Developing the analytical solution for the nonlinear bioheat transfer equation through homotopy analysis method along with an optimal convergence-control parameter

Abstract

The Homotopy Analysis Method (HAM) is an effective technique to achieve the analytical solution of a broad range of problems, mainly with nonlinear governing equations. The solution of Pennes’ bioheat equation in nonlinear form arising from the linear temperature-dependent nature of specific heat capacity of a biological tissue using the Homotopy Analysis Method has been obtained analytically and validated with the numerical results obtained from the Finite Difference Method (FDM) the first time in this study. The analysis demonstrated that considering the various values of the convergence parameter and computing the Mean Squared Error (MSR) to achieve the optimum values ensures accurate results even at the low-order approximations of the solution. Investigating the effect of the nonlinear term’s magnitude on the solution indicated a direct relationship; However, the effect was not remarkable even at the major values, thus it is possible to consider the specific heat capacity of a living tissue, a constant value through thermal simulations. According to this research, the Homotopy Analysis Method can be a proper method to derive the analytical solution of either the linear or nonlinear form of Pennes’ bioheat equation.

Keywords

• Homotopy Analysis Method (HAM)
• Mean Squared Error
• Optimum Convergence-Control Parameter
• Pennes

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