Nonlinear Fractional Model Research Articles

Modeling and Application of a New Nonlinear Fractional Financial Model

The paper proposes a new nonlinear dynamic econometric model with fractional derivative. The fractional derivative is defined in the Jumarie type. The corresponding discrete financial system is considered by removing the limit operation in Jumarie derivative’s. We estimate the coefficients and parameters of the model by using the least squared principle. The new approach to financial system modeling is illustrated by an application to model the behavior of Japanese national financial system which consists of interest rate, investment, and inflation. The empirical results with different time step sizes of discretization are shown, and a comparison of the actual data against the data estimated by empirical model is illustrated. We find that our discrete financial model can describe the actual data that include interest rate, investment, and inflation accurately.

On the numerical handling of fractional viscoelastic material models in a FE analysis

The paper presents the finite element (FE) implementation of linear and nonlinear fractional viscoelasticity models. To this end, a short introduction on fractional calculus is given. In addition to the fractional operators, this includes analytical and numerical solution schemes for selected fractional integral and differential equations. The presented rheological model is based on state of the art approaches and has been adopted to model the strain rate dependent material behavior of polymers. To this end, two approaches with constant and overstress dependent viscous properties resulting in linear and nonlinear evolution equations are discussed. The uniaxial constitutive relations are generalized to the multiaxial case and processed to be implemented in a FE code. The model behavior of both approaches is demonstrated and compared for selected uniaxial and multiaxial load cases.

Individualization of a pharmacokinetic model by fractional and nonlinear fit improvement

This study presents application of a new linear and nonlinear fractional derivative two compartmental model to the evaluation of individual pharmacokinetics. In the model, the integer order derivatives are replaced by derivatives of real order. A specific nonlinear function is used for the fit improvement of a fractional derivative two compartmental model with the mass balance conservation. The agreement of the values predicted by the proposed model with the values obtained through experiments with bumetanide tablets in human volunteers is shown to be good. Thus, pharmacokinetics of bumetanide can be described well by a linear or a nonlinear two compartmental model with fractional derivatives of the same order proposed here. Parameters in the model are determined by the least squares method and the particle swarm optimization numerical procedure is used. The results show that the linear fractional order two compartmental model for bumetanide is useful improvement of the classical (integer order) two compartmental model and that the nonlinear fractional order model is useful improvement of the linear model in a great number of volunteers.

Fractional sub-equation method and its applications to nonlinear fractional PDEs

A fractional sub-equation method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, the nonlinear time fractional biological population model and (4+1)-dimensional space–time fractional Fokas equation are considered. As a result, three types of exact analytical solutions are obtained.

Implicit Fractional Model Order Estimation Using Interacting Multiple Model Kalman Filters

AbstractThe paper deals with implicit and explicit approaches for fractional nonlinear model order estimation using a benchmark model relating applied angular rate and neuron's firing intensity within the vestibular system. The implicit approach is based on an interacting multiple models scheme, where several extended Kalman filters with fixed fractional order nonlinear models are running in parallel. An alternative method based on augmented Unscented Kalman filter is proposed where, where the fractional order of the model is estimated explicitly within the filter state. The performance of the estimators is compared with respect to parameters of estimators and fractional derivative approximation.

Reliable analysis for the nonlinear fractional calculus model of the semilunar heart valve vibrations

AbstractThe aim of this paper is to solve the equation of motion of semilunar heart valve vibrations using the homotopy perturbation method. The vibrations of the closed semilunar valves were modeled with fractional derivatives. The fractional derivatives are described in the Caputo sense. The methods give an analytic solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. Analytical solution is obtained for the equation of motion in terms of Mittag–Leffler function with the help of Laplace transformation. These solutions can be interesting for a better fit of experimental data. Copyright © 2010 John Wiley & Sons, Ltd.

Nonlinear Fractional Derivative Models of Viscoelastic Impact Dynamics Based on Entropy Elasticity and Generalized Maxwell Law

Two types of models are proposed for describing nonlinear fractional derivative dynamical behavior of viscoelastic materials subject to impulse forces. The models are derived based on the thermodynamic elasticity in terms of entropy and on the “scale-free response of the material” under the basic assumption that the viscoelastic materials consist of stable coils of polymers, which we refer to as blobs. The blobs, which may be connected to each other by chemical bonds or physical bonds, are considered here as the elementary constituent of viscoelastic materials from which the nonlinear fractional derivative models are derived. Responses of individual blobs can determine the net collective response of the viscoelastic material to impulse forces. From the above consideration, two types of models are proposed in which the force elements or the stress elements are connected by the generalized Maxwell law.

Linear programming method for MADM with interval-valued intuitionistic fuzzy sets

Fuzziness is inherent in decision data and decision making process. In this paper, interval-valued intuitionistic fuzzy (IVIF) sets are used to capture fuzziness in multiattribute decision making (MADM) problems. The purpose of this paper is to develop a methodology for solving MADM problems with both ratings of alternatives on attributes and weights being expressed with IVIF sets. In this methodology, a weighted absolute distance between IF sets is defined using weights of IF sets. Based on the concept of the relative closeness coefficients, we construct a pair of nonlinear fractional programming models which can be transformed into two simpler auxiliary linear programming models being used to calculate the relative closeness coefficient intervals of alternatives to the IVIF positive ideal solution, which can be employed to generate ranking order of alternatives based on the concept of likelihood of interval numbers. The proposed method is illustrated with a real example.

A nonlinear fractional derivative model of impulse motion for viscoelastic materials

Generally, force can be described as a function of displacement in the mechanical model. A nonlinear fractional derivative model with respect to displacement is proposed to describe force for a viscoelastic material based on the measured data of impulsive motion. In the model, the nonlinearity is assumed to appear in the term of the fractional derivative. Three types of nonlinearity in the fractional derivative term are considered as candidates for a suitable model for reproducing the impulsive responses of the measured data. The first one is the case where the nonlinearity appears in the coefficient of the fractional derivative and the second in the fractionally differentiated term. The third one is the case where the nonlinearity appears as the combination of the above two types. The equation of motion and the initial conditions are derived by employing the above nonlinear models for head-on collisions of a rigid body onto the viscoelastic material. The property of the impulsive responses for the system that is derived above is characterized by the time when the acceleration shows its maximum. The symmetry property of increasing and decreasing acceleration response about the time of maximum acceleration is also considered. The second-type nonlinearity in the model seems to be adequate for reproducing the measured response.

Long time behavior for a nonlinear fractional model

The asymptotic behavior for solutions of a weighted Cauchy-type nonlinear fractional problem is investigated. We find bounds for solutions on infinite time intervals and also provide sufficient conditions assuring decay to zero. This work improves earlier results by the same authors.

Modelling infiltration by means of a nonlinear fractional diffusion model

The classical Richards equation describes infiltration into porous soil as a nonlinear diffusion process. Recent experiments have suggested that this process exhibits anomalous scaling behaviour. These observations suggest generalizing the classical Richards equation by introducing fractional time derivatives. The resulting fractional Richards equation with appropriate initial and boundary values is solved numerically in this paper. The numerical code is tested against analytical solutions in the linear case. Saturation profiles are calculated for the fully nonlinear fractional Richards equation. Isochrones and isosaturation curves are given. The cumulative moisture intake is found as a function of the order of the fractional derivative. These results are compared against experiment.

A nonlinear fractional derivative model for large uni-axial deformation behavior of polyurethane foam

In uni-axial quasi-static tests, flexible polyurethane foam undergoing large compressive loading and unloading deformation exhibits highly nonlinear and viscoelastic behavior. In particular, the response in the first cycle is observed to be significantly different from the response in subsequent cycles. In addition, the stresses in the loading paths are higher than those in unloading paths. This quasi-static response is modeled as an additive sum of a nonlinear elastic and a linear viscoelastic response. The nonlinear elastic behavior is modeled as a polynomial function of compressive strain, while the viscoelastic behavior is modeled as a parallel combination of five-parameter fractional derivative models. The focus of this paper is to develop a multi-element fractional derivative model that can capture the multi-cycle behavior. A parameter estimation procedure based on separating the contributions of the viscoelastic elements and the nonlinear elastic component is developed. This approach is applied to both simulation and experimental data. The combination of a nonlinear elastic component and a two-element fractional derivative model is found to predict the observed responses reasonably well. The results for two distinct foams from tests with different compression rates are compared and discussed.