On the stable computational,semi-analytical,and numerical solutions of the Langmuir waves in an ionized plasma

Abstract

This research paper investigates the stable computational, semi-analytical, and numerical solutions of the nonlinear complex fractional generalized–Zakharov system. This system describes the nonlinear interactions between the low-frequency, acoustic waves and high-frequency, electromagnetic waves. The modified Khater method is applied to find the analytical solutions then the stability property of these solutions is discussed by using the Hamiltonian system properties. Moreover, stable computational solutions are used as the initial condition in the semi-analytical and numerical schemes. The Adomian decomposition and septic B–spline schemes are used to find the semi-analytical and numerical. For more explanation of the obtained analytical solutions, some sketched are plotted in different types. Also, the comparison between the distinct types of obtained solutions is shown by calculating the absolute value of error. The performance of the used method explains the powerful, effective, and the ability for applying to different forms of nonlinear evolution equation.

Keywords

The nonlinear complex fractional generalized–Zakharov system modified Khater method Adomian decomposition method Septic B–spline scheme computational,semi-analytical,and numerical solutions

1 Introduction

Recently, plasma is one of the most common fields for studying, where it is found more than the rest of the ordinary matters in the universe. Chemist Irving Langmuir discovered it in 1920 and described it with the matter that consists of a gas of atoms, ions which have free electrons and some of their orbital electrons removed. It also can be artificially generated by subjecting or heating a neutral gas to become increasingly electrically conductive. The resulting charged electrons and ions make the plasma behavior more sensitive to the electromagnetic fields than a neutral gas. The plasma properties are different from those in the other states where it depends on the density and temperature of the environment. The plasma and ionized plasma have many forms such as the Earth’s ionosphere, magnetosphere, the solar corona, stars, and so on. Sometimes, the instability of the dielectric function of a free electron gas causes a fast oscillation of electron density in plasma or metals in the ultraviolet region fast that called by Langmuir waves (Figs. 1–2). Quantization of these oscillations is the plasmon that is a quantum of plasma oscillation and also is known with a quasiparticle [1 –3].

Fig.1

Condensed matter physics that deals with the macroscopic and microscopic physical properties of matter where A, B, C, D respectively, represent the ordered state, classical critical, quantum critical, and disordered state.

Fig.2

The electromagnetic field that is considered as a mixed combination between both of electric and magnetic fields where electromagnetic force is determined with the following law F = q E + qv × B where q, v, E, B respectively, represent particle of charge, velocity, electric field, and magnetic field.

Irving Langmuir and Lewi Tonks have derived the Langmuir waves in 1920 which parallel in form to the Jeans instability waves. These waves are generated by the gravitational instabilities in a static medium. Based on the properties of these waves, many researchers have been investigating it such as Euler–Maxell whose have derived a mathematical system to describe the laser–plasma interaction. This system is given by [4].

$\emptyset_{t} + \frac{1}{ℏ^{2}} Ξ (ℏ, ℏ u, ℏ \partial_{x}) \emptyset = \frac{1}{ℏ} B (\emptyset, \emptyset) .$ (1)

Equation 1 has also the following formula

$\begin{matrix} c ▿ \times E^{ϱ} + B_{t}^{ϱ} = 0, \\ c ▿ \times E^{ϱ} - B_{t}^{ϱ} = 4 π e ((η_{0} + η_{i}^{ϱ}) v_{i}^{ϱ} - (η_{0} + η_{e}^{ϱ})), \\ m_{e} (η_{0} + η_{e}^{ϱ}) \times ((v_{e}^{ϱ})_{t} + (v_{e}^{ϱ} \cdot ▿) v_{e}^{ϱ}) = - e (η_{0} + η_{e}^{ϱ}) \times (E^{ϱ} + \frac{1}{c} v_{e}^{ϱ} \times B^{ϱ}) - γ_{e} T_{e} ▿ η_{e}^{ϱ}, \\ m_{i} (η_{0} + η_{i}^{ϱ}) \times ((v_{i}^{ϱ})_{t} + (v_{i}^{ϱ} \cdot ▿) v_{i}^{ϱ}) = - e (η_{0} + η_{e}^{ϱ}) \times (E^{ϱ} + \frac{1}{c} v_{i}^{ϱ} \times B^{ϱ}) - γ_{i} T_{i} ▿ η_{i}^{ϱ}, \\ ▿ \cdot ((η_{0} + η_{e}^{ϱ}) \times v_{e}^{ϱ}) = - (η_{e}^{ϱ})_{t}, \\ ▿ \cdot ((η_{0} + η_{e}^{ϱ}) \times v_{i}^{ϱ}) = - (η_{i}^{ϱ})_{t}, \end{matrix}$ (2) where $[(B^{ϱ}, E^{ϱ}) & (v_{e}^{ϱ}, v_{i}^{ϱ}) & (η_{e}^{ϱ}, η_{i}^{ϱ}) & (- e, + e) & (γ_{e}, γ_{i}) & (T_{i}, T_{e}) & η_{0}]$ respectively, represent the electromagnetic field, the velocity of the ions and electrons, the density fluctuations, the electric charge of the electrons, the specific heat ration, the temperature of both species, and density of the plasma at equilibrium. System (2) shows the Maxwell’s equations are just two equations of the Euler’s system. While, the non–dimensional system formula from system (2) is given by

${(E M)}^{℘} {\begin{matrix} ▿ \times E + B_{t} = 0, \\ Θ_{i} ▿ \cdot ((1 + η_{i}^{℘}) \times v_{i}) + (η_{i}^{℘})_{t} = 0, \\ Θ_{e} ▿ \cdot ((1 + η_{e}^{℘}) \times v_{e}) + (η_{e}^{℘})_{t} = 0, \\ ▿ \times B - E_{t} = \frac{1}{ℏ} \frac{Θ_{i}}{Θ_{e}} (1 + η_{i}^{℘}) v_{i} - \frac{1}{ℏ} (1 + η_{e}^{℘}) v_{e}, \\ Θ_{i} (v_{i} \cdot ▿) v_{i} + (v_{i})_{t} = - ϑ^{2} Θ_{i} \frac{▿ η_{i}^{℘}}{1 + η_{i}^{℘}} + \frac{1}{ℏ} \frac{Θ_{i}}{Θ_{e}} (E + Θ_{i} v_{i} \times B), \\ Θ_{e} (v_{e} \cdot ▿) v_{e} + (v_{e})_{t} = - Θ_{e} \frac{▿ η_{e}^{℘}}{1 + η_{e}^{℘}} - \frac{1}{ℏ} (E + Θ_{e} v_{e} \times B) . \end{matrix}$ (3)

Changing the variables with small amplitudes, yields

$(E M) {\begin{matrix} ▿ \times E + B_{t} = 0, \\ ▿ \times B - E_{t} = \frac{1}{ℏ} (e^{η_{e}} v_{e} - \frac{Θ_{i}}{Θ_{e}} e^{η_{i}} v_{i}), \\ (v_{e})_{t} + Θ_{e} (v_{e} \cdot ▿) v_{e} = - Θ_{e} ▿ η_{e} - \frac{1}{ℏ} (E + Θ_{e} v_{e} \times B), \\ (v_{e})_{t} + Θ_{i} (v_{i} \cdot ▿) v_{i} = - ϑ^{2} Θ_{i} ▿ η_{i} - \frac{Θ_{i}}{ℏ Θ_{e}} (E + Θ_{i} v_{i} \times B), \\ (η_{i})_{t} + Θ_{i} ▿ \cdot v_{i} + Θ_{i} v_{i} \cdot ▿ η_{i} = 0, \\ (η_{e})_{t} + Θ_{e} ▿ \cdot v_{e} + Θ_{e} v_{e} \cdot ▿ η_{e} = 0, \end{matrix}$ (4) where, $[(B, E, v_{e}, η_{e}, v_{i}, \frac{η_{i}}{ϑ}) \in R^{14} & (B, E) \in R^{3 + 3} & (v_{e}, v_{i}) \in R^{3 + 3} & (η_{e}, η_{i}) \in R^{1 + 1} & ℏ : = \frac{\sqrt{m_{e}}}{\sqrt{4 π e^{2} η_{0}} t_{0}}]$ . Thus, using the following initial data of size φ (ℏ) that is defined over the times diffraction $φ (\frac{1}{ℏ})$ , leads to ∅ (x, t) : = ℏ ∅ ^* (ℏ t, x). Zakharov–system is considered as a simple system of all above mentioned systems which is used to describe the nonlinear interactions between Λ (the envelope of the electric field) and ℧ (the mean mode of the ionic fluctuations of the density in the plasma) and is given by

${\begin{matrix} ι Λ_{t} + △ Λ = ϱ Λ, \\ ℧_{t t} - △ ϱ = △ | Λ |^{2} . \end{matrix}$ (5) This research paper forces on investigating the generalized complex fractional formula of system (5) which is given by [5 –8]: ${\begin{matrix} i \frac{D^{α} Λ}{D t^{ϑ}} + Λ_{x x} - 2 ϱ {| Λ |}^{2} Λ + 2 Λ ℧ = 0, \\ \frac{D^{2 ϑ} ℧}{D t^{2 ϑ}} - ℧_{x x} + {({| Λ |}^{2})}_{x x} = 0 . \end{matrix}$ (6) Applying the next formula of the wave transformation to (6) ${\begin{matrix} Λ (x, t) = H (ð) e^{i ℘}, ℧ (x, t) = ℧ (ð), \\ ℘ = λ x + \frac{ω t^{ϑ}}{ϑ}, ð = x - \frac{2 λ t^{ϑ}}{ϑ}, \end{matrix}$ (7) leads to ${\begin{matrix} H^{″} - (ω + λ^{2}) H - 2 ϱ H^{3} + 2 H ℧ = 0, \\ (4 λ^{2} - 1) ℧^{″} + (H^{2})^{″} = 0 . \end{matrix}$ (8) Twice integration with zero constant of the second equation in the previous system, gets [℧ (ð) = $\frac{H^{2}}{1 - 4 λ^{2}}]$ . This research paper tests the computational and numerical of the system (6). Substituting the value of ℧ (ξ) into first equation of the system (8), gives: $\begin{matrix} H^{″} - (ω + λ^{2}) H + (\frac{2}{1 - 4 λ^{2}} - 2 ϱ) \\ H^{3} = 0 . \end{matrix}$ (9) using the homogeneous balance rule on Equation 9, yields N = 1.

This kind of models attracts the attention of the mathematicians and physics where they can use them to discover more properties of them. In the context of the mathematical view of this survey, many computational, semi-analytical, and numerical schemes are derived to find distinct types of solutions such as The variational iteration method (VIM.), Adomian decomposition method,the generalized Kudryashov method, Riccti equation method, $\frac{G^{'}}{G}$ –expansion method, Khater method, the modified Khater method (modified auxiliary equation method) and so on [21 –35].

The rest of this research paper is ordered in the following order: Section 2 applies the modified Khater method [36 –40], the property of the Hamiltonian system to check the stability property of the computational obtained solution [15 –20], Adomian decomposition method [10 –14], and septic B-spline schemes [41 –45] to the modified BBM equation. Moreover, the comparison between the computational and numerical obtained solutions is explained, and some sketches are plotted to show more physical properties of this system. Section 3 discusses the obtained computational results and explain the comparison between them and that obtained in previous work. Moreover, it show the comparison between the obtained numerical results. Section gives the conclusion of the whole research.

2 Computational and numerical solutions of the nonlinear complex fractional generalized–Zakharov system.

This section applies the computational, semi–analytical and numerical schemes to the nonlinear complex fractional generalized–Zakharov system. to show more physical properties of the model in the optical illusions field by explaining the behavior the Langmuir waves in an ionized plasma. waves.

2.1 Solitary wave solutions

Implementation of the modified Khater method to the nonlinear complex fractional generalized–Zakharov system, leads to derive the general form of solution of Equation 9 in the following formula $\begin{matrix} H (ð) = \sum_{i = 1}^{N} a_{i} K^{i ℘ (ð)} + \sum_{i = 1}^{N} b_{i} K^{- i ℘ (ð)} + a_{0} \\ = a_{1} K^{℘ (ð)} + a_{0} + b_{1} K^{- ℘ (ð)}, \end{matrix}$ (10)

where a₀, a₁, b₁ are arbitrary constants and ℘ (ð) is the solution function of the following equation $℘^{'} (ð) = \frac{β + α K^{- ℘ (ð)} + σ K^{℘ (ð)}}{ln (K)},$ (11) where K, α, β, σ are arbitrary constants. Substituting Equation (10) along (11) into Equation (9), yields a polynomial of K^℘(ð). Gathering the coefficients with the same power of K^℘(ð), leads to a system of algebraic equation. Solving this system by the Mathematica 11.2, yields

Family I

$\begin{matrix} a_{1} \to \frac{2 a_{0} σ}{β}, b_{1} \to 0, ω \to \frac{1}{2} (4 α σ - β^{2} - 2 λ^{2}), \\ ϱ \to \frac{- 4 a_{0}^{2} + 4 β^{2} λ^{2} - β^{2}}{4 a_{0}^{2} (4 λ^{2} - 1)} . \end{matrix}$

Thus, the solitary wave solutions of the nonlinear complex fractional generalized–Zakharov system are in the following formulas

When β² - 4ασ < 0 & σ ≠ 0 $Λ_{1} (x, t) = \frac{a_{0} \sqrt{4 α σ - β^{2}} exp (i λ x - \frac{i t^{ϑ} (- 4 α σ + β^{2} + 2 λ^{2})}{2 ϑ}) tan (\frac{1}{2} \sqrt{4 α σ - β^{2}} (x - \frac{2 λ t^{ϑ}}{ϑ}))}{β},$ (12) $℧_{1} (x, t) = \frac{a_{0}^{2} (β^{2} - 4 α σ) {tan}^{2} (\frac{1}{2} \sqrt{4 α σ - β^{2}} (x - \frac{2 λ t^{ϑ}}{ϑ}))}{β^{2} (4 λ^{2} - 1)},$ (13) $Λ_{2} (x, t) = \frac{a_{0} \sqrt{4 α σ - β^{2}} exp (i λ x - \frac{i t^{ϑ} (- 4 α σ + β^{2} + 2 λ^{2})}{2 ϑ}) cot (\frac{1}{2} \sqrt{4 α σ - β^{2}} (x - \frac{2 λ t^{ϑ}}{ϑ}))}{β},$ (14) $℧_{2} (x, t) = \frac{a_{0}^{2} (β^{2} - 4 α σ) {cot}^{2} (\frac{1}{2} \sqrt{4 α σ - β^{2}} (x - \frac{2 λ t^{ϑ}}{ϑ}))}{β^{2} (4 λ^{2} - 1)} .$ (15) When β² - 4ασ > 0 & σ ≠ 0 $Λ_{3} (x, t) = - \frac{a_{0} \sqrt{β^{2} - 4 α σ} exp (i λ x - \frac{i t^{ϑ} (- 4 α σ + β^{2} + 2 λ^{2})}{2 ϑ}) tanh (\frac{1}{2} \sqrt{β^{2} - 4 α σ} (x - \frac{2 λ t^{ϑ}}{ϑ}))}{β},$ (16) $℧_{3} (x, t) = - \frac{a_{0}^{2} (β^{2} - 4 α σ) {tanh}^{2} (\frac{1}{2} \sqrt{β^{2} - 4 α σ} (x - \frac{2 λ t^{ϑ}}{ϑ}))}{β^{2} (4 λ^{2} - 1)},$ (17) $Λ_{4} (x, t) = - \frac{a_{0} \sqrt{β^{2} - 4 α σ} exp (i λ x - \frac{i t^{ϑ} (- 4 α σ + β^{2} + 2 λ^{2})}{2 ϑ}) coth (\frac{1}{2} \sqrt{β^{2} - 4 α σ} (x - \frac{2 λ t^{ϑ}}{ϑ}))}{β},$ (18) $℧_{4} (x, t) = - \frac{a_{0}^{2} (β^{2} - 4 α σ) {coth}^{2} (\frac{1}{2} \sqrt{β^{2} - 4 α σ} (x - \frac{2 λ t^{ϑ}}{ϑ}))}{β^{2} (4 λ^{2} - 1)} .$ (19) When β = σ = κ & α = 0 $\begin{matrix} Λ_{5} (x, t) = a_{0} (- e^{i (λ x - \frac{(κ^{2} + 2 λ^{2}) t^{ϑ}}{2 ϑ})}) \\ coth (\frac{1}{2} κ (x - \frac{2 λ t^{ϑ}}{ϑ})), \end{matrix}$ (20) $℧_{5} (x, t) = - \frac{a_{0}^{2} {coth}^{2} (\frac{1}{2} κ (x - \frac{2 λ t^{ϑ}}{ϑ}))}{4 λ^{2} - 1} .$ (21) When α = 0 & β ≠ 0 & σ ≠ 0 $Λ_{6} (x, t) = - \frac{a_{0} e^{i λ x - \frac{i (β^{2} + 2 λ^{2}) t^{ϑ}}{2 ϑ}} (σ e^{β (x - \frac{2 λ t^{ϑ}}{ϑ})} + 2)}{σ e^{β (x - \frac{2 λ t^{ϑ}}{ϑ})} - 2},$ (22)

$℧_{6} (x, t) = - \frac{a_{0}^{2} {(σ e^{β (x - \frac{2 λ t^{ϑ}}{ϑ})} + 2)}^{2}}{(4 λ^{2} - 1) {(σ e^{β (x - \frac{2 λ t^{ϑ}}{ϑ})} - 2)}^{2}} .$ (23) When β² - 4ασ = 0 $\begin{matrix} Λ_{7} (x, t) = \\ a_{0} e^{\frac{i λ (x ϑ - λ t^{ϑ})}{ϑ}} (\frac{4 α σ (- β - \frac{2 ϑ}{x ϑ - 2 λ t^{ϑ}})}{β^{3}} + 1), \end{matrix}$ (24) $℧_{7} (x, t) = \frac{a_{0}^{2} {(\frac{4 α σ (- β - \frac{2 ϑ}{x ϑ - 2 λ t^{ϑ}})}{β^{3}} + 1)}^{2}}{1 - 4 λ^{2}} .$ (25)Family II $a_{1} \to 0, b_{1} \to \frac{2 α a_{0}}{β}, ω \to \frac{1}{2} (4 α σ - β^{2} - 2 λ^{2}), ϱ \to \frac{- 4 a_{0}^{2} + 4 β^{2} λ^{2} - β^{2}}{4 a_{0}^{2} (4 λ^{2} - 1)} .$

Thus, the solitary wave solutions of the nonlinear complex fractional generalized–Zakharov system are in the following formulas

When β² - 4ασ < 0 & σ ≠ 0 $Λ_{8} (x, t) = a_{0} exp (i (λ x - \frac{t^{ϑ} (- 4 α σ + β^{2} + 2 λ^{2})}{2 ϑ})) (1 - \frac{4 α σ}{β^{2} - β \sqrt{4 α σ - β^{2}} tan (\frac{1}{2} \sqrt{4 α σ - β^{2}} (x - \frac{2 λ t^{ϑ}}{ϑ}))}),$ (26) $℧_{8} (x, t) = \frac{a_{0}^{2} {(1 - \frac{4 α σ}{β^{2} - β \sqrt{4 α σ - β^{2}} tan (\frac{1}{2} \sqrt{4 α σ - β^{2}} (x - \frac{2 λ t^{ϑ}}{ϑ}))})}^{2}}{1 - 4 λ^{2}},$ (27) $Λ_{9} (x, t) = a_{0} exp (i (λ x - \frac{t^{ϑ} (- 4 α σ + β^{2} + 2 λ^{2})}{2 ϑ})) (1 - \frac{4 α σ}{β^{2} - β \sqrt{4 α σ - β^{2}} cot (\frac{1}{2} \sqrt{4 α σ - β^{2}} (x - \frac{2 λ t^{ϑ}}{ϑ}))}),$ (28) $℧_{9} (x, t) = \frac{a_{0}^{2} {(1 - \frac{4 α σ}{β^{2} - β \sqrt{4 α σ - β^{2}} cot (\frac{1}{2} \sqrt{4 α σ - β^{2}} (x - \frac{2 λ t^{ϑ}}{ϑ}))})}^{2}}{1 - 4 λ^{2}} .$ (29) When β² - 4ασ > 0 & σ ≠ 0 $Λ_{10} (x, t) = a_{0} exp (i (λ x - \frac{t^{ϑ} (- 4 α σ + β^{2} + 2 λ^{2})}{2 ϑ})) (1 - \frac{4 α σ}{β^{2} + β \sqrt{β^{2} - 4 α σ} tanh (\frac{1}{2} \sqrt{β^{2} - 4 α σ} (x - \frac{2 λ t^{ϑ}}{ϑ}))}),$ (30) $\begin{matrix} ℧_{10} (x, t) = \\ \frac{a_{0}^{2} {(1 - \frac{4 α σ}{β^{2} + β \sqrt{β^{2} - 4 α σ} tanh (\frac{1}{2} \sqrt{β^{2} - 4 α σ} (x - \frac{2 λ t^{ϑ}}{ϑ}))})}^{2}}{1 - 4 λ^{2}}, \end{matrix}$ (31) $Λ_{11} (x, t) = a_{0} exp (i (λ x - \frac{t^{ϑ} (- 4 α σ + β^{2} + 2 λ^{2})}{2 ϑ})) (1 - \frac{4 α σ}{β^{2} + β \sqrt{β^{2} - 4 α σ} coth (\frac{1}{2} \sqrt{β^{2} - 4 α σ} (x - \frac{2 λ t^{ϑ}}{ϑ}))}),$ (32) $\begin{matrix} ℧_{11} (x, t) = \\ \frac{a_{0}^{2} {(1 - \frac{4 α σ}{β^{2} + β \sqrt{β^{2} - 4 α σ} coth (\frac{1}{2} \sqrt{β^{2} - 4 α σ} (x - \frac{2 λ t^{ϑ}}{ϑ}))})}^{2}}{1 - 4 λ^{2}} . \end{matrix}$ (33) When $β = \frac{α}{2} = κ & σ = 0$ $\begin{matrix} Λ_{12} (x, t) = \\ e^{i (λ x - \frac{(κ^{2} + 2 λ^{2}) t^{ϑ}}{2 ϑ})} (\frac{4}{e^{κ (x - \frac{2 λ t^{ϑ}}{ϑ})} - 2} + 1), \end{matrix}$ (34) $℧_{12} (x, t) = \frac{a_{0}^{2} {(\frac{4}{e^{κ (x - \frac{2 λ t^{ϑ}}{ϑ})} - 2} + 1)}^{2}}{1 - 4 λ^{2}} .$ (35) When σ = 0 & β ≠ 0 & α ≠ 0 $\begin{matrix} Λ_{13} (x, t) = \\ a_{0} e^{i (λ x - \frac{(β^{2} + 2 λ^{2}) t^{ϑ}}{2 ϑ})} (1 - \frac{2 α}{α - β e^{β (x - \frac{2 λ t^{ϑ}}{ϑ})}}), \end{matrix}$ (36) $℧_{13} (x, t) = \frac{a_{0}^{2} {(1 - \frac{2 α}{α - β e^{β (x - \frac{2 λ t^{ϑ}}{ϑ})}})}^{2}}{1 - 4 λ^{2}} .$ (37) When β² - 4ασ = 0 $Λ_{14} (x, t) = \frac{2 a_{0} ϑ e^{\frac{i λ (x ϑ - λ t^{ϑ})}{ϑ}}}{ϑ (β x + 2) - 2 β λ t^{ϑ}},$ (38) $℧_{14} (x, t) = - \frac{4 a_{0}^{2} ϑ^{2}}{(4 λ^{2} - 1) {(ϑ (β x + 2) - 2 β λ t^{ϑ})}^{2}} .$ (39)

Fig.3

Solitary wave sketch for the absolute, real, and imaginary of Equation (16) in three–dimensional plots, respectively. When [α = 2, a₀ = 5, β = 3, λ = 4, σ = 1, ϑ = 0.5].

Fig.4

Solitary wave sketch for the absolute, real, and imaginary of Equation (16) in two–dimensional plots, respectively. When [α = 2, a₀ = 5, β = 3, λ = 4, σ = 1, ϑ = 0.5].

Fig.5

Periodic wave sketch for the absolute, real, and imaginary of Equation (24) in three–dimensional plots, respectively. When $[α = \frac{9}{4}, a_{0} = 5, β = 3, λ = 4, σ = 1, ϑ = 0.5]$ .

Fig.6

Periodic wave sketch for the absolute, real, and imaginary of Equation (24) in two–dimensional plots, respectively. When $[α = \frac{9}{4}, a_{0} = 5, β = 3, λ = 4, σ = 1, ϑ = 0.5]$ .

2.2 Stability property

The investigation of the stability property of the computational obtained solutions is studied here by using the momentum M in the Hamiltonian system that is given by $M = \frac{1}{2} \int_{- k}^{k} H^{2} (ð) d ð,$ (40) where $H (ð)$ is the solution of the model. Consequently, the condition for stability of the solutions can be formulated as $\frac{\partial M}{\partial λ} > 0,$ (41) where λ is the wave velocity. The momentum in the Hamiltonian system for Equation (12) is given by $\begin{matrix} M = \frac{1}{9 λ} \times [32 (25 λ + log (e^{5 - 5 λ} + e^{5 λ}) \\ - log (e^{- 5 λ} + e^{5 λ + 5}))] \end{matrix}$ (42) and, thus $\frac{\partial M}{\partial ω} |_{λ = 5} = 0.7111111111 > 0 .$ (43) We conclude that this solution is unstable on the interval x ∈ [-5, 5] & t ∈ [-5, 5], and λ = 5. This result shows the ability of the solutions for their application. Using the same steps, allows checking the stability property of all other obtained solutions.

2.3 Semi-analytical and numerical solutions

This section studies the semi-analytical and numerical solutions of Equation (9) by applying the Adomian decomposition and septic B-spline techniques that are considered as the most accurate semi-analytical and numerical tools to get this type of solutions. Using the computational solution of (9) with the following initial conditions [ $H_{exact}$ = $\frac{1}{3}$ (-4) tanh $(\frac{ð}{2})$ , ϱ = $\frac{827}{6336}$ , α = 2, a₀ = 4, β = 3, λ = 5, σ = 1, ω = $- \frac{51}{2}],$ allows applying the following schemes, as follows:

2.3.1 Adomian Decomposition method

Based on the previous mentioned initial conditions, we obtain $H (0) = 0, H^{'} (0) = \frac{- 2}{3} .$ Implementation of the Adomian decomposition method on Equation (9), yields $H_{0} = - \frac{2 ð}{3},$ (44) $H_{1} = \frac{ð^{3}}{18} - \frac{ð^{5}}{240},$ (45) $H_{2} = \frac{ð^{10}}{86400} - \frac{ð^{8}}{4032} + \frac{ð^{7}}{20160} - \frac{ð^{5}}{720},$ (46) $\begin{matrix} H_{3} = - \frac{ð^{13}}{15974400} - \frac{ð^{12}}{22809600} + \frac{ð^{11}}{422400} + \frac{ð^{10}}{725760} \\ - \frac{67 ð^{9}}{1451520} + \frac{31 ð^{7}}{60480}, \end{matrix}$ (47) According to (44)–(47), we get $\begin{matrix} H_{Semi - - analy .} = - \frac{ð^{13}}{15974400} - \frac{ð^{12}}{22809600} + \frac{ð^{11}}{422400} \\ + \frac{47 ð^{10}}{3628800} - \frac{67 ð^{9}}{1451520} - \frac{ð^{8}}{4032} + \frac{17 ð^{7}}{30240} \\ - \frac{ð^{5}}{180} + \frac{ð^{3}}{18} - \frac{2 ð}{3} + . . . \end{matrix}$ (48)

Thus, calculating the absolute error between the computational and semi–analytical solutions shows the accuracy of the answers that calculating is shown in Table 1. This table explains the accuracy of the semi-analytical solution is always very small for every value of ð near to zero, and when these values go away from the zero point, the absolute error begins in increasing.

Table 1

Analytical, semi-analytical, and absolute value of error by using the Adomian decomposition method

Value of ð	Analytical	Semi-analytical	Abs. Value of error
0.1	0.0666112	0.0666112	2.46811 × 10^-12
0.2	0.132891	0.132891	6.28138× 10^-10
0.3	0.198513	0.198513	1.59894× 10^-8
0.4	0.263167	0.263167	1.58492× 10^-7
0.5	0.326558	0.326559	9.36693× 10^-7
0.6	0.388417	0.388421	3.99041× 10^-6
0.7	0.448501	0.448514	1.35593× 10^-5
0.8	0.506599	0.506638	3.90394× 10^-5
0.9	0.562532	0.562631	9.90239× 10^-5
1	0.616156	0.616383	0.000227249

2.3.2 Septic B-Spline

According to the septic B-spline, the numerical solution of Equation (9) is given by $H (ð) = \sum_{i = - 1}^{n + 1} ℧_{i} ℵ_{i},$ (49) where ℧_i, ℵ _i satisfies the next conditions $L H (ð) = \emptyset (ð_{i}, H (x_{i})) where (i = 0, 1, . . ., n)$ and $ℵ_{i} (ð) = \frac{1}{h^{5}} {\begin{matrix} (ð - ð_{i - 4})^{7}, & ð \in [ð_{i - 4}, ð_{i - 3}], \\ (ð - ð_{i - 4})^{7} - 8 (ð - ð_{i - 3})^{7}, & ð \in [ð_{i - 3}, ð_{i - 2}], \\ (ð - ð_{i - 4})^{7} - 8 (ð - ð_{i - 3})^{7} + 28 ð (ð - ð_{i - 2})^{7}, & ð \in [ð_{i - 2}, ð_{i - 1}], \\ (ð - ð_{i - 4})^{7} - 8 (ð - ð_{i - 3})^{7} + 28 (ð - ð_{i - 2})^{7} + 56 (ð - ð_{i - 1})^{7}, & ð \in [ð_{i - 1}, ð_{i}], \\ (ð_{i + 4} - ð)^{7} - 8 (ð_{i + 3} - ð)^{7} + 28 (ð_{i + 2} - ð)^{7} + 56 (ð_{i + 1} - ð)^{7}, & ð \in [ð_{i}, ð_{i + 1}], \\ (ð_{i + 4} - ð)^{7} - 8 (ð_{i + 3} - ð)^{7} + 28 (ð_{i + 2} - ð)^{7}, & ð \in [ð_{i + 1}, ð_{i + 2}], \\ (ð_{i + 4} - ð)^{7} - 8 (ð_{i + 3} - ð)^{7}, & ð \in [ð_{i + 2}, ð_{i + 3}], \\ (ð_{i + 4} - ð)^{7}, & ð \in [ð_{i + 3}, ð_{i + 4}], \\ 0, & otherwise, \end{matrix}$ (50) where i ∈ [-3, n + 3]. Thus, the approximate solution is given by $\begin{matrix} v_{i} (ð) = ℧_{i - 3} + 120 ℧_{i - 2} + 1191 ℧_{i - 1} \\ + 2416 ℧_{i} + 1191 ℧_{i + 1} + 120 ℧_{i + 2} + ℧_{i + 3} . \end{matrix}$ (51) Substituting Equation (51) into Equation (9), obtains a system of equations. Solving this system, gives the following data that are shown in the next Table 2.

Fig.7

Singular solitary wave in three, two-dimensional, contour, and stream plots of Equation (17), respectively. When [α = 2, a₀ = 5, β = 3, λ = 4, σ = 1, ϑ = 0.5].

Fig.8

Singular solitary wave in three, two-dimensional, contour, and stream plots of Equation (25), respectively. When $[α = 2, a_{0} = 5, β = 3, λ = 4, σ = \frac{9}{4}, ϑ = 0.5]$ .

Fig.9

Distinct types of plots for analytical and semi–analytical to show the convergence between both of them, according to the values in Table 1 This show the accuracy of the Adomian decomposition method on Equation (9).

Table 2

Exact, numerical, and absolute value of error by using septic B-spline scheme on Equation (9)

Value of ð	Approx. Sol.	Analytical Sol.	Abs. Value of error
0	0	0	0
0.1	–0.0666112	–0.0666112	6.56523× 10^-12
0.2	–0.132891	–0.132891	1.47184× 10^-11
0.3	–0.198513	–0.198513	1.928× 10^-11
0.4	–0.263167	–0.263167	2.29722× 10^-11
0.5	–0.326558	–0.326558	2.39532× 10^-11
0.6	–0.388417	–0.388417	2.30321× 10^-11
0.7	–0.448501	–0.448501	1.94107× 10^-11
0.8	–0.506599	–0.506599	1.50028× 10^-11
0.9	–0.562532	–0.562532	6.78468× 10^-12
1	–0.616156	–0.616156	1.11022 × 10^-16

Fig.10

Distinct types of plots for analytical and numerical to show the convergence between both of them, according to the values in Table 2 This show the accuracy of the septic B–spline scheme on Equation (9).

3 Result and discussion

This section is divided into two main parts. First, one is studying the computational solution and make a comparison between them and other obtained results in previous work. While the second part is making a comparison between the obtained numerical and semi–analytical solutions in our paper to show and explain which one of them is more accurate than the other.

Analytical solutions:

In [d1] Dianchen Lu et al., applied theg generalized Kudryashov and a novel $(\frac{G^{'}}{G})$ –expansion methods to Equation (9) and they obtained many kind of solutions and the investigation in our and their solution, shows Equation (5) is equal to Equation 24 when $[α = ϑ, λ = k, c = \frac{2 k^{2} - α^{2}}{2}, β = α, σ = α^{2} + \frac{1}{e^{α (x - \frac{2 k t^{α}}{α})}}]$ and all other obtained solutions in our paper is different from those obtained in [d1]. That shows the superiority of our used computational method in this paper, which obtains many solutions and covers the solutions that can be obtained by other methods.

Numerical and semi-analytical solutions: The comparison between these types of solution depends on showing which one of the used schemes get the smallest value of the absolute value of error. To figure out these values, Fig. 11 shows the comparison between the obtained value of the absolute value of error in each used methods, explains the septic B-spline scheme is the more accurate than the Adomian decomposition method for this model.

Fig.11

Distinct types of plots for semi–analytical and numerical to show the accuracy between both of them, according to the values in Tables 1 and 2.

4 Conclusion

This article studied the stable computational and numerical solutions of the nonlinear complex fractional generalized Zakharov system by applying the modified Khater method, Adomian decomposition, and B-spline schemes. The computational solutions are successfully obtained, tested the stability property of them, and some of them are sketched to explain more physical properties of the model (Figs. 3–8). Moreover, stable computational solutions are used to find the semi-analytical and approximate solutions. The comparison between the obtained distinct types of solutions is explained and investigated to show the absolute value of error between them and that is described in the shown Tables 1–2. The performance of both computational and numerical schemes shows powerful, effective, and its ability for applying to many and various forms of nonlinear evolution equations.

Footnotes

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through General Research Project under grant number G.R.P.1-162-40.

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