Fibonacci wavelets and their applications for solving two classes of time‐varying delay problems

Abstract

Summary In this paper, a numerical method for solving time‐varying delay equations and optimal control problems with time‐varying delay systems is discussed. This method is based upon Fibonacci wavelets and Petrov‐Galerkin method. To solve these problems, first, the Fibonacci wavelets are presented. With the aid of operational matrices of integration and delay for Fibonacci wavelets and using Petrov‐Galerkin method and Newton's iterative method, we solve two classes of time‐varying delay problems, numerically. The approximate solutions achieved by this method satisfy all the initial conditions. In addition, an estimation of the error is given. Numerical results are included to demonstrate the accuracy and applicability of the present technique.

Fibonacci wavelets and Galerkin method to investigate fractional optimal control problems with bibliometric analysis

“…Fibonacci wavelets are introduced on the interval [0, 1) (Sabermahani et al, 2020a). Here, we consider the wavelets in [0, t f ) aswhere m = 0, 1, …, M − 1, n = 1, 2, …, 2 k −1 .…”

A Fibonacci Wavelet Method for Solving Dual-Phase-Lag Heat Transfer Model in Multi-Layer Skin Tissue during Hyperthermia Treatment

“…Some of the commonly used wavelet families for solving different physical, engineering, and biological problems include Haar wavelets, Legendre wavelets, Laguerre wavelets, Chebyshev wavelets, harmonic wavelets, Euler wavelets, Bernoulli wavelets, and ultraspherical wavelets. A recent addition to the class of wavelet families is the Fibonacci wavelets, which have attained considerable attention from researchers working across various disciplines in physical and engineering sciences, mainly due to their several distinguishing features [24,25]. For instance, these wavelets are not based on orthogonal polynomials, however, one can differentiate and integrate these wavelets to obtain the corresponding operational matrices.…”

“…The obtained results shows that the proposed method is an appropriate tool for the approximation of smooth and piecewise smooth functions. Following the strategy of Chen and Haiso [27], we construct operational matrices of integration associated with the Fibonacci wavelets, which have less errors in comparison to the operational matrices based on Legendre wavelets [24]. The operational matrices of integration are then employed for converting the models at hand into a system of algebraic equations, which are subsequently solved by the usual Newton method.…”

Fibonacci polynomials for the numerical solution of variable‐order space‐time fractional Burgers‐Huxley equation

“…It is noteworthy that the FPs have many advantages in comparison with the classical polynomials, such as the Legendre polynomials (LPs). Some of these advantages are given in Sabermahani et al 13 as follows: the FPs include less terms than the LPs. Therefore, less runtime is spent using the FPs in comparison with utilizing the LPs.…”