@article{Withers2025, 
author = {Christopher S. Withers and Saralees Nadarajah},
title = {A class of multivariate linear partial differential equations on        C    p},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {7},
pages = {15588-15618},
keywords = {Bell polynomial, expansion, commute, linear partial differential equation},
url = {https://www.sciopen.com/article/10.3934/math.2025698},
doi = {10.3934/math.2025698},
abstract = {We used a new type of characteristics to solve a class of homogeneous linear multivariate partial differential equations on        C    p  . For    x in        R    p   and    n in        Z    p  , set        ∂    x    n    =      ∏          j      =      1        p              (              ∂                  /                ∂                  x          j                    )                      n        j            . Given square matrices        {          N      j        }   and        {          S      n        }   in        C          s      ×      s      , set                       Y        (        x        )        =        exp        ⁡                  (                                    ∑                              j                =                1                            p                                      x              j                                      N              j                                )                in                  C                      s            ×            s                              and        T    n    (  x  )  =  Y  (  x  )        S    n      Y  (  −  x  ) in        C          s      ×      s      . When        {          N      j        }   commute, we show that the linear partial differential equation                                 ∑                      n            =                          0              p                                q                          T                      n                          (        x        )                          ∂          x          n                f        (        x        )        =                  0          s                for        f        (        x        )        in                  C          s                    has solutions    f  (  x  )  =      f    n    (  x  ,  ν  ) for each admissible    n  ≤  s  q and any    ν in        C    p   such that    d  (  ν  )  =  0, where                       d        (        ν        )        =                  d          e          t                        D        (        ν        )        ,                D        (        ν        )        =                  ∑                      n            =                          0              p                                q                          S          n                                  ∏                      j            =            1                    p                                      (                                          ν                j                                            I                s                            +                              N                j                                      )                                              n              j                                      .            The research aims to develop a new method, based on a novel type of characteristics, for solving a broad class of multivariate homogeneous linear partial differential equations with matrix coefficients of a specific exponential-conjugate form, extending classical Cauchy characteristic techniques beyond the univariate case and providing explicit basis solutions parameterized over complex surfaces.}
}