<![CDATA[Purpose:This paper aims to solve the BS second-order parabolic equation in Sobolev spaces to obtain weak solutions for financial applications, extending previous work in this field.Methodology:This paper constructs a set of functions that transforms the Black-Scholes partial differential equation into weak formulations. This study focuses on the Mathematics of finance, particularly the evolution of option pricing. An option's underlying asset involves agreements to buy or sell at a future strike price. The Black-Scholes (BS) equation, widely used in financial applications, models this.Findings:The analytical solutions, existence, uniqueness and other estimates were also obtained in weak form using boundary conditions to establish the effects of their financial implications in Sobolev spaces.Implication:The problem's regularity conditions were considered, and the coefficients and boundary of the domain are all smooth functions. To this end, this paper illustrates the definitions and assumptions that led to valuable assertions.]]>
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