The focus of this investigation centers on the formulation of a novel spectral method deployed to numerically solve partial integro-differential equations that possess memory features. The approach leverages the hexic shifted Chebyshev polynomials, and addresses the nonlinearity of the computational output using iterative methods. A comprehensive explanation of the methodology is presented, and a thorough convergence analysis is conducted. This technique has the capacity to be effortlessly modified and used to solve a plethora of linear and nonlinear issues, while minimizing computational time. Ultimately, the effectiveness of this novel strategy is demonstrated through the successful resolution of two exemplary problems. The findings of this study suggest that this spectral approach holds significant promise for solving partial integro-differential equations.

这项研究的重点是制定一种新颖的谱方法，用于数值求解具有记忆特征的偏积分微分方程。该方法利用十六进制移位切比雪夫多项式，并使用迭代方法解决计算输出的非线性问题。对该方法进行了全面的解释，并进行了彻底的收敛分析。该技术能够轻松修改并用于解决大量线性和非线性问题，同时最大限度地减少计算时间。最终，通过成功解决两个典型问题证明了这一新颖策略的有效性。这项研究的结果表明，这种谱方法对于求解偏积分微分方程具有重要的前景。