A generalized Gegenbauer wavelet collocation method for solving p-type fractional neutral delay differential and delay partial differential equations

In this work, we have investigated p-type fractional neutral delay differential equations (p-FNDDE) and p-type fractional neutral delay partial differential equations (p-FNDPDE) via generalized Gegenbauer wavelet. Generalized Gegenbauer scaling function fractional integral operator (GGSFIO) is constructed using the Riemann-Liouville definition of fractional integral to handle the fractional derivatives present in p-FNDDE and p-FNDPDE. The operation of Gegenbauer wavelet basis and GGSFIO to p-FNDDE and p-FNDPDE returns a system of equations which is later solved by Newton's method for unknown wavelet coefficients. With the help of these coefficients, we get the approximate solution. We have established the convergence analysis to assure the theoretical authenticity of the present method. The developed scheme is tested on several examples of p-FNDDE and p-FNDPDE to ensure computational convergence which validated the theoretical findings. The comparison of the numerical results of our method with the existing methods concludes the superiority of the proposed method.