Analytical Solutions of Some Two-Point Non-Linear Elliptic Boundary Value Problems

Abstract

Several problems arising in science and engineering are modeled by differential equations that involve conditions that are specified at more than one point. The non-linear two-point boundary value problem (TPBVP) (Bratu's equation, Troesch's problems) occurs engineering and science, including the modeling of chemical reactions diffusion processes and heat transfer. An analytical expression pertaining to the concentration of substrate is obtained using Homotopy perturbation method for all values of parameters. These approximate analytical results were found to be in good agreement with the simulation results.

Analysis of non-linear reaction-diffusion processes with Michaelis-Menten kinetics by a new Homotopy perturbation method

“…Most models of real life problems are still very difficult to solve. Therefore, approximate analytical solutions such as Homotopy perturbation method (HPM) [10][11][12][13][14][15][16][17][18][19][20][21][22][23] were introduced. This method is the most effective and convenient ones for both linear and non-linear equations.…”

“…The HPM is unique in its applicability, accuracy and efficiency. The HPM [18][19][20][21][22][23] uses the imbedding parameter p as a small parameter, and only a few iterations are needed to search for an asymptotic solution. The law of mass action of oxygen uptake leads to the following non-linear equations…”

Analytical Solution of Non-Isothermal Diffusion-Reaction Processes and Effectiveness Factors

“…Case 1. When the reaction orders and , the analytical solution of (11) to (12) using homotopy perturbation method [35,[40][41][42][43][44] is…”

Analytical expression of concentrations of adsorbed CO molecules, O atoms and oxide oxygen

“…Ji Huan He used the HPM to solve the light hill equation [20], the Duffing equation [21] and the Blasius equation [22]. The idea has been used to solve non-linear boundary value problems, integral equations and many other problems [23][24][25][26][27]. The HPM is unique in its applicability, accuracy and efficiency.…”