Fractional-order Pell wavelets method for distributed-order time-fractional diffusion equations

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This article presents a wavelet-based approximation method using Pell wavelets to estimate solutions for distributed-order time-fractional diffusion equations (DO-FDEs). The regularized beta function is employed to derive an exact formula for the Riemann-Liouville fractional integral operator associated with these wavelets. By combining this exact formula with the Gauss-Legendre quadrature method, an approximate solution to the DO-FDEs is obtained. For Pell polynomials, even-degree polynomials contain only odd-power terms, and odd-degree polynomials include only even-power terms. Therefore, each Pell polynomial has half as many terms as its degree, which reduces computational complexity and minimizes errors. The article also provides an error analysis of the method. To highlight its advantages over existing approaches in the literature, several examples are included.