We propose optimal variational asymptotic method to solve time fractional nonlinear partial differential equations. In the proposed method, an arbitrary number of auxiliary parameters
Nonlinear problems have posed a challenge to the scientific world since long and many scientists and researchers have been working hard to find methods to solve these problems. Quite a remarkable progress has been made to achieve qualitative as well as quantitative solutions of some tough nonlinear problems of significance in the field of physical and biological sciences, as well as in engineering and technology. There are several analytical methods, such as homotopy analysis method (HAM) [
The aim of the present paper is to propose an optimal variational asymptotic method (OVAM) to solve the following initial value nonlinear time fractional PDE:
To test the method, we apply it to solve two important classes of nonlinear partial differential equations:
The variational iteration method (VIM) is a wellestablished iteration method [
First we generalize the correction functional of our earlier work [
To illustrate the method, we consider the general form of the initial value fractional partial differential equation described by (
The new generalized correction functional for (
Writing
To determine the optimal value of Lagrange multiplier
Consider
Taking
Substituting
Writing
Combining the first and second step, the new generalized correction functional (
Truncating the solution series (
Substituting (
At the
The novelty of our proposed algorithm is that
Now we apply our proposed method to solve the following two problems in Sections
Advectiondiffusion equation (ADE) describes the solute transport due to combined effect of diffusion and convection in a medium. It is a partial differential equation of parabolic type, derived on the principle of conservation of mass using Fick’s law. Due to the growing surface and subsurface hydro environment degradation and the air pollution, the advectiondiffusion equation has drawn significant attention of hydrologists, civil engineers, and mathematical modellers. Its analytical/numerical solutions along with an initial condition and two boundary conditions help to understand the contaminant or pollutant concentration distribution behaviour through an open medium like air, rivers, lakes, and porous medium like aquifer, on the basis of which remedial processes to reduce or eliminate the damages may be enforced. It has wide applications in other disciplines too, like soil physics, petroleum engineering, chemical engineering, and biosciences. In 2002, Inc and Cherruault [
The fractional order forms of the ADE are similarly useful. The most important advantage of using fractional order differential equation in mathematical modelling is their nonlocal property. It is a wellknown fact that the integer order differential operator is a local operator whereas the fractional order differential operator is nonlocal in the sense that the next state of the system depends not only upon its current state but also upon all of its proceeding states. In the last decade, many authors have made notable contribution to both theory and application of fractional differential equations in areas as diverse as finance [
In recent past several papers [
As the first illustration of our proposed method, we apply it to solve the FADE described by (
Taking
Optimal value of
Auxiliary 



































We define the absolute errors
Table
Comparison between our solution and that of Momani [













































































































Figure
The error
The fifth order error, Momani [
The error
Cross section of
Density gradientdriven fluid convection arises in geophysical fluid flows in the atmosphere, oceans, and in the earth’s mantle. The RayleighBenard convection is a prototype model for fluid convection, aiming at predicting spatiotemporal convection patterns. The mathematical model for the RayleighBenard convection involves the NavierStokes equations coupled with the transport equation for temperature. When the Rayleigh number is near the onset of convection, the RayleighBenard convection model may be approximately reduced to an amplitude or order parameter equation, as derived by Swift and Hohenberg [
The SwiftHohenberg (SH) equation is defined as
We consider the following time fractional SwiftHohenberg equation [
The optimal values of
Auxiliary 




































































































Figures
(a) Profiles of
(a) Profiles of
Figures
(a) Profiles of
(a) Profiles of
We have proposed, for the first time, a new concept which is the generalization of our previous new iterative method [
We give some basic definitions and properties of fractional calculus [
A real function
The RiemannLiouville fractional integral operator of order
The fractional derivative of order
The following properties of the operator
The authors declare that there is no conflict of interests regarding the publication of this paper.