Homotopy perturbation method to space–time fractional solidification in a finite slab

Abstract

a b s t r a c tA mathematical model describing the space and time fractional solidification of fluid initially at its freezing temperature contained in a finite slab under the constant wall temperature is presented. The approximate analytical solution of this problem is obtained by the homotopy perturbation method. The results thus obtained are compared with exact solution of integer order (b = 1,a = 2) and are good agreement. The problem has been studied in detail by considering different order time and space fractional derivatives. The temperature distribution and the moving interface position for different fractional order space and time derivatives are shown graphically. The model and the solution are the generalization of the previous works and include them as special cases.

Analytical expressions for the concentrations of substrate, oxygen and mediator in an amperometric enzyme electrode

“…In this section, the homotopy perturbation method [24][25][26][27][28][29][30][31][32][33][34] is used to solve the non linear equations. In this method, according to the homotopy technique, a homotopy with an imbedding parameter p 2 [0, 1] is constructed, and the imbedding parameter is considered as a ''small parameter''.…”

“…This non-linear boundary value problem does not have a general analytical solution [20]. While no general method of solving these non-linear problems has been proposed, several vigorous procedure such as homotopy perturbation method [24][25][26][27][28][29][30][31][32][33][34], Adomian decomposition method [35] and homotopy analysis method [36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51], etc., have been analyzed. Here, homotopy perturbation method and homotopy analysis method are used to solve these non-linear differential equations.…”

An efficient approach for solving the Riccati equation with fractional orders

“…In this method we do not need the Lagrange multiplier, correction functional, stationary conditions, and calculating integrals, which eliminate the complications that exist in the VIM. In contrast to the HPM [27], in this method, it is not required to solve the functional equation in each iteration. Moreover, the present technique requires less work if compared with the Taylor matrix method.…”

Two methods to solve a fractional single phase moving boundary problem

“…From a mathematical point of view, this is a moving boundary problem and it is difficult to obtain its exact solutions [19]. For fractional moving boundary problems, Liu and Xu [12], Voller [13] and Sigh [20] present the analytical solutions for different problems. If we only consider the early stage of loss before the interface moves to R, the semi-infinite assumption can be used.…”