This study presents a robust kernel-based collocation method (KBCM) for solving multi-term variable-order time fractional partial differential equations (VOTFPDEs). In the proposed method, Radial ...basis functions (RBFs) and Muntz polynomials basis (MPB) are implemented to discretize the spatial and temporal derivative terms in the VOTFPDEs, respectively. Due to the properties of the RBFs, the spatial discretization in the proposed method is mathematically simple and truly meshless, which avoids troublesome mesh generation for high-dimensional problems involving irregular geometries. Due to the properties of the MPB, only few temporal discretization is required to achieve the satisfactory accuracy. Numerical efficiency of the proposed method is investigated under several typical examples.
We first show that four fractional integro-differential inclusions have solutions. Also, we show that dimension of the set of solutions for the second fractional integro-differential inclusion ...problem is infinite dimensional under some different conditions.
In this paper, we treat some fractional differential equations on the sequence Lebesgue spaces ℓp(N0) with p≥1. The Caputo fractional calculus extends the usual derivation. The operator, associated ...to the Cauchy problem, is defined by a convolution with a sequence of compact support and belongs to the Banach algebra ℓ1(Z). We treat in detail some of these compact support sequences. We use techniques from Banach algebras and a Functional Analysis to explicity check the solution of the problem.
A localized meshless collocation method, namely the generalized finite difference method (GFDM), is introduced to cope with the inverse Cauchy problem associated with the fractional heat conduction ...model under functionally graded materials (FGMs). The variable-order time-fractional heat conduction equation under the Caputo definition is employed to describe anomalous heat conduction problems. In the present numerical framework, temporal-discretization is implemented by using the standard implicit finite difference method with the spatial-discretization through the GFDM. On the basis of moving least squares and Taylor series expansion, the GFDM is capable of avoiding the ill-posedness in inverse Cauchy problems and solving the time fractional heat conduction equations (TFHCE) under FGMs. To give evidence of the efficiency and accuracy of the proposed approach for solving inverse heat conduction of FGMs, three numerical experiments are considered in the results and discussions section.
This paper introduces a novel model for image fusion that is based on a fractional-order osmosis approach. The model incorporates a definition of osmosis energy that takes into account nonlocal pixel ...relationships using fractional derivatives and contrast change. The proposed model was subjected to theoretical and experimental investigation. The semigroup theory was used to demonstrate the existence and uniqueness of the evolution equation solution. Additionally, the model was validated and tested using numerical experiments and compared to local image fusion methods. The findings demonstrate that the proposed model outperforms the competitive local image fusion models.
A model of the self-similar process of relaxation is given, and a method of derivation of the kinetic equations for the total polarization based on the ideas of fractional kinetics is suggested. The ...derived kinetic equations contain integro-differential operators having non-integer order. They lead to the Cole–Cole expression for the complex dielectric permittivity. It is shown rigorously that the power-law exponent α in the Cole–Cole expression coincides with the dimension of the mixed space-temporal fractal ensemble. If the discrete scale invariance for the temporal-space structure of the dielectric medium considered becomes important, then the expression for the complex dielectric permittivity contains log-periodic corrections (oscillations) and, hence, it generalizes the conventional Cole–Cole expression. The corrections obtained in this model suggest another way of interpretation and analysis of dielectric spectra for different complex materials.
► The physical meaning of power-law exponent in the Cole–Cole law is given. ► Log-periodic corrections (oscillations) to the Cole–Cole law are obtained. ► Log-periodic corrections suggest another way of analysis of dielectric spectra.
Successive approximation method for Caputo q-fractional IVPs Salahshour, Soheil; Ahmadian, Ali; Chan, Chee Seng
Communications in nonlinear science & numerical simulation,
July 2015, 2015-07-00, 20150701, Volume:
24, Issue:
1-3
Journal Article
Peer reviewed
•This paper studies the convergence of the Caputo q-fractional initial value problem.•It is studied under the q-Krasnoselskii–Krein type condition for the first time.•The aim is to handle the missing ...convergence investigation in ref 1.
Recently, Abdeljawad and Baleanu (2011) introduced Caputo q-fractional derivatives and used it to solve Caputo q-fractional initial value problem. For this purpose, they applied successive approximation method to obtain an explicit solution; but did not clarify under which conditions that this method will be convergence. In this paper, we propose q-Krasnoselskii–Krein type condition to investigate the convergence of the method.
Actuators made of dielectric elastomers are used in soft robotics for a variety of applications. However, due to their mechanical properties, they exhibit viscoelastic behaviour, especially in the ...initial phase of their performance, which can be observed in the first cycles of dynamic excitation. A fully fractional generalised Maxwell model was derived and used for the first time to capture the viscoelastic effect of dielectric elastomer actuators. The Laplace transform was used to derive the fully fractional generalised Maxwell model. The Laplace transform has proven to be very useful and practical in deriving fractional viscoelastic constitutive models. Using the global optimisation procedure called Pattern Search, the optimal parameters, as well as the number of branches of the fully fractional generalised Maxwell model, were derived from the experimental results. For the fully fractional generalised Maxwell model, the optimal number of branches was determined considering the derivation order of each element of the branch. The derived model can readily be implemented in the simulation of a dielectric elastomer actuator control, and can also easily be used for different viscoelastic materials.
The theoretical generalization of the Jonscher's relationship for the complex conductivity of carriers moving in self-similar medium is derived. It is shown that the correction derived enters to more ...general expression, which, in turn, we define as the generalized Jonscher's relationship. The basic idea which was used for the derivation of the relationship is based on the supposition that disordered medium has self-similar property. The derived relationship is confirmed on dielectric spectroscopy data related to sodium nitrite embedded to porous glasses. Based on new relationship there is a possibility to extract additional information about relaxation processes of a system of dipoles from the processes related to conductivity. It is important in the cases when the contribution to relaxation peaks is small and unnoticeable on the background of essential domination of processes related to conductivity.
► The generalized expression for the Jonscher's conductivity was obtained. ► This new expression was confirmed experimentally on dielectric spectra of the sodium nitrate embedded in porous glass. ► In low-frequency region the suggested expression is reduced to the conventional Jonscher's expression.
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