**1. Introduction**

Partial differential equations play a dominant role in applied mathematics. The classical heat conduction equation is second order linear partial differential equation. The solutions of which are obtained by using various analytical and numerical methods [1–3]. This equation describes the heat distribution in each domain over some time. Jean-Joseph Fourier was the first to formulate and describe the heat conduction process [1, 4]. Perturbation methods depending upon small/large parameters have been encountered from past few years. Perturbation methods are analytical approximation method to understand physical phenomena which depends on perturbation quantity. But these methods do not provide an easy way to find out the rapid convergence of approximate series. Therefore, this method is simple, suitable and appropriate method to provide the rapid convergence of series [5–7]. The perturbation method along with the homotopy method has been employed to develop a hybrid method known as homotopy perturbation method (HPM) [1–4]. Ji-Huan was the first to introduce HPM. Homotopy perturbation method provides analytical approximation to linear/nonlinear problems without linearization or discretization. It helps in formulating simpler equations by breaking down the complex problems, which can be solved easily. Since HPM does not depend on small parameters, therefore drawbacks of the existing perturbation

methods can be abolished [8–11]. The solution obtained by HPM converges to exact solution, which are in the form of an infinite function series. Various problems are modeled by linear and non-linear partial differential equations problems in the fields of physics, engineering etc. To solve such kind of partial differential equations (PDE), many methods are used to find the numerical or exact solutions. Homotopy perturbation method (HPM) is one of the methods used in recent years to solve various linear and non-linear PDE [12–15]. Initial and boundary value problems can be solved using HPM extensively. Many researchers and scientists show great interest in homotopy perturbation method. Huan was the first who described homotopy perturbation method. He showed that this method is a one of the powerful tools used to investigate various problems which are arising nowadays. HPM is used for solving linear and non-linear ordinary and partial differential equations [16].

In HPM, complex linear or non-linear problem can be continuously distorted into simpler ones. Perturbation theory and homotopy theory in topology is combined to develop homotopy perturbation method [1]. HPM is applicable to linear and non-linear boundary value problems. The solution obtained by HPM gives the solution approximately near to the universally accepted method of separation of variable [17–19].

Recently, Biazar and Eslami proposed the new homotopy perturbation method (NHPM). Construction of an appropriate homotopy equation and selection of appropriate initial approximation guess are two important steps of NHPM [19, 20]. The study reveals that with less computational work, we can construct proper homotopy by decomposition of source function in a correct way. New homotopy perturbation method is the most powerful tool which can be used to obtain analytical solution of various kinds of linear and nonlinear PDE's. This method is widely used by researchers to obtain solution of various functional Equations [20–22].

To develop this new technique, HPM is combined with the decomposition of source function. The decomposition of a source function is the basis of homotopy used in this method because convergence of a solution is affected by the decomposition of source functions [23]. Different kind of homotopy can be formed using various decomposition of a source functions. This study is aimed at constructing suitable homotopy by decomposition of a source function which requires less computational efforts and made calculations in simpler form unlike other perturbation methods. The obtained results directly imply the fact that NHPM is very influential as compared to HPM or any other perturbation technique. To establish exact solution of linear and non-linear problem with boundary and initial condition, new homotopy method is most appropriate method to apply [23].

The two most important steps in application of new homotopy perturbation method to construct a suitable homotopy equation and choose a suitable initial guess, we aim in this work to effectively employ the (NHPM) to establish exact solution for two-dimensional Laplace equation with Dirichlet and Neumann boundary condition, the difference between (NHPM) and standard (HPM) is starts from the form of initial approximation of the solution.

In this chapter, the semi analytic solution of one-dimensional heat conduction equation is obtained by means of homotopy perturbation method and new homotopy perturbation method. These methods are effectively applied to obtain the exact solution for the problem in hand which reveals the effectiveness and simplicity of the method. Numerical results have also been analyzed graphically to show the rapid convergence of infinite series expansion. The obtained analytic solution for one dimensional heat conduction equation with boundary and initial conditions using NHPM is same as the universally accepted exact solution. This tells us about the capability and reliability of this method. The solution obtained using NHPM is

considered in the form of an infinite series. The convergence of solution to the exact solution is very rapid.
