
We focused on the analytical solution of strong nonlinearity and complicated time-fractional evolution equations, including the Sawada-Kotera equation, Ito equation, and Kaup-Kupershmidt equation, using an effective and accurate method known as the Aboodh residual power series method (ARPSM) in the framework of the Caputo operator. Therefore, the Caputo operator and the ARPSM are practical for figuring out a linear or nonlinear system with a fractional derivative. This technique was effectively proposed to obtain a set of analytical solutions for various types of fractional differential equations. The derived solutions enabled us to understand the mechanisms behind the propagation and generation of numerous nonlinear phenomena observed in diverse scientific domains, including plasma physics, fluid physics, and optical fibers. The fractional property also revealed some ambiguity that may be observed in many natural phenomena, and this is one of the most important distinguishing factors between fractional differential equations and non-fractional ones. We also helped clarify fractional calculus in nonlinear dynamics, motivating researchers to work in mathematical physics.
Citation: Humaira Yasmin, Aljawhara H. Almuqrin. Efficient solutions for time fractional Sawada-Kotera, Ito, and Kaup-Kupershmidt equations using an analytical technique[J]. AIMS Mathematics, 2024, 9(8): 20441-20466. doi: 10.3934/math.2024994
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We focused on the analytical solution of strong nonlinearity and complicated time-fractional evolution equations, including the Sawada-Kotera equation, Ito equation, and Kaup-Kupershmidt equation, using an effective and accurate method known as the Aboodh residual power series method (ARPSM) in the framework of the Caputo operator. Therefore, the Caputo operator and the ARPSM are practical for figuring out a linear or nonlinear system with a fractional derivative. This technique was effectively proposed to obtain a set of analytical solutions for various types of fractional differential equations. The derived solutions enabled us to understand the mechanisms behind the propagation and generation of numerous nonlinear phenomena observed in diverse scientific domains, including plasma physics, fluid physics, and optical fibers. The fractional property also revealed some ambiguity that may be observed in many natural phenomena, and this is one of the most important distinguishing factors between fractional differential equations and non-fractional ones. We also helped clarify fractional calculus in nonlinear dynamics, motivating researchers to work in mathematical physics.
The study on flow of fluids which are electrically conducting is known as magnetohydrodynamics (MHD). The magnetohydrodynamics have important applications in the polymer industry and engineering fields (Garnier [1]). Heat transfer caused by hydromagnetism was discussed by Chakrabarthi and Gupta [2]. Using an exponentially shrinking sheet, Nadeem et al. [3] investigated the MHD flow of a Casson fluid. Krishnendu Bhattacharyya [4] examined the effect of thermal radiation on MHD stagnation-point Flow over a Stretching Sheet.
Mixed convection magnetohydrodynamics flow is described by Ishak on a vertical and on a linearly stretching sheet [5,6,7]. Hayat et al. [10] examine a mixed convection flow within a stretched sheet of Casson nanofluid. Subhas Abel and Monayya Mareppa, examine magnetohydrodynamics flow on a vertical plate [11]. Shen et al. [12], examined a vertical stretching sheet which was non-linear. Ishikin Abu Bakar [13] investigates how boundary layer flow is affected by slip and convective boundary conditions over a stretching sheet. A vertical plate oscillates with the influence of slip on a free convection flow of a Casson fluid [14].
Nasir Uddin et al. [15] used a Runge-Kutta sixth-order integration method. Barik et al. [16] implicit finite distinction methodology of Crank Sir Harold George Nicolson sort Raman and Kumar [17] utilized an exact finite distinction theme of DuFort–Frankel. Mondal et al. [18] used a numerical theme over the whole vary of physical parameters. With the laplace transform method, we can determine the magnetohydrodynamic flow of a viscous fluid [19]. Thamizh Suganya et al. [20] obtained that the MHD for the free convective flow of fluid is based on coupled non-linear differential equations. In this study, the analytical approximation of concentration profiles in velocity, temperature and concentration using homotopy perturbation method (HPM).
The governing differential equations in dimensionless form [19] as follows:
d2udy2−Hu+Grθ+Gmϕ=0, | (2.1) |
1Fd2θdy2=0, | (2.2) |
1Scd2ϕdy2−Sr∂2ϕ∂y2−γϕ=0. | (2.3) |
The dimensionless boundary conditions given by:
u=0, θ=1,ϕ=0 at y=0 | (2.4) |
and
u=0, θ=0,ϕ=0 at y→∞. | (2.5) |
He [21,22] established the homotopy perturbation method, which waives the requirement of small parameters. Many researchers have used HPM to obtain approximate analytical solutions for many non-linear engineering dynamical systems [23,24]. The basic concept of the HPM as follows:
d2udy2−Hu+Grθ+Gmϕ=0 | (3.1) |
1Fd2θdy2=0 | (3.2) |
1Scd2ϕdy2−Sr∂2ϕ∂y2−γϕ=0 | (3.3) |
with initial and boundary conditions given by:
y=0 at u=0, θ=1,C=1y→∞ as u=0, θ=1,ϕ=0. | (3.4) |
Homotopy for the above Eqs (3.1) to (3.4) can be constructed as follows:
(1−p)[d2udy2−Hu+Grθ+Gmϕ]+p[d2udy2−Hu+Grθ+Gmϕ]=0 | (3.5) |
(1−p)[1Fd2θdy2−θ]+p[1Fd2θdy2−θ+θ]=0 | (3.6) |
(1−p)[1Scd2ϕdy2−γϕ]+p[1Scd2ϕdy2−Sr∂2θ∂y2−γϕ]=0 | (3.7) |
The approximate solution of the Eqs (3.5) to (3.7) are
u=u0+pu1+p2u2+p3u3+... | (3.8) |
θ=θ0+pθ1+p2θ2+p3θ3+... | (3.9) |
ϕ=ϕ0+pϕ1+p2ϕ2+p3ϕ3+... | (3.10) |
Substitution Eqs (3.5) to (3.7) in Eqs (3.8) to (3.10) respectively. We obtain the following equations
(1−p)[d2(u0+pu1+p2u2+p3u3+...)dy2−H(u0+pu1+p2u2+p3u3+...)+Gr(θ0+pθ1+p2θ2+p3θ3+...)+Gm(ϕ0+pϕ1+p2ϕ2+p3ϕ3+...)]+p[d2(u0+pu1+p2u2+p3u3+...)dy2−H(u0+pu1+p2u2+p3u3+...)+Gr(θ0+pθ1+p2θ2+p3θ3+...)+Gm(ϕ0+pϕ1+p2ϕ2+p3ϕ3+...)]=0 | (3.11) |
and
(1−p)[1Fd2(θ0+pθ1+p2θ2+p3θ3+...)dy2−(θ0+pθ1+p2θ2+p3θ3+...)]+p[1Fd2(θ0+pθ1+p2θ2+p3θ3+...)dy2−(θ0+pθ1+p2θ2+p3θ3+...)+(θ0+pθ1+p2θ2+p3θ3+...)]=0, | (3.12) |
and
(1−p)[1Scd2(ϕ0+pϕ1+p2ϕ2+p3ϕ3+...)dy2−γ(ϕ0+pϕ1+p2ϕ2+p3ϕ3+...)]+p[1Scd2(ϕ0+pϕ1+p2ϕ2+p3ϕ3+...)dy2−Sr∂2(θ0+pθ1+p2θ2+p3θ3+...)∂y2−γ(ϕ0+pϕ1+p2ϕ2+p3ϕ3+...)]=0. | (3.13) |
Equating the coefficient of p on both sides, we get the following equations
P0:d2u0dy2−Hu0+Grθ0+Gmϕ0=0; | (3.14) |
p0:1Fd2θ0dy2−θ0=0; | (3.15) |
P1:1Fd2θ0dy2−θ0+θ1=0; | (3.16) |
P1:1Scd2ϕ0dy2−γϕ0=0; | (3.17) |
P1:1Scd2ϕ0dy2−Sr∂2θ0∂y2−γϕ0=0. | (3.18) |
The boundary conditions are
u0=0, θ0=1,ϕ0=1 at y=0u0=0, θ0=1,ϕ0=1 at y→∞. | (3.19) |
and
u1=0, θ1=0,ϕ1=0 at y=0u1=0, θ1=0,ϕ1=0 at y→∞. | (3.20) |
Solving the Eqs (3.9)–(3.14), we obtain
u0(t)=Gr√1F+H[e−y√1F−e−y√H]+Gm√γSc+H[e−y√γSc−e−y√H]; | (3.21) |
θ0(y)=e−y√1F; | (3.22) |
ϕ0(y)=e−y√γSc; | (3.23) |
ϕ1(y)=ScSr√F+FγSc[e−y√1F−e−y√γSc]. | (3.24) |
Considering the iteration, we get,
u(t)=Gr√1F+H[e−y√1F−e−y√H]+Gm√γSc+H[e−y√γSc−e−y√H]; | (3.25) |
θ(y)=e−y√F; | (3.26) |
ϕ(y)=e−y√γSc+ScSr√F+FγSc[e−y√1F−e−y√γSc]. | (3.27) |
From the Eqs (3.25)–(3.27), we obtain
Cf=−(∂u∂y)y=0=(Gm√γSc−Gm√H+Gr(γSc+H)√1+RPr+GmH√γSc+(−Gm−Gr)H32−γGr√HSc)(√1+RPr+H(γSc+H)); | (4.1) |
Nu=−(∂θ∂y)y=0=√1+RPr; | (4.2) |
Sh=−(∂ϕ∂y)y=0=ScSr(1+R)√1+RPr−√γSc((−R−1)√1+RPr+((1+R)Sc−Prγ)Sc)(1+R)√1+RPrPrγSc | (4.3) |
The combined impacts of transient MHD free convective flows of an incompressible viscous fluid through a vertical plate moving with uniform motion and immersed in a porous media are examined using an exact approach. The approximate analytical expressions for the velocity u, temperature θ, and concentration profile ϕ are solved by using the homotopy perturbation method for fixed values of parameters is graphically presented.
Velocity takes time at first, and for high values of y, it takes longer, and the velocity approaches zero as time increases. The velocity of fluid rises with Gr increasing, as exposed in Figure 1.
The variations of parameter Gm are depicted in Figure 2. It has been established that as the value of increases, neither does the concentration. Gm. This is because increasing the number of 'Gm' diminishes the slog energy, allow the fluid to transfer very rapidly.The dimensionless Prandtl number is a number which combines the viscosity of a fluid with its thermal conductivity. For example, Figure 3 shows how a decrease in 'Pr'increases the concentration of velocity profile.
The Figure 4 shows how a decrease in concentration occurs when the value of H increases.As shown in Figure 5, when the Schmidt number Sc increases, the concentration of velocity profiles decreases, while the opposite is true for the Soret number Sr, as shown in Figures 6, 7 and 8, represented the radiation parameter R, chemical reaction parameter γ is increasing when it implies a decrease in concentration.
Based on Figure 9, it is evident that the thickness of the momentum boundary layer increases for fluids with Pr<1. When Pr<0.015, the heat diffuses rapidly in comparison to the velocity.
Figure 10 depicts the impact of the radiation parameter R on temperature profiles. The temperature profiles θ, which are a decreasing function of R, are found to decrease the flow and lower fluid velocity. As the radiation parameter R is increased, the fluid thickens, temperatures and thermal boundary layer thickness to decrease.
This statement is justified because the thermal conductivity of a fluid declines by the growing Prandtl number Pr and hence the thickness of thermal boundary layers and temperature profiles decrease as well. Based on Figure 9, we see an increase in fluid concentration with large Prandtl numbers Pr. Radiation parameter R and temperature profiles are illustrated in Figure 10. The temperature profiles θ, which are a decreasing function of R, are initiate to reduction the flow and decline the fluid velocity. Radiation parameter R increases as fluid thickness increases, temperature increases, and thickness of thermal boundary layer decreases.
The influence of Pr, R, γ, Sc, and Sr on the concentration profiles ϕ is shown in Figures 11–15. The fluid concentration rises on highest values of Pr, as shown in Figure 11. The profile of temperature is affected by the radiation parameter R which is shown in Figure 12. As a function of R, the concentration profiles reduce the flow and decrease fluid velocity.
The growing values of γ and Sc lead to falling in the concentration profiles, is described from Figures 13 and 15. The concentration profiles increase as the number of sorts (Sr) increases, as shown in Figure 14.
A free convection magnetohydrodynamic (MHD) flow past a vertical plate embedded in a porous medium was offered in this paper. Homotopy perturbation method is used to find approximate analytical solutions for the concentration of species. The effects of system parameters on temperature and velocity profiles were investigated using these analytical expressions. The graphic representation of the impact of several physical parameters attempting to control the velocity, temperature, and concentration profiles and a brief discussion. Analytical expressions were also developed for the Skin-friction and Nusselt and Sherwood numbers.
The authors declare that they have no conflict of interest.
The authors are thankful to the reviewers for their valuable comments and suggestions to improve the quality of the paper. The work of H. Alotaibi is supported by Taif University Researchers Supporting Project Number (TURSP-2020/304), Taif University, Taif, Saudi Arabia.
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