Fractional Differential Equations

International Journal of Computer and Applications

During the last decade, a big progress has been achieved in the analysis and numerical treatment of Initial Value Problems (IVPs) in Differential Algebraic Equations (DAEs) and Ordinary Differential Equations (ODEs). In spite of the rich variety of results available in the literature, there are still many specific problems that require special attention. Two of such, which are considered in this work, are the optimization of order of accuracy and reduction of cost of functional evaluations of Explicit Runge - Kutta (ERK) methods.

Traditionally, the maximum attainable order p of an s-stage ERK method for advancing the solution of an IVP is such that
p(s)
 4
 
In 1999, Goeken presented an s-stage ERK Method of order p(s)=s +1,s>2.
However, this work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher 
U n and Jonhson [94]; and the classical ERK methods. The order p of 
the new scheme called Multiderivative Explicit Runge-Kutta (MERK) Methods is such that p(s) 2. The stability, convergence and implementation for the optimization of IVPs in DAEs and ODEs systems are also considered.

Numerical solution of initial value problems in differential - algebraic equations

To classify brain images into pathological or healthy is a key pre-clinical state for patients. Manual classification is tiresome, expensive, time-consuming, and irreproducible. In this study, we aimed to present an automatic computer-aided system for brain-image classification. We used 90 T2-weighted images obtained by magnetic resonance images. First, we used weighted-type fractional Fourier transform WFRFT to extract spectrums from each magnetic resonance image. Second, we used principal component analysis PCA to reduce spectrum features to only 26. Third, those reduced spectral features of different samples were combined and were fed into support vector machine SVM and its two variants: generalized eigenvalue proximal SVM and twin SVM. A 5 × 5-fold cross-validation results showed that this proposed "WFRFT+PCA+generalized eigenvalue proximal SVM" yielded sensitivity of 99.53%, specificity of 92.00%, precision of 99.53%, and accuracy of 99.11%, which are comparable with the proposed "WFRFT+PCA+twin SVM" and better than the proposed "WFRFT+PCA+SVM." Besides, all three proposed methods were superior to eight state-of-the-art algorithms. Thus, WFRFT is effective, and the proposed methods can be used in practical. © 2015 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 25, 317-327, 2015

Magnetic resonance brain image classification based on weighted-type fractional Fourier transform and nonparallel support vector machine

One of the important classes of coupled systems of algebraic, differential and fractional differential equations (CSADFDEs) is fractional differential algebraic equations (FDAEs). The main difference of such systems with other class of CSADFDEs is that their singularity remains constant in an interval. However, complete classifying and analyzing of these systems relay mainly to the concept of the index which we introduce in this paper. For a system of linear differential algebraic equations (DAEs) with constant coefficients, we observe that the solvability depends on the regularity of the corresponding pencils. However, we show that in general, similar properties of DAEs do not hold for FDAEs. In this paper, we introduce some practical applications of systems of FDAEs in physics such as a simple pendulum in a Newtonian fluid and electrical circuit containing a new practical element namely fractors. We obtain the index of introduced systems and discuss the solvability of these systems. We numerically solve the FDAEs of a pendulum in a fluid with three different fractional derivatives (Liouville–Caputo’s definition, Caputo–Fabrizio’s definition and with a definition with Mittag–Leffler kernel) and compare the effect of different fractional derivatives in this modeling. Finally, we solved some existing examples in research and showed the effectiveness and efficiency of the proposed numerical method.

System of fractional differential algebraic equations with applications

In this paper, a novel and effective stiction detection method is proposed by combining K-means clustering and the moving window approach As a byproduct, the proposed stiction detection method off

Valve Stiction Detection and Quantification Using a K-Means Clustering Based Moving Window Approach

Numerical solution of multi-order fractional differential equations using generalized triangular function operational matrices

A Gaussian mixture model based virtual sample generation approach for small datasets in industrial processes

Modern Machine Learning Tools for Monitoring and Control of Industrial Processes: A Survey

This article introduces a new application of piecewise linear orthogonal triangular functions to solve fractional order differential-algebraic equations. The generalized triangular function operational matrices for approximating Riemann-Liouville fractional order integral in the triangular function (TF) domain are derived. Error analysis is carried out to estimate the upper bound of absolute error between the exact Riemann-Liouville fractional order integral and its TF approximation. Using the proposed generalized operational matrices, linear and nonlinear fractional order differential-algebraic equations are solved. The results show that the TF estimate of Riemann-Liouville fractional order integral is accurate and effective.

Seshu Kumar Damarla

Papers

Valve Stiction Detection and Quantification Using a K-Means Clustering Based Moving Window Approach

Numerical solution of multi-order fractional differential equations using generalized triangular function operational matrices

A Gaussian mixture model based virtual sample generation approach for small datasets in industrial processes

Modern Machine Learning Tools for Monitoring and Control of Industrial Processes: A Survey

Numerical Solution of Fractional Order Differential-Algebraic Equations Using Generalized Triangular Function Operational Matrices