New study of weakly singular kernel fractional fourth-order partial integro-differential equations based on the optimum q-homotopic analysis method

Numerical Solution of Variable-Order Time Fractional Weakly Singular Partial Integro-Differential Equations With Error Estimation

“…In the first and second inequality, the values of M and N are constant, respectively. According to this fact, in the first inequality, the rate of convergence of u Mn to u M is faster than 1 N to the power of N 2 + 1 2 − r. And also, in the second inequality, the rate of convergence of u mN to u N is faster than 1 M to the power of M − 2k + 3 2 .…”

“…Various numerical and analytical methods have been used for solving integro-differential equations. For example, Baleanu et al [3] utilized the optimum q-homotopic analysis method for solving weakly singular kernel fractional fourth-order partial integro-differential equations, Tang [34] applied a finite difference scheme for partial integrodifferential equations with a weakly singular kernel, McLean et al [19] used Laplace transformation of an integro-differential equation of parabolic types, Nemati et al [25] discussed the numerical solution of two-dimensional nonlinear Volterra integral equations by the Legendre polynomials, Fairweather [12] introduced spline collocation methods for a class of hyperbolic partial integro-differential equations, Dehestani et al [9] proposed an efficient computational approach based on the Genocchi hybrid functions for solving a class of fractional Fredholm-Volterra functional integro-differential equations, Patel et al [26] applied two-dimensional wavelets collocation scheme for linear and nonlinear Volterra weakly singular partial integro-differential equations and many other methods have been used in this type of equation, which can refer to [4,10,25,30,31,32,35].…”

Approximate solution for high-order fractional integro-differential equations via trigonometric basic functions

“…Due to the difficulty and impossibility in obtaining precise solutions for a lot of FDEs and FIDEs, some researchers and scholars use numerical or approximate solution approaches to get the solution to these problems. Among the approximate approaches used by some researchers for FDEs and FIDEs are homotopic analysis (HAM) and optimum q-homotopic analysis method (Oq-HAM) [11,12], optimal homotopic perturbation (OHPM) and homotopic perturbation method (HPM) [13,14], Adomian's decomposition method (ADM) [15], variational iteration method (VIM) [13,15] and collocation approach [16,17], etc. [18,19].…”

A generalized Chebyshev operational method for Volterra integro-partial differential equations with weakly singular kernels

“…A Crank–Nicolson time-stepping approach was adopted to address a WSIPDE in [22] . Baleanu et al harnessed the homotopy analysis method to tackle fourth-order fractional WSIPDEs in [23] . Additionally, a bivariate wavelets collocation method was employed for VWSIPDEs [24] .…”