分数阶广义扩散模型在市场营销中的应用

The Application of Fractional-Order Generalized Diffusion Model in Marketing

摘要: 本文提出了一种基于分数阶微积分的广义扩散模型(GDM-R)，用于描述包含重复购买行为的市场扩散过程，该模型扩展了传统Bass模型，通过引入Caputo分数阶导数和记忆积分项，更准确地刻画消费者决策的历史依赖性和复购动态。我们详细推导了模型的数学形式，设计了高效的数值求解算法，并验证了模型的有效性。最后，提出了针对性的营销策略优化建议。

Abstract: This paper presents a generalized diffusion model (GDM-R) based on fractional calculus, which is used to describe the market diffusion process involving repeat purchasing behavior. This model extends the traditional Bass model by introducing Caputo fractional derivatives and memory integral terms, thereby more accurately depicting the historical dependence of consumer decisions and the dynamics of repeat purchases. We have derived the mathematical form of the model in detail, designed an efficient numerical solution algorithm, and verified the model’s effectiveness. Finally, specific marketing strategy optimization suggestions are proposed.