A new non-polynomial spline method for solution of linear and non-linear third order dispersive equations

In this paper, a new three-level implicit method is developed to solve linear and non-linear third order dispersive partial differential equations. The presented method is obtained by using exponential quartic spline to approximate the spatial derivative of third order and finite difference discretization to approximate the first order spatial and temporal derivative. The developed method is tested on four examples and the results are compared with other methods from the literature, which shows the applicability and feasibility of the presented method. Furthermore, the truncation error and stability analysis of the presented method are investigated, and graphical comparison between analytical and approximate solution is also shown for each example.


Introduction
In [4] Boussinesq and Korteweg-de Vries (KdV) equations described the problem of water waves and the long waves in which dispersive effects are present. We use an exponential quartic spline function to develop a numerical method to approximate the solution of third order homogeneous and non-homogeneous linear dispersive equation in one space dimension with f (x, t) as a source term: with y(x, 0) = g 1 (x), a ≤ x ≤ b, (1.2) and y(a, t) = γ 0 (t), t > 0, y x (a, t) = γ 1 (t), t > 0, y xx (a, t) = γ 2 (t), t > 0, where γ 0 (t), γ 1 (t), γ 2 (t), and g 1 (x) are assumed to be continuous functions, t is time and x is space variable [1,7]. In [1] theorems for the existence and uniqueness of solution of such dispersive equations are given. In [20,21], the criteria for deriving stability conditions of difference method were considered for the numerical solution of a third order linear dispersive equation. In [25], the analytical solution was obtained of such equations by using Adomian decomposition method, and in [19], such equations were solved numerically. The authors in [17] solved fourth order parabolic partial differential equation numerically by using parametric septic spline. Djidjeli and Twizell [7] developed numerical method for solution of third order linear dispersive equation with time-dependent boundary conditions.
We have also solved the third order non-linear dispersive equation named as Kortewegde Vries (KdV) equation: and y(x, t) = γ 3 (t), x ∈ ∂ , t > 0, y x (b, t) = γ 4 (t), t > 0, (1.6) where = [a, b] ⊂ R, ε and μ are positive parameters, and g 2 (x), γ 3 (t), γ 4 (t) are known functions. This equation shows both dispersion and non-linearity [4,23,27]. The solution of Eq. (1.4) may exhibit solitons. Solitons are localized waves that propagate without change in their shape and velocity and are stable in mutual interaction just like the phenomenon of totally elastic collision in kinetics. The KdV type equations have been an important class of non-linear evolution equations with numerous applications in physical sciences and engineering fields see [5,8,12,18,23,24,26].
The existence and uniqueness of solutions of the KdV equation for appropriate conditions have shown in [11] . Many well-known numerical methods, such as finite difference scheme, finite element schemes, Fourier spectral methods and mesh-free radial basis functions (RBF), collocation method, multiquadric (MQ), multiquadric quasi-interpolation [3, 5, 10, 13-15, 22, 23, 27, 28], have been used to solve the KdV equation. Authors in [16] also used decomposition method for solution of KdV equation. The numerical solution was presented in [6] for the first and fifth order KdV equations. In [2], authors solved coupled Burgers' equations using non-polynomial spline method. In [9], authors presented non-polynomial spline method for solving the generalized regularized long wave (GRLW) equations.
The purpose of this paper is to present a new method to solve third order linear and nonlinear dispersive partial differential equations based on spline function approximation. In our method, the third order spatial and first order temporal derivatives are approximated by exponential quartic spline and finite difference respectively. Besides, we have used the first order central difference discretization to approximate the first order spatial derivative in a non-linear term.
Here, we obtain a derivation of exponential quartic spline and its relations in Sect. 2.
In Sect. 3, we present the formulation of our methods along with boundary equations for both the linear and non-linear dispersive equations. In Sect. 4, a class of methods and truncation error are given. Stability analysis is discussed in Sect. 5. Numerical evidences and comparison with other available methods are included in Sect. 6 to show the accuracy of our method. The presented method is tested on four examples. Finally, conclusion is presented in Sect. 7.

Exponential quartic spline
Let a set of grid points in the interval [a, b] be such that We also denote the function value y(x j ) by y j .
Let E j (x, t) be the exponential quartic spline at the grid point (x j , t) given by for each j = 0, 1, . . . , n, where a 1j , a 2j , a 3j , a 4j , a 5j are unknown coefficients and τ is a free parameter. We determine the unknown coefficients in (2.2) from the interpolatory condi- , and E (3) j (x j+1 , t) = T j+1 (t) as given below: Applying the continuity conditions of first and second derivatives of E j (x, t) at the knots, , and using (2.3) yields the following relations: where α 1 = θ (e θe -θ ) -2(e θ + e -θ ) + 4 θ 3 (e θe -θ ) , (2.6) Using Eqs. (2.4) and (2.5), we obtain the following method: where the coefficients p = β 1 and q = - 11 24 ). Now, the operator x for any function W is supposed to have the following form according to Eq. (2.7): (2.8)

Spline solution for linear dispersive equation
In this section we develop an approximation for (1.1) in which the time and space derivatives are replaced by a finite difference and exponential quartic spline respectively. Equation (1.1) is discretized as: , y m j is the approximate solution of (1.1) at (x j , t m ), δ t is the central difference operator w.r.t. t and σ is a parameter such that finite difference approximation to the time derivative is of O(k) for arbitrary σ .
Operating x on both sides of (3.1) and after some simplifications, we obtain the following method: The final method (3.2) may be written in the schematic form as follows: Relation (3.2) gives (n -2) equations in (n -1) unknowns y j , j = 1(1)(n -1). We require one more equation at j = 1, i.e., at the end of the range of integration in order to have a closed form solution for y j . We discretize the boundary conditions in (1.3) and develop the following boundary equation of accuracy O(k + h 2 ): where

Spline solution for non-linear dispersive equation
In the similar manner, Eq.(1.4) is discretized as follows: where F = ε 2 y 2 . Operating x on both sides of (3.5) and after some simplifications, we obtain the following method: The schematic form of method (3.6) is given by Relation (3.6) gives (n -4) equations in (n -1) unknowns y j , j = 1(1)(n -1). We require three more equations each at j = 1, 2, (n -1), i.e., at the ends of the range of integration in order to have a closed form solution for y j . We discretize the boundary conditions in (1.

Truncation error and a class of methods
Expanding (3.2) or (3.6) in a Taylor series in terms of y(x j , t m ) and its derivatives and using (1.1) or (1.4) respectively, we obtain the truncation error as follows:
The present method is convergent by Lax theorem as the stability criterion is satisfied.

Numerical simulation and comparison
In this section, the presented three-level implicit method based on exponential quartic spline is tested on four examples. The following norms are used in this paper: where y ana is analytical and y app is approximate solution of third order dispersive equation for our method.
Example 2 (Linear non-homogeneous case) Consider the following linear non-homogeneous dispersive equation [25] ∂y ∂t The analytical solution is y(x, t) = sin(πx) cos t.

Conclusion
The class of a new three-level implicit methods has been obtained using exponential quartic spline for numerical approximation of third order linear and non-linear dispersive partial differential equations and is tested on four examples using MATLAB. The performance of these methods have been examined for different values of parameters. Having compared the solutions with available results in the literature, we found them to be better. The comparison between analytical and approximate solutions is also shown graphically in Figs. 1-4. Tables and figures show the feasibility and applicability of our method.