Solution of the nonlinear fractional diffusion equation with absorbent term and external force

Abstract

a b s t r a c tThe article presents the approximate analytical solutions of general nonlinear diffusion equation with fractional time derivative in the presence of an absorbent term and a linear external force obtained with the help of powerful mathematical tool like Homotopy Perturbation Method. By using initial value, the approximate analytical solutions of the equation are derived. The fractional derivatives are described in the Caputo sense. Numerical results for different particular cases are presented graphically. The anomalous behavior of nonlinear diffusivity in the presence or absence of external force and reaction term are calculated numerically and presented graphically.

Approximate Solution of Fractional Nonlinear Partial Differential Equations by the Legendre Multiwavelet Galerkin Method

“…Substituting (22) into (21) and using (18) and (19) we obtained the solution of (21) for different values of = {1/3, 1/2, 2/3, 1}. Table 1 shows the approximate solutions for (21) obtained for different values of using the Legendre multiwavelet method and the Homotopy perturbation method [37]. The values of = 1 are the only case for which we know the exact solution ( , ) = + and our approximate solution using Legendre multiwavelet method coincides with the approximate solution obtained using the Homotopy perturbation method [37].…”

“…Table 1 shows the approximate solutions for (21) obtained for different values of using the Legendre multiwavelet method and the Homotopy perturbation method [37]. The values of = 1 are the only case for which we know the exact solution ( , ) = + and our approximate solution using Legendre multiwavelet method coincides with the approximate solution obtained using the Homotopy perturbation method [37]. It is noted that only two bases of Legendre multiwavelet and fourth-order term of Homotopy perturbation method were used in evaluating the approximate solution of Table 1.…”

“…Consider the nonlinear time-fractional diffusion equation in absence of both external force and reaction term[37] − ( ) = 0, 0 < ≤ 1, > 0,…”

Gegenbauer wavelet operational matrix method for solving variable-order non-linear reaction–diffusion and Galilei invariant advection–diffusion equations

“…where R(c, t) denotes the reaction term at time t. The extension of the reaction-diffusion equation in fractional-order system can be found in the articles (Das et al 2011;Jaiswal et al 2018;Das et al 2018;Tripathi et al 2016). In nature, many of the beautiful systems in biology, physics,chemistry, and physiology can be described by reaction-diffusion equations.…”

Solution of the Nonlinear Fractional Diffusion Equation with Absorbent Term and External Force Using Optimal Homotopy-Analysis Method

“…Das and Gupta [10] have solved the similar type of linear fractional diffusion equation by the homotopy perturbation method (HPM). Recently, Das et al [11] have solved the approximate analytical solution of the general nonlinear diffusion equation with fractional time derivative in the presence of different types of absorbent terms and a linear external force using HPM. In another recent article of Yao [12], it is seen that the fractal geometry theory is combined with seepage flow mechanism to establish the nonlinear diffusion equation of fluid flow in fractal reservoir.…”