A Numerical Study for the Solution of Time Fractional Nonlinear Shallow Water Equation in Oceans

Abstract

In this paper, an analytical solution for the coupled one-dimensional time fractional nonlinear shallow water system is obtained by using the homotopy perturbation method (HPM). The shallow water equations are a system of partial differential equations governing fluid flow in the oceans (sometimes), coastal regions (usually), estuaries (almost always), rivers and channels (almost always). The general characteristic of shallow water flows is that the vertical dimension is much smaller than the typical horizontal scale. This method gives an analytical solution in the form of a convergent series with easily computable components, requiring no linearization or small perturbation. A very satisfactory approximate solution of the system with accuracy of the order 10 −4 is obtained by truncating the HPM solution series at level six.

Numerical method for fractional dispersive partial differential equations

“…Because of its fabulous opportunity and uses in numerous fields, an extensive consideration has been given to exact and numerical solutions of fractional differential equations. An abundant deal of researchers has exposed the beneficial use of the fractional calculus in the forming and mechanism of numerous dynamical structures [3][4][5][6][7][8][9][10][11][12][13][14]. In addition to forming of different modeling features of these fractional ordered differential equations, the solution procedures and their consistency are rather more vital characteristics.…”

A novel computational framework to approximate analytical solution of nonlinear fractional elastic beam equation

“…In recent years, it has been discovered that di erential equations involving derivatives of a non-integer order can be adequate models for various physical phenomena [3][4][5][6][7][8][9][10]. The present paper is a study on developing a fairly complete theoretical understanding of the solutions for beam equations with fractional coordinate derivatives.…”

The power series method for a long term solution of Duffing oscillator

“…Chen [14] has used a target function method to solve Duffing equation, while the Laplace decomposition methods were introduced by Yusufoglu [15], and Khuri [16]. Recently, Kumar has developed powerful method to solve gas dynamics equation arising in shock fronts [17], telegraph equation via Laplace transform [18], time-fractional Fokker-Planck equation arising in solid state physics and circuit theory [19], and time fractional nonlinear shallow-water equation in oceans [20]. The power series method (PSM) is a classical method to solve ordinary differential equations (ODEs), which is closely related to the Taylor series method, but does not need an elaborate differential process to derive the expansion coefficients.…”