Two-dimensional Müntz–Legendre hybrid functions: theory and applications for solving fractional-order partial differential equations

Abstract

In this manuscript, we present a new numerical technique based on two-dimensional Müntz-Legendre hybrid functions to solve fractional-order partial differential equations (FPDEs) in the sense of Caputo derivative, arising in applied sciences. First, one-dimensional (1D) and two-dimensional (2D) Müntz-Legendre hybrid functions are constructed and their properties are provided, respectively. Next, the Riemann-Liouville operational matrix of 2D Müntz-Legendre hybrid functions is presented. Then, applying this operational matrix and collocation method, the considered equation transforms into a system of algebraic equations. Examples display the efficiency and superiority of the technique for solving these problems with a smooth or non-smooth solution over previous works. Keywords Müntz-Legendre polynomial • Hybrid functions • Operational matrix • Two-dimensional hybrid functions Mathematics Subject Classification 65M70 • 65N35 Recently, fractional calculus is one of the worthy and essential tools for modeling many phenomena in physics (Hilfer 2000), engineering applications (Magin 2004), viscoelasticity Communicated by José Tenreiro Machado.

Optimal control of linear fractional-order delay systems with a piecewise constant order based on a generalized fractional Chebyshev basis

“…Many research works have been allocated to providing efficient numerical procedures for solving constant order fractional models. For example, a general formulation based on the Hamiltonian function associated with optimal control of fractional problems without delay (Agrawal 2004), a collocation method based on the Bessel functions (Tohidi and saberi 2015), two-dimensional Müntz-Legendre hybrid functions (Sabermahani et al 2020), Müntz-Legendre polynomials (Kheyrinataj and Nazemi 2020a), a hybrid of orthonormal Taylor polynomials (Marzban and Malakoutikhah 2019, a hybrid of the conventional Legendre polynomials (Marzban 2021a), combining fractional-order Legendre functions with the block-pulse functions (Marzban 2021b), fractional Chebyshev functions (Kheyrinataj and Nazemi 2020b), Genocchi polynimials (Chang et al 2018), a neural network scheme (Yavari and Nazemi 2019), Bernstein polynomials (Nemati 2018), two-dimensional Müntz-Legendre wavelets (Sabermahani 2020), Ritz’s method (Jahanshahi and Torres 2017), a hybrid method with the use of Hermite cubic spline multi-wavelets (Mohammadzadeh and Lakestani 2018), Bernoulli wavelets (Rahimkhani et al 2017), Euler–Lagrange equation (Rakhshan and Effati 2020), second Chebyshev wavelets technique (Baghani 2021), Legendre wavelet approach (Yuttanan et al 2021), Hartley series (Dadkhah and Mamehrashi 2021). The main idea and fundamental concepts of variable-order fractional operators have been extended and studied in the excellent works (Samko and Ross 1993) and (Lorenzo and Hartley 2002).…”

Solving nonlinear systems of fractional-order partial differential equations using an optimization technique based on generalized polynomials

“…Sabermahani et al (2020b) used hybrid functions of block-pulse and fractional-order Fibonacci polynomials for obtaining the approximate solution of fractional delay differential equations. Sabermahani et al (2020c) presented a numerical technique based on two-dimensional Müntz-Legendre hybrid functions to solve fractional-order partial differential equations in the sense of the Caputo derivative. Several other studies on systems of differential equations can be found in Schittkowski (1997), Zhou and Casas (2014), Owolabi (2018), Feng et al (2019), and Jaros and Kusano (2014).…”