A semi-analytic method with an effect of memory for solving fractional differential equations

In this paper, we propose a new modification of the multistage generalized differential transform method (MsGDTM) for solving fractional differential equations. In MsGDTM, it is the key how to impose an initial condition in each sub-domain to obtain an accurate approximate solution. In several literature works (Odibat et al. in Comput. Math. Appl. 59:1462-1472, 2010; Alomari in Comput. Math. Appl. 61:2528-2534, 2011; Gokdoğan et al. in Math. Comput. Model. 54:2132-2138, 2011), authors have updated an initial condition in each sub-domain by using the approximate solution in the previous sub-domain. However, we point out that this approach is hard to apply an effect of memory which is the basic property of fractional differential equations. Here we provide a new algorithm to impose the initial conditions by using the integral operator that enhances accuracy. Several illustrative examples are demonstrated, and it is shown that the proposed technique is robust and accurate for solving fractional differential equations.close0

Variational Problems with Fractional Derivatives: Euler-Lagrange Equations

We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Also, for the standard case when these two bounds coincide, we derive a new form of Euler-Lagrange equations. We use approximations for fractional derivatives in the Lagrangian and obtain the Euler-Lagrange equations which approximate the initial Euler-Lagrange equations in a weak sense

A numerical study of fractional relaxation–oscillation equations involving ψ-Caputo fractional derivative

We provide a numerical method to solve a certain class of fractional differential equations involving ψ -Caputo fractional derivative. The considered class includes as particular case fractional relaxation–oscillation equations. Our approach is based on operational matrix of fractional integration of a new type of orthogonal polynomials. More precisely, we introduce ψ -shifted Legendre polynomial basis, and we derive an explicit formula for the ψ -fractional integral of ψ -shifted Legendre polynomials. Next, via an orthogonal projection on this polynomial basis, the problem is reduced to an algebraic equation that can be easily solved. The convergence of the method is justified rigorously and confirmed by some numerical experiments.publishe

Geometry and field theory in multi-fractional spacetime

We construct a theory of fields living on continuous geometries with fractional Hausdorff and spectral dimensions, focussing on a flat background analogous to Minkowski spacetime. After reviewing the properties of fractional spaces with fixed dimension, presented in a companion paper, we generalize to a multi-fractional scenario inspired by multi-fractal geometry, where the dimension changes with the scale. This is related to the renormalization group properties of fractional field theories, illustrated by the example of a scalar field. Depending on the symmetries of the Lagrangian, one can define two models. In one of them, the effective dimension flows from 2 in the ultraviolet (UV) and geometry constrains the infrared limit to be four-dimensional. At the UV critical value, the model is rendered power-counting renormalizable. However, this is not the most fundamental regime. Compelling arguments of fractal geometry require an extension of the fractional action measure to complex order. In doing so, we obtain a hierarchy of scales characterizing different geometric regimes. At very small scales, discrete symmetries emerge and the notion of a continuous spacetime begins to blur, until one reaches a fundamental scale and an ultra-microscopic fractal structure. This fine hierarchy of geometries has implications for non-commutative theories and discrete quantum gravity. In the latter case, the present model can be viewed as a top-down realization of a quantum-discrete to classical-continuum transition.Comment: 1+82 pages, 1 figure, 2 tables. v2-3: discussions clarified and improved (especially section 4.5), typos corrected, references added; v4: further typos correcte