Exact Travelling Wave Solutions of Richards’ Equation by Modified F-expansion Method

Patel, Parimal; Pradhan, V. H.

doi:10.1016/j.proeng.2015.11.410

Cited by 5 publications

(3 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Also, comparison was made between the solution of the standard tanh method and F-Expansion method under special conditions. We observe that our results represented by equations (15) and (21) are similar to the results obtained by F-Expansion method if the condition h=1, C=-1, and B=2 in equation (46) [15]. The closed-form solution obtained via this method is in good agreement with the F-Expansion method.…”

Section: Resultssupporting

confidence: 85%

“…numerous methods to find exact solution of nonlinear PDFs, have been suggested in the literature like: the tanh-coth method [1,2], sine-cosine method [3], homogeneous balance method [4,5], exp-function method [6], first-integral method [7], Jacobi elliptic function method [8], and ( ′ / )-Expansion method [9,10,11]. The standard method offered by Malfliet [12] is a strong technique to compute exact traveling waves solutions of non-linear partial differential equations, we will utilize the standard tanh method to find the traveling wave for a nonlinear diffusionconvection equation for different values of m and n [13,14,15].…”

Section: -Introductionmentioning

confidence: 99%

See 1 more Smart Citation

New traveling wave solutions of a nonlinear diffusion–convection equation by using standard tanh method

Aziz¹,

Taha²,

Hameed³

et al. 2018

Tikrit J. Pure Sci.

View full text Add to dashboard Cite

Exact solutions of traveling wave are acquired by employ a relatively new technique which is called standard tanh method for a nonlinear diffusion-convection equation. The tanh method is applied for the first time for finding travelling wave solutions for the nonlinear diffusionconvection equation = ( ) + ( ) , where n and m, are integers, and ≥ > 1, of order (m=4, n=7) and (m=5, n=9). Analytical solutions of a nonlinear diffusion-convection equations are obtained as a polynomial in tanh(x), and the plots for exact solutions are given. The obtained results are compared with F-Expansion method to validate the proposed approach.

show abstract

Section: Resultssupporting

confidence: 85%

Section: -Introductionmentioning

confidence: 99%

New traveling wave solutions of a nonlinear diffusion–convection equation by using standard tanh method

Aziz¹,

Taha²,

Hameed³

et al. 2018

Tikrit J. Pure Sci.

View full text Add to dashboard Cite

show abstract

“…numerous methods to find exact solution of nonlinear PDFs, have been suggested in the literature like: the tanh-coth method [1,2], sine-cosine method [3], homogeneous balance method [4,5], exp-function method [6], first-integral method [7], Jacobi elliptic function method [8], and (𝐺 ′ /𝐺)-Expansion method [9,10,11]. The standard method offered by Malfliet [12] is a strong technique to compute exact traveling waves solutions of non-linear partial differential equations, we will utilize the standard tanh method to find the traveling wave for a nonlinear diffusionconvection equation for different values of m and n [13,14,15]. This equation arises in many physical phenomena for different values of 𝑛 and .…”

Section: -Introductionmentioning

confidence: 99%

New traveling wave solutions of a nonlinear diffusion–convection equation by using standard tanh method

Aziz¹,

Taha²,

Hameed³

et al. 2023

Tikrit J. Pure Sci.

View full text Add to dashboard Cite

Exact solutions of traveling wave are acquired by employ a relatively new technique which is called standard tanh method for a nonlinear diffusion–convection equation. The tanh method is applied for the first time for finding travelling wave solutions for the nonlinear diffusion–convection equation , where n and m, are integers, and , of order (m=4, n=7) and (m=5, n=9). Analytical solutions of a nonlinear diffusion–convection equations are obtained as a polynomial in tanh(x), and the plots for exact solutions are given. The obtained results are compared with F-Expansion method to validate the proposed approach.

show abstract

Comparing and Contrasting Travelling Wave Behaviour for Groundwater Flow and Foam Drainage

Boakye-Ansah

Grassia

2021

Transp Porous Med

View full text Add to dashboard Cite

Liquid drainage within foam is generally described by the foam drainage equation which admits travelling wave solutions. Meanwhile, Richards’ equation has been used to describe liquid flow in unsaturated soil. Travelling wave solutions for Richards equation are also available using soil material property functions which have been developed by Van Genuchten. In order to compare and contrast these solutions, the travelling waves are expressed as dimensionless height, $$ {\hat{\xi }} $$ ξ ^ , versus moisture content, $$ \varTheta $$ Θ . For low moisture content, $$ {{\hat{\xi }}} $$ ξ ^ exhibits an abrupt change away from the dry state in Richards equation compared to a much more gradual change in foam drainage. When moisture content nears saturation, $$ {\hat{\xi }} $$ ξ ^ reaches large values (i.e. $$ {\hat{\xi }} \gg 1 $$ ξ ^ ≫ 1 ) for both Richards equation and foam drainage, implying a gradual approach of $$ \varTheta $$ Θ towards the saturated state. The $$ {\hat{\xi }} $$ ξ ^ values in Richards equation tend, however, to be larger than those in the foam drainage equation, implying an even more gradual approach towards saturation. The reasons for the difference between foam drainage and Richards equation solutions are traced back to soil material properties and in particular a soil specific parameter “m” which is determined from the soil-water retention curve.

show abstract

Exact Travelling Wave Solutions of Richards’ Equation by Modified F-expansion Method

Cited by 5 publications

References 9 publications

New traveling wave solutions of a nonlinear diffusion–convection equation by using standard tanh method

New traveling wave solutions of a nonlinear diffusion–convection equation by using standard tanh method

New traveling wave solutions of a nonlinear diffusion–convection equation by using standard tanh method

Comparing and Contrasting Travelling Wave Behaviour for Groundwater Flow and Foam Drainage

Contact Info

Product

Resources

About