Fractional Differential Equations

For the statistical consultant working with social science researchers the estimation of reliability and validity is a task frequently encountered. Measurement issues differ in the social sciences in that they are related to the quantification of abstract, intangible and unobservable constructs. In many instances, then, the meaning of quantities is only inferred. Let us begin by a general description of the paradigm that we are dealing with. Most concepts in the behavioral sciences have meaning within the context of the theory that they are a part of. Each concept, thus, has an operational definition which is governed by the overarching theory. If a concept is involved in the testing of hypothesis to support the theory it has to be measured. So the first decision that the research is faced with is \" how shall the concept be measured? \" That is the type of measure. At a very broad level the type of measure can be observational, self-report, interview, etc. These types ultimately take shape of a more specific form like observation of ongoing activity, observing video-taped events, self-report measures like questionnaires that can be open-ended or close-ended, Likert-type scales, interviews that are structured, semi-structured or unstructured and open-ended or close-ended. Needless to say, each type of measure has specific types of issues that need to be addressed to make the measurement meaningful, accurate, and efficient. Another important feature is the population for which the measure is intended. This decision is not entirely dependent on the theoretical paradigm but more to the immediate research question at hand. 6/14/2016 2 A third point that needs mentioning is the purpose of the scale or measure. What is it that the researcher wants to do with the measure? Is it developed for a specific study or is it developed with the anticipation of extensive use with similar populations? Once some of these decisions are made and a measure is developed, which is a careful and tedious process, the relevant questions to raise are \" how do we know that we are indeed measuring what we want to measure? \" since the construct that we are measuring is abstract, and \" can we be sure that if we repeated the measurement we will get the same result? \". The first question is related to validity and second to reliability. Validity and reliability are two important characteristics of behavioral measure and are referred to as …

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Validity and reliability.

Foundations of Science focuses on significant methodological and philosophical topics concerning the structure and the growth of science. It serves as a forum for exchange of views and ideas among working scientists and theorists of science, and promotes interdisciplinary cooperation.  The journal presents foundational issues of science in a way that is free from unnecessary technicalities, yet faithful to the scientific content. Its aim is not simply to identify and highlight foundational issues and problems, but to suggest constructive solutions.  While acknowledging that various sciences have their own approaches and methods, the editors hold that important truths can be discovered about and by the sciences and that these transcend cultural and political contexts. The editors believe that the central foundational questions of contemporary science can be posed and answered without recourse to sociological or historical methods.  Manuscripts should be submitted as an attached file to an e-mail directed to the Editor-inChief, Diederik Aerts at the address: FOSFoundationsOfScience@gmail.com with a cc to the address: diraerts@vub.ac.be  The file should be an electronic copy of the original, not blinded, and a pdf version of the article, also not blinded.  Authors have to additionally give the names and e-mail addresses of five potential reviewers, who need to be scientists considered to have expertise in the domain where the subject of the manuscript is situated.

Foundations of Science

Solitons, Introduction to.

Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology

The Boussinesq equation simulates weakly nonlinear and long wave approximation that can be used in water waves, coastal engineering, and numerical models for water wave simulation in harbors and shallow seas. In this article, the sine-Gordon expansion (SGE) approach and the generalized Kudryashov (GK) scheme are used to establish broad-spectral solutions including unknown parameters and typical analytical solutions are recovered as a special case. The well-known bell-shape soliton, kink, singular kink, compacton, contracted bell-shape soliton, periodic soliton, anti-bell shape soliton, and other shape solitons are retrieved for the definite value of these constraints. The 3D and contour plots of some of the results obtained are sketched by assigning individual values of the parameter and analyzed the dynamical behavior of the waves. Furthermore, the compatibility of the two approaches has been compared and examined the efficiency to ascertain soliton solutions.

Soliton solutions to the Boussinesq equation through sine-Gordon method and Kudryashov method

We construct new analytical solutions of the ( 3 + 1 )-dimensional modified KdV-Zakharov-Kuznetsev equation by the Exp-function method. Plentiful exact traveling wave solutions with arbitrary parameters are effectively obtained by the method. The obtained results show that the Exp-function method is effective and straightforward mathematical tool for searching analytical solutions with arbitrary parameters of higher-dimensional nonlinear partial differential equation.

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New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp-Function Method

Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and the modified KdV-Zakharov-Kuznetsov equations using the modified simple equation method

A generalized and improved -expansion method is proposed for finding more general type and new travelling wave solutions of nonlinear evolution equations. To illustrate the novelty and advantage of the proposed method, we solve the KdV equation, the Zakharov-Kuznetsov-Benjamin-Bona-Mahony (ZKBBM) equation and the strain wave equation in microstructured solids. Abundant exact travelling wave solutions of these equations are obtained, which include the soliton, the hyperbolic function, the trigonometric function, and the rational functions. Also it is shown that the proposed method is efficient for solving nonlinear evolution equations in mathematical physics and in engineering.

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A Generalized and Improved (G\'/G)-Expansion Method for Nonlinear Evolution Equations

In this paper, the general solutions of the Duffing equation with third degree nonlinear term is obtain using the Exp-function method. Using the Duffing equation and its general solution, the new and general exact solution with free parameter and arbitrary functions of the (2+1) dimensional dispersive long wave equation are obtained. Setting free parameters as special values, hyperbolic as well as trigonometric function solutions are also derived. With the aid of symbolic computation, the Exp-function method serves as an effective tool in solving the nonlinear equations under study. Key words: Exp-function method; Duffing equation; Exact solutions; Nonlinear evolution equations

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M. Ali Akbar

Papers

Soliton solutions to the Boussinesq equation through sine-Gordon method and Kudryashov method

New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp-Function Method

Exact and solitary wave solutions for the Tzitzeica-Dodd-Bullough and the modified KdV-Zakharov-Kuznetsov equations using the modified simple equation method

A Generalized and Improved (G\'/G)-Expansion Method for Nonlinear Evolution Equations

Exp-Function Method for Duffing Equation and New Solutions of (2+1) Dimensional Dispersive Long Wave Equations