Solution of Cauchy and Boundary Problems of Discrete Multiplicative-Poverative-Additive Derivative Third-Order Equation

Abstract

Introduction: Within the framework of Cauchy and boundary issues, the authors of this paper investigate the third-order discrete mixed derivative equation. For the purpose of removing these derivatives and establishing the general solution of the related discrete multiplicative-poverative-additive derivative equation, it makes use of the definitions of discrete additive, multiplicative, and poverative derivatives. There are three arbitrary constants that are dependent on this answer. The subsequent step entails finding these constants by making use of primary or boundary conditions. In order to correctly handle the Cauchy and boundary issues, the essay attempts to solve for these constants.

Objectives: To study third-order discrete mixed derivative equations in the framework of Cauchy and boundary problems, and to find a general solution through eliminating mixed derivatives with the use of discrete derivative definitions.

Methods: The problem is stated as a new discrete additive, multiplicative, and poverative derivative definitions. We formulate the general solution and identify the arbitrary constants from primary or boundary conditions.

Results: The general solution of discrete equation has successfully been obtained containing three arbitrary constants determined by the imposed boundary or Cauchy conditions.

Conclusions: This study enhances the efficient solution of third-order discrete mixed derivative equations, consolidating the utility of the discrete definitions of the derivatives for solving Cauchy and boundary problems in a systematic approach.