A study of moisture content in unsaturated porous medium by using homotopy perturbation method (HPM) and variational iteration method (VIM)

Rehman, Gohar; Qin, Shengwu; Ain, Qura Tul; Ullah, Zaheen; Zaheer, Muhammad; Talib, Muhammad Afnan; Mehmood, Qaiser; Baloch, Muhammad Yousuf Jat; Rahman, Naveed Ur

doi:10.1007/s13137-021-00193-z

Cited by 5 publications

(5 citation statements)

References 26 publications

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“…where p, q, and r are real parameters and M is a positive integer number and will be defined later in (19). We note that equation (17) has different kind of solutions depending on p, q, and r. snðηÞ = snðη, mÞ, cnðηÞ = cnðη, mÞ, dnðη, mÞ = dnðη, mÞ are the Jacobi elliptic functions (JEFs) for 0 < m < 1: When m ⟶ 1, the JEFs are transformed into the following hyper-bolic functions:…”

Section: Analytical Solutions Of Sfrklementioning

confidence: 99%

“…First set: if p > 0 and ℏ 1 > 0, then the solutions φðηÞ of equation (17) corresponding to Table 1 are as follows.…”

Section: Analytical Solutions Of Sfrklementioning

confidence: 99%

“…As a result, solving nonlinear problems is crucial in nonlinear sciences. Some of these methods, such as Darboux transformation [1], sine-cosine [2,3], exp ð−ϕðς ÞÞ-expansion [4], ðG′/GÞ-expansion [5,6], Hirota's function [7], perturbation [8,9], Jacobi elliptic function [10,11], trial function [12], tanh-sech [13], fractal semi-inverse method [14,15], F-expansion method [16], and homotopy perturbation method [17], have been recently developed. However, it is completely obvious that the phenomena that happen in the environment are not always deterministic.…”

Section: Introductionmentioning

confidence: 99%

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The Analytical Solutions of the Stochastic Fractional RKL Equation via Jacobi Elliptic Function Method

Al‐Askar

Mohammed

2022

Advances in Mathematical Physics

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This article considers the stochastic fractional Radhakrishnan-Kundu-Lakshmanan equation (SFRKLE), which is a higher order nonlinear Schrödinger equation with cubic nonlinear terms in Kerr law. To find novel elliptic, trigonometric, rational, and stochastic fractional solutions, the Jacobi elliptic function technique is applied. Due to the Radhakrishnan-Kundu-Lakshmanan equation’s importance in modeling the propagation of solitons along an optical fiber, the derived solutions are vital for characterizing a number of key physical processes. Additionally, to show the impact of multiplicative noise on these solutions, we employ MATLAB tools to present some of the collected solutions in 2D and 3D graphs. Finally, we demonstrate that multiplicative noise stabilizes the analytical solutions of SFRKLE at zero.

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Section: Analytical Solutions Of Sfrklementioning

confidence: 99%

“…First set: if p > 0 and ℏ 1 > 0, then the solutions φðηÞ of equation (17) corresponding to Table 1 are as follows.…”

Section: Analytical Solutions Of Sfrklementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Analytical Solutions of the Stochastic Fractional RKL Equation via Jacobi Elliptic Function Method

Al‐Askar

Mohammed

2022

Advances in Mathematical Physics

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“…13 , the variational method of iteration (V.I.M.) 14,15 , and the perturbation method of homotopy (H.P.M.) 16 .…”

Section: Introductionmentioning

confidence: 99%

الحلول شبه التحليلية لمعادلات فيشر الكسرية للزمن عبر الطريقة التكرارية الجديدة

Tarate,

Bhadane,

Gaikwad

et al. 2024

Baghdad Sci.J

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An effective method for resolving non-linear partial differential equations with fractional derivatives is the New Sumudu Transform Iterative Method (NSTIM). It excels at solving difficult mathematical puzzles and offers insightful information about the behaviour of time-fractional Fisher equations. The method, which makes use of Caputo's sense derivatives and Wolfram in Mathematica, is reliable, simple to use, and gives a visual depiction of the solution. The analytical findings demonstrate that the proposed approach is effective and simple in generating precise solutions for the time-fractional Fisher equations. The results are made more reliable and applicable by including Caputo's sense derivatives. Mathematical modelling relies on the effectiveness and simplicity of the NSTIM approach to solve time-fractional Fisher equations since it enables precise solutions without the use of a lot of processing power. The NSTIM approach is a useful tool for researchers in a variety of domains since it also offers a flexible framework that is easily adaptable to other fractional differential equations. It now becomes possible to examine the dynamics and behaviour of complex systems governed by time-fractional Fisher equations with efficiency and reliability, opening up new research avenues. The ability to solve time-fractional Fisher equations efficiently and reliably using the NSTIM approach has significant implications for various fields such as population dynamics, mathematical biology, and epidemiology. Researchers can now analyze the spread of diseases or study the population dynamics of species with higher accuracy and less computational effort. This advancement in solving fractional differential equations paves the way for deeper insights into the behavior and patterns of complex systems, ultimately advancing scientific understanding and offering new possibilities for practical applications.

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“…Rehman et al's [41] unsaturated porous media were analyzed by solving Burger's equation using the variational iterative modeling and homotopy perturbation method. Wang and Wang [42] described two different types of plasma models with variable coefficients by using the fractal derivative.…”

Section: Introductionmentioning

confidence: 99%

The $K$ Extended Laguerre Polynomials Involving $\{A \begin{matrix} () \end{matrix}\}$
Khan
¹
,
Mateen
²
,
Akgül
³

et al. 2022
Advances in Mathematical Physics
1
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0
0
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In this manuscript, we present the generalized hypergeometric function of the type F r r , r > 2 and extension of the K Laguerre polynomial for the K extended Laguerre polynomials A r , n , k α x . Additionally, we describe the K generating function, K recurrence relations, and K S Rodrigues formula.

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A study of moisture content in unsaturated porous medium by using homotopy perturbation method (HPM) and variational iteration method (VIM)

Cited by 5 publications

References 26 publications

The Analytical Solutions of the Stochastic Fractional RKL Equation via Jacobi Elliptic Function Method

The Analytical Solutions of the Stochastic Fractional RKL Equation via Jacobi Elliptic Function Method

الحلول شبه التحليلية لمعادلات فيشر الكسرية للزمن عبر الطريقة التكرارية الجديدة

The $K$ Extended Laguerre Polynomials Involving $\{A \begin{matrix} () \end{matrix}\}$
Khan
¹
,
Mateen
²
,
Akgül
³

et al. 2022
Advances in Mathematical Physics
1
0
0
0
View full text Add to dashboard Cite

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A study of moisture content in unsaturated porous medium by using homotopy perturbation method (HPM) and variational iteration method (VIM)

Cited by 5 publications

References 26 publications

The Analytical Solutions of the Stochastic Fractional RKL Equation via Jacobi Elliptic Function Method

The Analytical Solutions of the Stochastic Fractional RKL Equation via Jacobi Elliptic Function Method

الحلول شبه التحليلية لمعادلات فيشر الكسرية للزمن عبر الطريقة التكرارية الجديدة

The K Extended Laguerre Polynomials Involving A Khan1, Mateen2, Akgül3 et al. 2022Advances in Mathematical Physics1000View full textAdd to dashboardCite

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The $K$ Extended Laguerre Polynomials Involving $\{A \begin{matrix} () \end{matrix}\}$
Khan
¹
,
Mateen
²
,
Akgül
³

et al. 2022
Advances in Mathematical Physics
1
0
0
0
View full text Add to dashboard Cite