In this work, a numerical scheme based on combined Lucas and Fibonacci polynomials is proposed for one- and two-dimensional nonlinear advection–diffusion–reaction equations. Initially, the given partial differential equation (PDE) reduces to discrete form using finite difference method and $$\theta -$$
 weighted scheme. Thereafter, the unknown functions have been approximated by Lucas polynomial while their derivatives by Fibonacci polynomials. With the help of these approximations, the nonlinear PDE transforms into a system of algebraic equations which can be solved easily. Convergence of the method has been investigated theoretically as well as numerically. Performance of the proposed method has been verified with the help of some test problems. Efficiency of the technique is examined in terms of root mean square (RMS), $$L_2$$
 and $$L_\infty $$
 error norms. The obtained results are then compared with those available in the literature.

/pdf/numerical-study-of-1d-and-2d-advection-diffusion-reaction-4o0msdj145.pdf

Numerical study of 1D and 2D advection-diffusion-reaction equations using Lucas and Fibonacci polynomials

The main goal of this paper is to develop a new formula of the fractional derivatives of the shifted Chebyshev polynomials of the third kind. This new formula expresses approximately the fractional derivatives of these polynomials in the Caputo sense in terms of their original ones. The linking coefficients are given in terms of a certain 
 
 
 
  
 
 
 4
 
 
 
 
 F
 
 
 3
 
 
 
 
 1
 
 
 
 terminating hypergeometric function. The integer derivatives of the shifted third-kind Chebyshev polynomials can be calculated using this formula after performing some reductions. To solve a nonlinear fractional pantograph differential equation with quadratic nonlinearity, the fractional derivative formula is used in conjunction with the tau technique. The role of the tau method is to convert the pantograph differential equation with its governing initial/boundary conditions into a nonlinear system of algebraic equations that can be treated with the aid of Newton’s iterative scheme. To test the method’s convergence, certain estimations are included. The proposed numerical method is demonstrated by numerical results to ensure its applicability and efficiency.

/pdf/new-fractional-derivative-expression-of-the-shifted-third-1tojlp14.pdf

New Fractional Derivative Expression of the Shifted Third-Kind Chebyshev Polynomials: Application to a Type of Nonlinear Fractional Pantograph Differential Equations

The article presents a method for solving the inverse problem of a two-dimensional anomalous diffusion equation with a Riemann–Liouville fractional-order derivative. In the first part of the present study, the authors present a numerical solution of the direct problem. For this purpose, a differential scheme was developed based on the alternating direction implicit method. The presented method was accompanied by examples illustrating its accuracy. The second part of the study concerned the inverse problem of recreating the model parameters, including the orders of the fractional derivative, in the anomalous diffusion equation. Equations of this type can be used to describe, inter alia, the heat conductivity in porous materials. The ant colony optimization algorithm was used to solve this problem. The authors investigated the impact of the distribution of measurement points, the use of different mesh sizes, and the input data errors on the obtained results.

/pdf/inverse-problem-for-a-two-dimensional-anomalous-diffusion-8v807wc5p2.pdf

Inverse Problem for a Two-Dimensional Anomalous Diffusion Equation with a Fractional Derivative of the Riemann–Liouville Type

This article deals with a Horadam type of generalized numbers involving four parameters. These numbers generalize several celebrated numbers in the literature such as the generalized Fibonacci, generalized Lucas, Fibonacci, Lucas, Pell, Pell-Lucas, Fermat, Fermat-Lucas, Jacobsthal, Jacobsthal-Lucas, balancing, and co-balancing numbers. We present some new identities concerned with the generalized numbers of four parameters. An explicit expression for these numbers is developed, and a mixed recurrence relation between two certain families of the generalized numbers is given, and after that, some new identities are presented and proved. A large number of identities between several celebrated numbers are obtained as special cases of our developed ones. Furthermore, some of the identities that were previously published in other articles can be deduced as special ones of our new identities.

Some new identities of a type of generalized numbers involving four parameters

In this article, the solution for Volterra–Fredholm integral equation (V–FIE) is investigated numerically by using the generalized Lucas polynomials. We approximate the solution of this equation as a base of the collocation method. This method depends on the operational matrices of these polynomials. These expansions and the properties of the generalized Lucas polynomials help us to solve the V–FIE. First, we approximate the unknown function and its integration in terms of the generalized Lucas polynomials with unknown coefficients. Then, by substituting these approximations into the equation and using the properties of these polynomials together with the collocation method, the problem is reduced to a system of algebraic equations in the expansion coefficients of the solution, which can be simply solved. An error estimate and convergence of the numerical solution for the generalized Lucas expansion are proved extensively. Eventually, some examples are included and compared with other methods to show the accuracy and validity of the proposed method.

Spectral Solutions with Error Analysis of Volterra–Fredholm Integral Equation via Generalized Lucas Collocation Method

This paper deals with the implementation and presentation of numerical solutions of fractional pantograph differential equations (FPDEs) using generalized Lucas polynomials (GLPs) The derivation of our proposed algorithms is built on introducing an operational matrix of derivatives (OMDs) of the GLPs and after that employing it to convert the problem into an algebraic system of equations whose solution can be found through some suitable algorithms such as Gauss elimination and Newton–Raphson methods Finally, by providing various illustrative examples, including comparisons with the results obtained by some other existing literature methods, the efficiency and applicability of our proposed algorithms are demonstrated

S. M. Sayed

Papers

Generalized Lucas Polynomial Sequence Treatment of Fractional Pantograph Differential Equation