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The new originally capital general solution of determining the prototype filter function as the response that satisfies the specifications of all pole low-pass continual time filter functions of odd and even order is presented in this article. In this article, two new classes of filter functions are proposed using orthogonal and orthonormal Jacobi polynomials. The approximation problem of filter function was solved mathematically, most directly applying the summed Christoffel–Darboux formula for the orthogonal polynomials. The starting point in solving the approximation problem is a direct application of the Christoffel–Darboux formula for the initial set of continual Jacobi orthogonal polynomials in the finite interval in full respect to the weighting function with two free real parameters. General solution of the filter functions is obtained in a compact explicit form, which is shown to enable generation the Jacobi filter functions in a simple way by choosing the numerical values of the free real parameters. For particular specifications of free parameters, the proposed solution is used with the same criterion of approximation to generate the appropriate particular filter functions as are: the Gegenbauer, Legendre and Chebyshev filter functions of the first and second kind as well. The examples of proposed filter functions of even and odd order are illustrated and compared with classical solutions. [ABSTRACT FROM AUTHOR]

In diffusion transport, the flux at a point is typically modeled in terms of the local gradient of a potential. When heterogeneities are present, this local model can break down and it may be more appropriate to model the diffusion flux as a weighted sum of gradients present throughout the domain. Here a discrete nonlocal flux model-consistent with control-volume implementations-is developed. This scheme is referred to as the control-volume weighted flux scheme (CVWFS). The key component is the modeling of the diffusion flux at a given control-volume face in terms of a weighted sum of gradients at that face and at faces up- and downstream. Criteria for choosing the weights are proposed. This results in numerical solution schemes in which the coefficient matrix is diagonally dominant, has positive off-diagonal elements, and zero row sums. For a particular power-law weighting scheme it is shown how the CVWFS is related to the definition of the Caputo fractional derivative and the one-shift Grunwald approximation of the Riemann-Liouville fractional derivative. On developing transients and boundary condition treatments, the accuracy and suitability of the CVWFS scheme is demonstrated by solving a number of problems governed by Caputo fractional diffusion equations. [ABSTRACT FROM AUTHOR]

This aritcle describes a simple alternative unified method of solving nonlinear regular perturbation problems. The procedure is based upon the manipulation of Taylor's approximation for the expansion of the nonlinear term in the perturbed equation. An essential feature of this technique is the relative simplicity used and the associated unified computational procedure that is employed. As such it should be of interest to teachers of applied mathematics courses, particularly those courses which include perturbation methods. One of the merits of this approach is that it leads on naturally to a scheme based on Taylor's expansions and, consequently, allows the regular perturbation method to be introduced into a course much earlier than is currently common. The method is illustrated by implementing it to four perturbation problems, including two algebraic equations and two initial-value problems. The new approach is compared and contrasted with the traditional perturbation scheme in order to demonstrate its relative merit. [ABSTRACT FROM AUTHOR]