xvii century. The fractional calculus may be considered an old and yet novel topic.
It starts with elementary calculus of q differences and integration of jacksons type before turning to q difference equations. Our analysis, based on the classical linear theory of viscoelsticity recalled in chapter 2, will start from the power law creep to. An introduction to the fractional calculus and fractional differential equations book. It starts with elementary calculus of qdifferences and integration of jacksons type. Annaby and others published q fractional calculus and equations find, read and cite all the research you need. Despite generalized would be a better option, the name fractional is used for denoting this kind of derivative. Features mathematical modeling techniques and realworld processes with applications in diverse fields. As in the classical theory of ordinary fractional differential equations, qdifference equations of fractional order are divided into linear, nonlinear, homogeneous, and inhomogeneous equations with constant and variable coefficients. Fractional calculus is the branch of calculus that generalizes the derivative of a function to noninteger order, allowing calculations such as deriving a function to 12 order.
Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation and integration operators. Pdf study of new class of qfractional derivative and its properties. This ninechapter monograph introduces a rigorous investigation of q difference operators in standard and fractional settings. The multiplicity solutions for nonlinear fractional differential. The number of studies in this field has increased after 1930, when e. A note on hadamard fractional differential equations with varying coefficients and their. Annaby and others published qfractional calculus and equations find, read and cite all the research you need on researchgate. Mathematical modeling with multidisciplinary applications details the interdisciplinary nature of mathematical modeling and numerical algorithms.