On Use of Expanding Parameters and Auxiliary Term in Homotopy Perturbation Method for Boussinesq Equation with Tidal Condition

Abstract

This paper uses the homotopy perturbation method for the analytical solution of groundwater table fluctuations, in response to the tidal boundary condition, for a coastal unconfined aquifer with sloping beach face. The Boussinesq equation for sloping beach contains two non-linear terms. The governing equation is reconstructed in homotopic form with two virtual perturbation parameters and an auxiliary term. The secular terms generated from the non-linear diffusion term and the slope term are eliminated by using parameter expansions based on two virtual parameters. Two non-dimensional parameters emerge from the solution in the process of eliminating secular terms: (i) parameter equivalent to amplitude parameter and (ii) parameter representing beach slope. The second-order (starting from zeroth-order) solution is presented. The higher-order solution efficiently captures the non-linearity of the problem.

Appendix: Secular Terms Removal from Inhomogeneous Terms

Solution for the ${p_{1}^{0}} {p_{2}^{0}}$ (15) is given by

$$ h_{0,0}(x^{*},t)= D+ A exp(-k_{1} x^{*}) \cos[\omega t- k_{1} x^{*}]. $$

(35)

By substituting (35) in the expression (18), the inhomogeneous term (I_1,0) can be written as

$$\begin{array}{@{}rcl@{}} I_{1,0}\!&=&\!\frac{A^{2} {k_{1}^{2}} K}{\eta_{e}}\left\{\mathit{exp}(-2 k_{1} x^{*})-2exp(-2 k_{1} x^{*}){\sin}[2 {\omega} t- 2 k_{1} x^{*}]\right\}\\ &&+\frac{2 A {k_{1}^{2}} }{\eta_{e}}[K D-\eta_{e} {\Gamma}_{1,0}] \underbrace{exp(- k_{1} x^{*})\sin[\omega t- k_{1} x^{*}]}_{secular term}. \end{array} $$

(36)

Requirement of no secular term in inhomogeneous term $I_{1,0}$ needs

$$ {\Gamma}_{1,0}=\frac{K D }{\eta_{e}}. $$

(37)

If ${\Gamma }_{0,0}$ is replaced with ${\Gamma }_{1,0}$ from expression (37), Eq. 15 represents linearized Boussinesq equation. Particular integral for the ${p_{1}^{1}} {p_{2}^{0}}$ equation is given as

$$\begin{array}{@{}rcl@{}} h_{1,0}^{PI}(x^{*},t)&=&-\frac{A^{2} {k_{1}^{2}} K }{2{ \eta}_{e} {\omega}} exp(-2 k_{1} x^{*})\\ &&\times\left\{1 + 2 {\cos}[2 {\omega} t - 2 k_{1} x^{*}]\right\}. \end{array} $$

(38)

Boundary condition can be satisfied by expressing the solution $h_{1,0}$ as addition of particular integral [$h_{1,0}^{PI}$] and complementary function ($h_{1,0}^{CF}$),

$$\begin{array}{@{}rcl@{}} h_{1,0}(0,t)&=&h_{1,0}^{PI}(0,t)+h_{1,0}^{CF}(0,t)= 0, \end{array} $$

(39)

$$\begin{array}{@{}rcl@{}} h_{1,0}^{CF}(x^{*},t)&=&\frac{A^{2} {k_{1}^{2}} K}{2 \eta_{e} \omega}\{1 + 2 exp(-\sqrt{2} k_{1} x^{*})\\&&\times \cos[2 \omega t-\sqrt{2} k_{1} x^{*}]\}. \end{array} $$

(40)

Solution for the ${p_{1}^{1}}{p_{2}^{0}}$ equation (h_1,0) can be given as

$$\begin{array}{@{}rcl@{}} \mathit{h}_{1,0}(\mathit{x}^{*},\mathit{t})\!&=&\!\frac{A^{2} {k_{1}^{2}} K}{2 \eta_{e} \omega} \left\{1-exp(-2 k_{1} x^{*})\right.\\ &&\!-2\ \mathit{exp}(-2 k_{1} x^{*}) \cos[2\omega t- 2 k_{1} x^{*}]\\ &&\left.\!+ 2 \mathit{exp}(-\sqrt{2} k_{1} x^{*}) \cos[2 \omega t-\sqrt{2} k_{1} x^{*}]\right\}. \end{array} $$

(41)

By substituting (41) in Eq. (21), the inhomogeneous term $I_{0,1}$ is given by

$$\begin{array}{@{}rcl@{}} I_{0,1}&=&\frac{A k_{1} s_{p} exp(-k_{1} x^{*})}{2}\{-\cos[k_{1} x^{*}]\\&&+\cos[2\omega t-k_{1} x^{*}]\\ &&+\sin[k_{1} x^{*}]+\sin[2 \omega t-k_{1} x^{*}]\}\\ &&+ 2 A {k_{1}^{2}} {\Gamma}_{0,1} exp(-k_{1} x^{*}) \sin[\omega t- k_{1} x^{*}]\\ &&+\alpha_{0,0}f(x,t). \end{array} $$

(42)

Eliminating the secular term in particular integral by forcing the coefficient of secular term to zero,

$$\begin{array}{@{}rcl@{}} {\Gamma}_{0,1}&=&0, \end{array} $$

(43)

$$\begin{array}{@{}rcl@{}} \alpha_{0,0}&=&0. \end{array} $$

(44)

Solution for the ${p_{1}^{0}}{p_{2}^{1}}$ equation (h_0,1) can be obtained as

$$\begin{array}{@{}rcl@{}} h_{0,1}(x^{*},t)\!&=&\!\frac{A k_{1} s_{p}}{2 \omega}\left\{1-exp(-k_{1} x^{*})\cos[k_{1} x^{*}]\right.\\ &&\!-exp(-k_{1} x^{*})\cos[2 \omega t-k_{1} x^{*}]\\ &&\!+exp(-\sqrt{2} k_{1} x^{*})\cos[2 \omega t-\sqrt{2} k_{1} x^{*}]\\ &&\!-exp(-k_{1} x^{*})\sin[k_{1} x^{*}]\\ &&\!+exp(-k_{1} x^{*})\sin[2\omega t-k_{1} x^{*}]\\ &&\left.\!-exp(-\sqrt{2} k_{1} x^{*})\sin[2\omega t-\sqrt{2} k_{1} x^{*}]\right\}. \end{array} $$

(45)

Expressions of the inhomogeneous terms $I_{2,0}$, $I_{1,1}$, and $I_{0,2}$ are provided in the Supplementary Material for brevity. By eliminating secular terms from $I_{2,0}$, the expressions for ${\Gamma }_{2,0}$ and $\alpha _{0,0}$ can be obtained as

$$\begin{array}{@{}rcl@{}} {\Gamma}_{2,0}&=&\frac{A^{2} {k_{1}^{2}} K^{2}}{2 {\eta_{e}^{2}} \omega}, \end{array} $$

(46)

$$\begin{array}{@{}rcl@{}} \alpha_{0,0}&=&0. \end{array} $$

(47)

Solution for the ${p_{1}^{2}} {p_{2}^{0}}$ equation (h_2,0) can be given as

$$\begin{array}{@{}rcl@{}} &&h_{2,0}(x^{*},t)\\ &\!\!=&\frac{A^{3} {k_{1}^{4}} K^{2}}{{\eta_{e}^{2}} \omega^{2}}\left\{\frac{3}{2}exp(-3 k_{1} x^{*})\cos[3 \omega t- 3 k_{1} x^{*}]\right.\\ &&-\frac{1}{4} (4 + 3\sqrt{2})exp\left[\!-(1+\sqrt{2}) k_{1} x^{*}\right]\cos[3 \omega t-(1+\sqrt{2}) k_{1} x^{*}]\\ &&+\frac{1}{4} (-2 + 3\sqrt{2})exp(-\sqrt{3} k_{1} x^{*})\cos[3 \omega t-\sqrt{3} k_{1} x^{*}]\\ &&+\frac{11}{5}exp\left( -3 k_{1} x^{*}\right)\cos\left[\omega t- k_{1} x^{*}\right]\\ &&-\frac{6}{5}exp(-k_{1} x^{*})\cos[\omega t-k_{1} x^{*}]\\ &&-exp[-(1+\sqrt{2}) k_{1} x^{*}] \cos[\omega t+(1-\sqrt{2}) k_{1} x^{*}]\\ &&+\frac{2}{5}exp(-3 k_{1} x^{*})\sin[\omega t-k_{1} x^{*}]\\ &&-\frac{1}{20}(-8 + 5\sqrt{2})exp(-k_{1} x^{*})\sin[\omega t-k_{1} x^{*}]\\ &&\left.+\frac{1}{2\sqrt{2}}exp(-(1+\sqrt{2}) k_{1} x^{*})\sin[\omega t+(1-\sqrt{2}) k_{1} x^{*}]\right\}. \end{array} $$

(48)

By substituting (35), (41), and (45) in Eq. 27 and eliminating the secular terms (by forcing the coefficient of secular terms to zero) in $I_{1,1}$,

$$\begin{array}{@{}rcl@{}} {\Gamma}_{1,1}&=&\frac{A k_{1} K s_{p}}{2 \eta_{e} \omega}, \end{array} $$

(49)

$$\begin{array}{@{}rcl@{}} \alpha_{1,0}&=&0. \end{array} $$

(50)

Solution for the ${p_{1}^{1}} {p_{2}^{1}}$ equation (h_1,1) can be given as

$$\begin{array}{@{}rcl@{}} &&h_{1,1}(x^{*},t)\\ &=&\frac{A^{2} {k_{1}^{3}} K s_{p}}{\eta_{e} \omega^{2}}\left\{\vphantom{\sqrt{2}}exp(-2 k_{1} x^{*})\cos[ \omega t]\right.\\&&+exp(-2 k_{1} x^{*})\cos[3 \omega t-2 k_{1} x^{*}]\\ &&-\frac{1}{\sqrt{2}} exp(-\sqrt{2} k_{1} x^{*})\cos[3 \omega t-\sqrt{2} k_{1} x^{*}]\\ &&+\frac{1}{8}(-4 + 7\sqrt{2})exp(-\sqrt{3} k_{1} x^{*})\cos[3 \omega t- \sqrt{3} k_{1} x^{*}]\\ &&-\frac{1}{8}(4 + 3\sqrt{2})exp[-(1+\sqrt{2}) k_{1} x^{*}]\cos[3 \omega t\\&&-(1+\sqrt{2}) k_{1} x^{*}]\\ &&+\frac{3}{8}(-4+\sqrt{2})exp(- k_{1} x^{*}) \cos[\omega t- k_{1} x^{*}] \end{array} $$

$$\begin{array}{@{}rcl@{}} &&+exp(-2 k_{1} x^{*})\cos[\omega t-2 k_{1} x^{*}]\\ &&-\frac{1}{\sqrt{2}}exp(-\sqrt{2} k_{1} x^{*})\cos[\omega t-\sqrt{2} k_{1} x^{*}]\\ &&+\frac{1}{8}(-4+\sqrt{2})exp(-(1+\sqrt{2}) k_{1} x^{*})\cos[\omega t\\&&+(1-\sqrt{2}) k_{1} x^{*}]\\ &&-exp(-2 k_{1} x^{*})\sin[3 \omega t-2 k_{1} x^{*}]\\ &&+\frac{1}{\sqrt{2}}exp(-\sqrt{2} k_{1} x^{*})\sin[3 \omega t-\sqrt{2} k_{1} x^{*}]\\ &&-\frac{1}{8}(-4 + 7\sqrt{2})exp(-\sqrt{3} k_{1} x^{*})\sin[3 \omega t-\sqrt{3} k_{1} x^{*}]\\ &&+\frac{1}{8}(4 + 3\sqrt{2})exp[-(1+\sqrt{2}) k_{1} x^{*}]\sin[3 \omega t\\&&-(1+\sqrt{2}) k_{1} x^{*}]\\ &&-\frac{1}{8}(-4 + 5\sqrt{2})exp(-k_{1} x^{*})\sin[\omega t-k_{1} x^{*}]\\ &&-exp(-2 k_{1} x^{*})\sin[\omega t-2 k_{1} x^{*}]\\ &&+\frac{1}{\sqrt{2}}exp(-\sqrt{2} k_{1} x^{*})\sin[\omega t-\sqrt{2} k_{1} x^{*}]\\ &&+\frac{1}{8}(4+\sqrt{2})exp^{-(1+\sqrt{2}) k_{1} x^{*}}\sin[\omega t\\&&\left.+(1-\sqrt{2}) k_{1} x^{*}]\right\}. \end{array} $$

(51)

Solution for the ${p_{1}^{0}} {p_{2}^{2}}$ order can be given as

$$\begin{array}{@{}rcl@{}} \mathit{h}_{0,2}(\mathit{x}^{*},\mathit{t})\!&=&\!\frac{A {k_{1}^{2}} {s_{p}^{2}}}{\omega^{2}}\left\{-\frac{1}{4}exp(-k_{1} x^{*})\sin[ 3 \omega t- k_{1} x^{*}]\right.\\ &&\!+\frac{1}{\sqrt{2}}exp(-\sqrt{2} k_{1} x^{*})\sin[3 \omega t-\sqrt{2} k_{1} x^{*}]\\ &&\!-\frac{1}{4}(-1 + 2\sqrt{2}) exp(-\sqrt{3} k_{1} x^{*})\\&&\!\times\sin[3 \omega t-\sqrt{3} k_{1} x^{*}]\\ &&\!-\frac{1}{\sqrt{2}}exp(-k_{1} x^{*})\sin[ \omega t- k_{1} x^{*}]\\ &&\!+\frac{1}{\sqrt{2}}\mathit{exp}(-\sqrt{2} k_{1} x^{*})\\&&\left.\times\sin[ \omega t-\sqrt{2} k_{1} x^{*}]\vphantom{\frac{1}{4}}\right\}. \end{array} $$

(52)

By substituting (35) and (45) in Eq. 30 and eliminating the secular term $exp(-k_{1} x^{*})\sin [\omega t-k_{1} x^{*}]$ from inhomogeneous term $I_{0,2}$,

$$ {\Gamma}_{0,2}= 0. $$

(53)

There is a new secular term $exp(-k_{1} x^{*})\cos [\omega t+k_{1} x^{*}]$ present in the inhomogeneous term $I_{0,2}$. By defining $f(x,t)$ and eliminating it by forcing its coefficient to zero, $\alpha _{0,1}$ can be obtained as

$$\begin{array}{@{}rcl@{}} f(x,t)&=&exp(-k_{1} x^{*})\cos[\omega t+k_{1} x^{*}], \end{array} $$

(54)

$$\begin{array}{@{}rcl@{}} \alpha_{0,1}&=&-\frac{A {k_{1}^{2}} {s_{p}^{2}}}{2 \omega}. \end{array} $$

(55)

Groundwater fluctuation due to tidal oscillations for a coastal aquifer with vertical beach face is given as

$$\begin{array}{@{}rcl@{}} h(x^{*},t)\!&=&\! D+ A exp(-k_{1} x^{*}) \cos[\omega t- k_{1} x^{*}]\\ &&\!\times\frac{A^{2} {k_{1}^{2}} K}{2 \eta_{e} \omega} \left\{1-exp(-2 k_{1} x^{*})\right.\\ &&\!-2 exp(-2 k_{1} x^{*}) \cos[2\omega t- 2 k_{1} x^{*}]\\ &&\left.\!+ 2 exp(-\sqrt{2} k_{1} x^{*}) \cos[2 \omega t-\sqrt{2} k_{1} x^{*}]\right\}\\ &&\!+\frac{A^{3} {k_{1}^{4}} K^{2}}{{\eta_{e}^{2}} \omega^{2}}\left\{\frac{3}{2}exp(-3 k_{1} x^{*})\cos[3 \omega t- 3 k_{1} x^{*}]\right.\\ &&\!\!-\frac{1}{4} (4 + 3\sqrt{2})exp[-(1+\sqrt{2}) k_{1} x^{*}]\\&&\cos[3 \omega t-(1+\sqrt{2}) k_{1} x^{*}]\\ &&\!+\frac{1}{4} (-2 + 3\sqrt{2})exp(-\sqrt{3} k_{1} x^{*})\cos[3 \omega t-\sqrt{3} k_{1} x^{*}]\\ &&\!+\frac{11}{5}exp(-3 k_{1} x^{*})\cos[\omega t- k_{1} x^{*}]\\ &&\!-\frac{6}{5}exp(-k_{1} x^{*})\cos[\omega t-k_{1} x^{*}]\\ &&\!-exp[-(1+\sqrt{2}) k_{1} x^{*}] \cos[\omega t+(1-\sqrt{2}) k_{1} x^{*}]\\ &&\!+\frac{2}{5}exp(-3 k_{1} x^{*})\sin[\omega t-k_{1} x^{*}]\\ &&\!-\frac{1}{20}(-8 + 5\sqrt{2})exp(-k_{1} x^{*})\sin[\omega t-k_{1} x^{*}]\\ &&\!+\frac{1}{2\sqrt{2}}exp(-(1+\sqrt{2}) k_{1} x^{*})\\&&\left.\sin[\omega t+(1-\sqrt{2}) k_{1} x^{*}]\vphantom{\frac{1}{2\sqrt{2}}}\right\}. \end{array} $$

(56)