1111 ISSN 1063-7761, Journal of Experimental and Theoretical Physics, 2016, Vol. 122, No. 6, pp. 1111–1116. © Pleiades Publishing, Inc., 2016. Time Fractional Effect on Ion Acoustic Shock Waves in Ion-Pair Plasma 1 H. G. Abdelwahed a,b, *, E. K. El-Shewy b , and A. A. Mahmoud b a College of Science and Humanitarian Studies, Physics Department, Prince Sattam Bin Abdulaziz University, Kingdom of Saudi Arabia b Theoretical Physics Group, Physics Department, Faculty of Science, Mansoura University, Mansoura, Egypt *e-mail: hgomaa_eg@hotmail.com Received October 23, 2015 Abstract—The nonlinear properties of ion acoustic shock waves are studied. The Burgers equation is derived and converted into the time fractional Burgers equation by Agrawal’s method. Using the Adomian decompo- sition method, shock wave solutions of the time fractional Burgers equation are constructed. The effect of the time fractional parameter on the shock wave properties in ion-pair plasma is investigated. The results obtained may be important in investigating the broadband electrostatic shock noise in D- and F-regions of Earth’s ionosphere. DOI: 10.1134/S1063776116050149 1. INTRODUCTION In last few years, ion acoustic waves in ion-pair plasmas were investigated both numerically and exper- imentally [1–3]. Many observations clearly indicate the presence of positive-negative ion structures in a variety of astrophysical plasma environments [4]. More specifically, negative ions are present at D- region altitudes of the ionosphere of Earth in coexis- tence with electrons as they are formed primarily by electrons added to electronegative species [5]. In the past decade, acoustic shocks in astrophysical plasma have been investigated [6–10]. The propagation of dust ion acoustic shocks in multi-ion plasmas was studied in [11]. It is noted that the acoustic shocks are modified by the heavy-to-light ion number density parameter. Also, potentials of both polarities exist in the plasma system [11]. On the other hand, applica- tions of nonlinear fractional partial differential equa- tions have received much attention in fluid mechanics and the physics of plasma [12–15]. In [16], electro- static Viking satellite electron acoustic solitons observed in the dayside auroral zone were investigated by using the nonlinear time fractional Korteweg–de Vries (KdV) equation. Accordingly, the effect of trapped hot electrons on the dusty ion acoustic waves was discussed using the modified KdV equation with a time fractional term [17]. The progress of ion waves in an ion-pair plasma model was studied in [18] by means of the Gardner equation with a time fraction term. The method of variational iterations was used in [18] to investigate the effect of nonthermal electrons on the produced wave. The studies on plasma physics using fractional nonlinear evolution equations have been discussed by many authors [19–22]. Later, the properties of dust acoustic shock waves have been studied in two-tem- perature dust plasmas using the Burgers equation with a time fractional order. The time fractional parameter effect on shock wave features was discussed using the variational iteration technique [23]. Furthermore, shock waves in dusty plasmas were studied in [24] using the space-time fractional KdV–Burgers equa- tion. It was noted that the space-time fractional parameter affects the coexistence of shocks [24]. In this paper, an ion acoustic model with nonthermal electrons and ion pairs is considered. The KdV equa- tion is derived and Agrawal’s method [12, 25–27] is applied to formulate the time fractional KdV equation, and the Adomian decomposition method [28, 29] is used to solve it. In Section 2, we present the basic set of fluid equations for the system, and the Burgers equation with a time-fractional term is derived in Sec- tion 3. In Section 4, the Adomian decomposition method is used to solve the time-fractional Burgers equation. Section 5 contains the results and a discus- sion. 2. BASIC EQUATIONS In the model considered here, a three-component collisionless nonmagnetized plasma consists of vis- cous fluids of positive and negative ions and a non- thermal electron density distribution n e . The normal- 1 The article is published in the original. STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS