A generalized differential transform method for linear partial differential equations of fractional order

In this letter we develop a new generalization of the two-dimensional differential transform method that will extend the application of the method to linear partial differential equations with space- and time-fractional derivatives. The new generalization is based on the two-dimensional differential transform method, generalized Taylor’s formula and Caputo fractional derivative. Several illustrative examples are given to demonstrate the effectiveness of the present method. The results reveal that the technique introduced here is very effective and convenient for solving linear partial differential equations of fractional order

Demining Quality Management: Case Studies from Jordan

Two case studies of clearance in the Jordan Valley and along Jordan’s northern border highlight the importance of quality management to ensure efficiency of clearance and credibility of land release

Finding Legacy Minefields in the Jordan Valley

Due to the many difficulties in accurately determining the location of legacy minefields, demining personnel need traditional and sometimes improvised methods for locating and verifying contamination. With a unique combination of terrain, vegetation, water resources and soil types, the Jordan Valley requires specialized minefield survey and clearance methods to avoid harming the environment

Fractional Variational Iteration Method for Fractional Nonlinear Differential Equations

Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to overcome such shortcomings, a fractional variational iteration method is proposed. The Lagrange multipliers can be identified explicitly based on fractional variational theory.Comment: 12 pages, 1 figure

Differential modeling for cancer microarray data

Capturing the changes between two biological phenotypes is a crucial task in understanding the mechanisms of various diseases. Most of the existing computational approaches depend on testing the changes in the expression levels of each single gene individually. In this work, we proposed novel computational approaches to identify the differential genes between two phenotypes. These approaches aim to quantitatively characterize the differences between two phenotypes and can provide better insights and understanding of various diseases. The purpose of this thesis is three-fold. Firstly, we review the state-of-the-art approaches for differential analysis of gene expression data. Secondly, we propose a novel differential network analysis approach that is composed of two algorithms, namely, DiffRank and DiffSubNet, to identify differential hubs and differential subnetworks, respectively. In this approach, two datasets are represented as two networks , and then the problem of identifying differential genes is transformed to the problem of comparing two networks to identify the most differential network omponents. Studying such networks can provide valuable knowledge about the data. The DiffRank algorithm ranks the nodes of two networks based on their differential behavior using two novel differential measures: differential connectivity and differential betweenness centrality for each node. These measures are propagated through the network and are optimized to capture the local and global structural changes between two networks. Then, we integrated the results of this algorithm into the proposed differential subnetwork algorithm which is called DiffSubNet. This algorithm aims to identify sets of differentially connected nodes. We demonstrated the effectiveness of these algorithms on synthetic datasets and real-world applications and showed that these algorithms identified meaningful and valuable information compared to some of the baseline methods that can be used for such a task. Thirdly, we propose a novel differential co-clustering approach to efficiently find arbitrarily positioned difeferntial (or discriminative) co-clusters from large datasets. The goal of this approach is to discover a distinguishing set of gene patterns that are highly correlated in a subset of the samples (subspace co-expressions) in one phenotype but not in the other. This approach is useful when the biological samples are assumed to be heterogenous or have multiple subtypes. To achieve this goal, we propose a novel co-clustering algorithm, Ranking-based Arbitrarily Positioned Overlapping Co-Clustering (RAPOCC), to efficiently extract significant co-clusters. This algorithm optimizes a novel ranking-based objective function to find arbitrarily positioned co-clusters, and it can extract large and overlapping co-clusters containing both positively and negatively correlated genes. Then, we extend this algorithm to discover discriminative co-clusters by incorporating the class information into the co-cluster search process. The novel discriminative co-clustering algorithm is called Discriminative RAPOCC (Di-RAPOCC), to efficiently extract the discriminative co-clusters from labeled datasets. We also characterize the discriminative co-clusters and propose three novel measures that can be used to evaluate the performance of any discriminative subspace algorithm. We evaluated the proposed algorithms on several synthetic and real gene expression datasets, and our experimental results showed that the proposed algorithms outperformed several existing algorithms available in the literature. The shift from single gene analysis to the differential gene network analysis and differential co-clustering can play a crucial role in future analysis of gene expression and can help in understanding the mechanism of various diseases

A multi-step differential transform method and application to non-chaotic or chaotic systems

International audienceThe differential transform method (DTM) is an analytical and numerical method for solving a wide variety of differential equations and usually gets the solution in a series form. In this paper, we propose a reliable new algorithm of DTM, namely multi-step DTM, which will increase the interval of convergence for the series solution. The multi-step DTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. This new algorithm is applied to Lotka-Volterra, Chen and Lorenz systems. Then, a comparative study between the new algorithm, multi- step DTM, classical DTM and the classical Runge-Kutta method is presented. The results demonstrate reliability and efficiency of the algorithm developed

Numerical methods for nonlinear partial differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order

Status of the differential transformation method

Further to a recent controversy on whether the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used, often only, as a technique for (analytically) calculating the power series of the solution (in terms of the initial value parameters). Sometimes, a piecewise analytic continuation process is implemented either in a numerical routine (e.g., within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basic ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the "traditional"-Taylor-method users (notably in the elaboration of software packages --numerical routines-- for automatically solving ordinary differential equations). At now, the defenders of the DTM still ignore the, though much better developed, studies of the "traditional"-Taylor-method users who, in turn, seem to ignore similarly the existence of the DTM. The DTM has been given an apparent strong formalization (set on the same footing as the Fourier, Laplace or Mellin transformations). Though often used trivially, it is easily attainable and easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence applications to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been sketched. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the "misunderstandings" which have caused the controversy, the preceding topics are concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages, references and further considerations adde

Exact solutions of the generalized K(m,m)K(m,m) equations

Family of equations, which is the generalization of the $K(m,m)$ equation, is considered. Periodic wave solutions for the family of nonlinear equations are constructed