Analytical Solutions of Heat-Like Equation Using Elzaki Transform Variational Iteration Method: Black–Scholes Equation

This paper presents the application of the Elzaki variational iteration method to solve the Black–Scholes model, which can be formulated as a heat-like partial differential equation with specified initial conditions. The Black–Scholes equation is fundamental in financial mathematics for option pricing, traditionally solved using numerical methods that are computationally intensive and prone to discretization errors. The proposed Elzaki variational iteration method combines the advantages of the Elzaki transform with the variational iteration technique to obtain exact analytical solutions. The Elzaki transform effectively converts the partial differential equation into a more tractable algebraic form, while the variational iteration method provides systematic solution construction. This hybrid approach yields solutions in the form of rapidly convergent infinite series. The analytical nature of our solutions offers significant computational advantages: option prices can be calculated within minutes with high precision, eliminating the need for time-consuming numerical iterations. The method provides exact formulas that can be evaluated to arbitrary accuracy, making it particularly valuable for real-time financial applications and high-frequency trading systems where speed and precision are critical. Numerical examples demonstrate the effectiveness and rapid convergence of the infinite series solutions across various parameter ranges. This work establishes the Elzaki variational iteration method as a powerful analytical tool for solving financial differential equations, offering superior efficiency compared to traditional numerical approaches.