Solutions of General Fractional-Order Differential Equations by Using the Spectral Tau Method

Here, in this article, we investigate the solution of a general family of fractional-order differential equations by using the spectral Tau method in the sense of Liouville–Caputo type fractional derivatives with a linear functional argument. We use the Chebyshev polynomials of the second kind to develop a recurrence relation subjected to a certain initial condition. The behavior of the approximate series solutions are tabulated and plotted at different values of the fractional orders ν and α. The method provides an efficient convergent series solution form with easily computable coefficients. The obtained results show that the method is remarkably effective and convenient in finding solutions of fractional-order differential equations.

A numerical technique for a general form of nonlinear fractional-order differential equations with the linear functional argument

In this paper, a numerical technique for a general form of nonlinear fractional-order differential equations with a linear functional argument using Chebyshev series is presented. The proposed equation with its linear functional argument represents a general form of delay and advanced nonlinear fractional-order differential equations. The spectral collocation method is extended to study this problem as a discretization scheme, where the fractional derivatives are defined in the Caputo sense. The collocation method transforms the given equation and conditions to algebraic nonlinear systems of equations with unknown Chebyshev coefficients. Additionally, we present a general form of the operational matrix for derivatives. A general form of the operational matrix to derivatives includes the fractional-order derivatives and the operational matrix of an ordinary derivative as a special case. To the best of our knowledge, there is no other work discussed this point. Numerical examples are given, and the obtained results show that the proposed method is very effective and convenient.

Spectral Tau method for solving general fractional order differential equations with linear functional argument

In this paper, a numerical technique for solving new generalized fractional order differential equations with linear functional argument is presented. The spectral Tau method is extended to study this problem, where the derivatives are defined in the Caputo fractional sense. The proposed equation with its functional argument represents a general form of delay and advanced differential equations with fractional order derivatives. The obtained results show that the proposed method is very effective and convenient.

The Functional Argument

This chapter discusses law’s capacity to fulfil its conduct-guiding function within different frameworks of practical reasoning. A functional argument of Raz is initially presented: according to this argument, authorities—including legal authorities—would not be able to fulfil their intended function if their directives operated as reasons for action that compete with opposing reasons in terms of their weight, rather than as pre-emptive reasons (Section 6.1). Several grounds for this argument are considered and found to be inadequate (Section 6.2). The spotlight is then directed onto another relevant consideration: law’s structural suitability to counteract several situational biases operative in contexts of individual and collective action (Sections 6.3.1–6.3.5). It is argued that law’s pivotal role in addressing practical problems linked with those biases strongly militate against the weighing model (Sections 6.3.6). Finally, the implications of those biases for the pre-emption thesis are discussed (Sections 6.3.7).