Traveling Wave Solutions of Fractional Differential Equations Arising in Warm Plasma

: This paper aims to study the fractional differential systems arising in warm plasma, which exhibits traveling wave-type solutions. Time-fractional Korteweg-De Vries (KdV) and time-fractional Kawahara equations are used to analyze cold collision-free plasma, which exhibits magnet-acoustic waves and shock wave formation respectively. The decomposition method is used to solve the proposed equations. Also, the convergence and uniqueness of the obtained solution are discussed. To illuminate the effectiveness of the presented method, the solutions of these equations are obtained and compared with the exact solution. Furthermore, solutions are obtained for different values of time-fractional order and represented graphically.


Introduction:
The study of nonlinear fractional systems becomes crucial in all fields of mathematics, engineering, physics, etc. Due to their nonlinear behavior, numerous applications of such fractional systems can be found in fluid dynamics, plasma physics, nonlinear biological systems, viscoelasticity, solid mechanics, quantum field theory, etc [1][2][3][4] . Finding the exact solutions to such differential equations is not a straightforward task so; researchers prefer the bestestimated solutions 5 . Tools like series solution methods and numerical methods are widely used for such determination [6][7][8][9] . The existence of traveling wave behavior occurs in many physical phenomena such as plasma physics, fluid mechanics, waves in shallow water, etc. The Adomian decomposition method which is invented by George Adomian is extensively used for solving nonlinear PDEs [10][11][12][13] . Moreover, solutions obtained by this method are convergent.
The KdV equation narrates the appearance of collision-free shock wave [14][15][16][17] . The behavior of weak non-linear dispersive waves arising in gravity waves, plasma waves, and lattice waves can be described by the KdV equation. The time-fractional KdV model for the potential ( , ) can express as follows 18 Time-fractional Kawahara equation is given by which is the fifth-order KdV equation. Due to the fifth-order term , it is used for analyzing the cold collision-free plasma having magneto-acoustic waves.
The Adomian decomposition method is explored to solve these equations. Many facts can be explored by incorporating time-fractional derivatives. The arrangement of the paper is as follows: Definitions related to fractional derivatives and integrals are given in section 2. A description of the fractional Adomian decomposition method is

Basic Preliminaries:
Some basic definitions related to fractional ordered calculus are given in this section.

Fractional Adomian Decomposition Method:
To demonstrate the proposed method, consider the non-linear PDE: = ( , ), = ⌈ ⌉ ∈ ℕ, the differential operator is the th order fractional derivative, is a nonlinear operator and is a linear differential operator. Applying to Eq.3 and from the above remarks, Eq.3 becomes, The method decomposes ( , ) into a sum = ∑ =0 ∞ and the nonlinear term can be expressed as where Adomian polynomials 's are determined as follows 20 : Substituting decomposition series for and in Eq.4, The values ( , ) are determined by the recurrence relation: Therefore, the required solution ( , ) can be obtained by calculating the values of ( , ), for ≥ 1.
. Hence, whenever 0 < < 1, would be a contraction mapping. Therefore, by the Banach fixed point theorem, the time-fractional partial differential Eq.5 has a unique solution. To prove the convergence, consider the following theorem.

Theorem. 2: Let
= ∑ =0 ( , ) be the ℎ the partial sum then the sequence { } is a Cauchy sequence in Banach space . Proof: For ∈ ℕ, consider The following inequality can be obtained similarly, This implies, lim →∞ ∥ ∥ − ∥ ∥ → 0. Therefore, is a Cauchy sequence in . Hence, the solution of the given equation is convergent. The required solution ( , ) is given by The numerical solution obtained using the fractional Adomian decomposition method is compared with the exact solution in Table 1, which shows the efficiency and effectiveness of the method. A comparison of the exact and approximate solutions is given in Fig. 1(a), and observe that the approximate solution is enormously agreed with the exact solution. In Fig. 1(b), the behavior of solutions for = 1.0,0.9,0.7 and observed that the obtained solutions are stable and sufficiently approximate to the exact solution. In Fig. 2, the obtained solution ( , ) is presented for the parameters = 1,0 ≤ ≤ 1, −10 ≤ ≤ 10, and observe the soliton solution exists in plasma waves which has infinite support or infinite tails. The exact solution to Eq.11 for = 1 is Apply the integral operator to Eq.11 inserting = ∑ =0 ∞ and = ∑ =0 ∞ , the following recurrence relation is obtained to estimate values of . The estimation obtained by the fractional Adomian decomposition method and the exact solution are compared in Table 2, which shows the efficiency of the method. The Kawahara equation is the key model for to study of magnet-acoustic waves. In Fig. 3, a comparison of the exact and approximate solutions is given, and observed that the approximate solution is enormously agreed with the exact solution. In Fig. 4, the obtained solution ( , ) is presented for the parameters = 0.9,0 ≤ ≤ 1, −20 ≤ ≤ 20, and observed the soliton-type solution in plasma waves which has infinite support or infinite tails.

Conclusion:
The fractional Adomian decomposition method is executed to solve the time-fractional Korteweg-De Vries equation and the time-fractional Kawahara equation. Furthermore, the solutions obtained by the proposed method converge to exact solutions and uniquely exist in Banach space. Also, an absolute error is obtained, which shows that the approximate solutions are well close to the exact solution with high precision. The obtained results will support to study of traveling wave solutions in an unmagnetized collisionless plasma as well as magnetacoustic waves in a cold collision-free plasma. This work conveys that the Adomian decomposition method is effective and suitable to obtain the traveling wave solutions for nonlinear fractional ordered partial differential equations.