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A large number of financial engineering problems involve non-linear equations with non-linear or timedependent boundary conditions. Despite available analytical solutions, many classical and modified forms of the well-known Black-Scholes (BS) equation require fast and accurate numerical solutions. This work introduces the radial basis function (RBF) method as applied to the solution of the BS equation with non-linear boundary conditions, related to path-dependent barrier options. Furthermore, the diffusional method for solving advective-diffusive equations is explored as to its effectiveness to solve BS equations. Cubic and Thin-Plate Spline (TPS) radial basis functions were employed and evaluated as to their effectiveness to solve barrier option problems. The numerical results, when compared against analytical solutions, allow affirming that the RBF method is very accurate and easy to be implemented. When the RBF method is applied, the diffusional method leads to the same results as those obtained from the classical formulation of Black-Scholes equation.Keywords: financial engineering; radial basis functions; diffusional method; barrier options. ResumoMuitos problemas de engenharia financeira envolvem equações não-lineares com condições de contorno não-lineares ou dependentes do tempo. Apesar de soluções analíticas disponíveis, várias formas clássicas e modificadas da conhecida equação de Black-Scholes (BS) requerem soluções numéricas rápidas e acuradas. Este trabalho introduz o método de função de base radial (RBF) aplicado à solução da equação BS com condições de contorno não-lineares relacionadas a opções de barreira dependentes da trajetória. Além disso, explora-se o método difusional para solucionar equações advectivo-difusivas quanto à sua efetividade para solucionar equações BS. Utilizam-se funções de base radial Cúbica e Thin-Plate Spline (TPS), aplicadas à solução de problemas de opções de barreiras. Os resultados numéricos, quando comparados com as soluções analíticas, permitem afirmar que o método RBF é muito acurado e fácil de ser implementado. O método difusional associado ao método RBF leva aos mesmos resultados obtidos pela formulação clássica da equação de Black-Scholes.

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The application of irreversible thermodynamics offers a formal treatment for drying analysis that allows the evaluation of intra-particle or intra-medium temperature and moisture profiles, and enthalpy, liquid, and vapor fluxes. However, researchers have claimed that its implementation is complex. This work presents a simple methodology for modeling, solving, and validating the drying equations, as applied to wheat kernels, and for obtaining the inherent and usually unavailable transport coefficients. To clarify and simplify the ensuing physical analysis, a spherical shape and isotropy were assumed. Additionally, solutions obtained with both Dirichlet and convective boundary conditions were analyzed and compared against experimental data. The thermal and hydro-stresses depend heavily on internal vapor and liquid fluxes and on the respective drying evaporation fronts, all of which were evaluated and compared.