An Integral Equation Formalism for Integrating a Nonlinear Initial-Boundary Value Problem for a Boussinesq Equation

In this paper, a new nonlinear initial-boundary value problem for a Boussinesq equation is formulated. And a coupled system of nonlinear integral equations, equivalent to the new initial-boundary value problem, is constructed for integrating the initialboundary value problem, but which is inherently different from other conventional formulations for integral equations. For the numerical solutions, successive approximations are applied, which leads to a functional iterative formula. A propagating solitary wave is simulated via iterating the formula, which is in good agreement with the known exact solution.


Introduction
We begin with formulating a nonlinear initial-boundary value problem for the nonlinear partial differential equation (PDE) [1]: in which constant h 0 denotes water depth and constant c 0 characteristic velocity, i.e., with gravitation acceleration denoted by g. e nonlinear PDE (1) describes evolution of nonlinear dispersive waves in shallow water propagating in a uniform channel of water depth h 0 under gravitation g, where u in (1) represents depth-average velocity in horizontal direction x. In fact, the model equation (1) is chosen for mathematical simplicity, as a starting point in this paper, which is the time evolution equation of the lowest order mathematical model for nonlinear and dispersive long wave; however, the physics represented by (1) is clearly included in higher order Boussinesq's models [2]. Generally, the differential equations of the Boussinesq type such as (1) indicate essential model equations in physics and applied mathematics, the time evolution of which is an issue of scientific as well as engineering importance. Next, we would like to solve the nonlinear initialboundary value problem formulated above through the construction of an (equivalent) integral equation formalism, a primary purpose of this paper. To this end, we first build an auxiliary (initial-boundary value) problem, associated with the (original) initial-boundary value problem. ereby, it is possible to build (regular) coupled nonlinear integral equations of second kind, being equivalent to the present initial-boundary value problem. Applying the method of successive approximation to the integral equations, we arrive at a functional iterative formula. It is a semianalytic and derivative-free recurrence relation, which generates a functional sequence of improving approximate solutions of the initial-boundary value problem. It is emphasized that the only thing needed for iterating the recurrence is simply the usual numerical regular (not singular) integration in each step of iteration. By iterating the recurrence, we simulate, for example, a moving solitary wave, which agrees with the analytical solution well.
In referring to the Boussinesq type equations such as (1), a considerable number of studies has been conducted over the past few decades, but most of them are limited to just their "initial" value problems (e.g., [3]), while the present study concerns the initial-"boundary" value problem as discussed at the beginning of this section. To name a few, El-Zoheiry [4] performed a numerical study of the classical Boussinesq equation using a three-level iterative scheme based on the compact implicit method, where a solitary wave solution of the equation was examined for the accuracy and efficiency. Wang et al. [5] analyzed the improved Boussinesq equation using an energy-preserving finite volume element method. Finally, in a recent year, Jang [6] investigated the classical Boussinesq equation by proposing a new dispersion-relation preserving (DRP) method and further improved the convergence characteristic of the DRP method for integrating the Boussinesq equation [7].

Initial-Boundary Value Problem and Auxiliary Problem
We consider the nonlinear PDE of fourth order (1), as discussed in the preceding section, but on the half interval R + � (0, ∞), with the dispersion relation [6,7] between wave frequency ω Β and wave number k: For PDE (1), we shall formulate a (initial-boundary value) problem subject to initial conditions: together with boundary condition at x � 0 + : and at (positive) infinity, u, u x ⟶ 0, as x ⟶ ∞.
Associated with the initial-boundary value problem formulated above, we will next build an auxiliary problem. Definition 1. As a counterpart of the original initialboundary problem of (1) with (4)-(6), let an auxiliary initialboundary value problem for u Aux be defined as subject to initial conditions at t � 0: as well as, boundary conditions at x � 0 − : and at negative infinity: Remark 1. e auxiliary initial-boundary value problem in Definition 1, defined on the half interval R − � (− ∞, 0), is in fact constructed in such a way that e original and auxiliary problems may be combined to form a new problem through the next definition.
Definition 2. u Ext is a function from R × [0, T] to R for T ∈ R + satisfying a new (extended) initial-boundary value problem, being governed by the same Boussinesq PDE (1) but in a weak (derivative) sense: jump discontinuity at x � 0 and localized conditions Remark 2. For x > 0, the new initial-boundary value problem for u Ext in Definition 2 reduces to the original initial-boundary value problem for u; while, for x < 0, the problem for u Ext remains the same as the auxiliary problem for u Aux .
e above remarks immediately give the following lemma. Lemma 1. u Ext in Definition 2 is the odd extension of u with respect to x in the original problem of (1) with (4)- (6); thus, u Ext is antisymmetric with respect to x � 0 (See Figure 1).
due to Remarks 1 and 2. is completes the proof.

□
It would be instructive to note that, as indicated in Figure 1, the original initial-boundary value problem for u(x > 0) corresponds to a physical problem, where a wavemaker, located at x � 0, generates a propagating (velocity) wave with specified wavemaker-velocity U(t).

Integral Formalism for u Ext
is section is devoted to the construction of an integral formulation which is equivalent to the extended initialboundary value problem (appearing in Definition 2) in the previous section.

Pseudodissipative Differential Formulation.
We start by introducing two pseudoparameters of α, β > 0 in (12), adding the sum α · (u Ext ) t + β · u Ext to both sides of (12) and arranging, which leaves us with and a forcing term φ with the functional form Notice that the pair of (17) and (18) is a pseudodissipative differential formulation, but which is mathematically still equivalent to the (extended) Boussinesq equation (12); both the parameters, α, β > 0, are directly related to a pseudodissipation (which will be discussed later). ereby, we would like to build an equivalent (nonlinear) pseudodissipative integral formalism in Section 3.5. We begin with integral transforms for (17), which are explained in the next section.

Image of u Ext under Laplace-Fourier Transform.
We first discover the image of Laplace transform L of u Ext in (17), with respect to time t, for a (Laplace transform) parameter s. Here, we assume that u Ext is Laplace transformable.

Lemma 2.
Given two positive pseudoparameters, α, β > 0, PDE (17) for u Ext can be converted into the following ordinary differential equation (ODE) for u * Ext : for a parameter s.
Proof. We remind the differentiation properties of Laplace transform due to the null initial conditions (13). Applying Laplace transform to (17), combined with the properties of (21) and (22), leads to (20), which proves the lemma.
for a parameter k and its inverse by F − 1 , i.e., in which u * Ext (x, s) and u * Ext (k, s) are assumed to be Fourier transformable and inverse Fourier transformable, respectively. en, applying Fourier transform F to ODE (20), we find that which becomes an algebraic equation for u * Ext because of the following lemma.

Lemma 3.
e boundary velocity U in (5) appears in Fourier transform of the second derivative, d 2 u * Ext /dx 2 , as follows: where U * stands for Laplace transform of (5), i.e., Proof. At first, we consider, where integration by parts is used with (14). Similarly, we further have Here, note that because of the property of odd extension by Lemma 1, i.e., and the chain rule from (16). Plugging (28) into (29) gives which completes the proof.

□
With the use of Lemma 3, (25) is rewritten as which can be solved for u * Ext , i.e., Even though the above expression presents the (desired) exact result for the image of u Ext under the Laplace-Fourier transform, however, the content seems to be hard to see. To overcome the difficulty, the next section discusses an appropriate arrangement of (35) suitable for finding the image of u Ext under Laplace-Fourier transform.

Pseudodissipative Parameter and Frequency.
is section gives an appropriate decomposition of the algebraic expression for u * Ext in (35) derived in the previous section, being realized by introducing a new concept of pseudodissipation. e new concept will make it more efficient (i.e., concise and clear) to analyze the problem both physically as well as mathematically. We need some definitions.
Definition 3 (see [6]). Given β > 0, we define a frequency function ω B : R ⟶ R + such that If we use ω B in the above Definition, the denominator of the second fraction in the right hand of (35) arranges to and (35) may be rewritten as by substituting (37) into (35).
Definition 4. ζ is defined as a (dimensionless) pseudodissipative parameter (dependent of frequency) whose absolute value is less than unity, relying on α, β > 0: is a function ω pD B : R ⟶ R + , termed as a pseudodissipative frequency, such that where ω B denotes the frequency function in Definition 3. e next lemma is immediately followed by the above definitions.

Lemma 4. Let A and Β be denoted by
respectively. en, the expression of u * Ext in (38) may be shortened to a simpler form via ζ and ω pD B of Definitions 4 and 5, respectively, i.e., Proof. Writing (38) in terms of A and B in the above as follows: Mathematical Problems in Engineering we are led to result (42) due to Definitions 4 and 5. is completes the proof.
Ext ] � u Ext ) based on the results in the preceding section and the (well known) identities (from mathematical  table): e first fraction involving A in the right hand of (42) can be expressed as A Proof. It is noted that where (44) is used. is immediately proves the lemma. □ Lemma 6. e second fraction involving B in the right hand of (42) decomposes into Proof. Letting F denote the second fraction involving B, we have Here, we note that from (40) and (44). And, (50) can be combined to (51), yielding 6 Mathematical Problems in Engineering (52) erefore, substituting (49) and (52) into F in (48) leads to (47), completing the proof.

□
It will be convenient, if we denote by f i , i � 1, 2, 3, 4, so that u * Ext in (42) can be simply represented as the sum of f i 's from Lemmas 5 and 6, i.e., Laplace transforms are calculated as Proof. To prove (58), note that L − 1 (f 1 ) has the explicit expression: where the notation ∘ denotes Laplace convolution and L − 1 L is the identity. is completes the proof of (58). □ e inverse Laplace transform of f 2 is written as (simply through B in (41)) by linearity of L − 1 , which proves (59).

□
Similarly as above, which completes the proof of (60). □ is finishes proving (61). □ We are now in a position to take the inverse Laplace-Fourier transform of u * Ext in (42). Mathematical Problems in Engineering Proof. To prove (66), we take the inverse Fourier transform for (58), leaving Here, we note that the F − 1 L − 1 (f 1 ) remains real because the real parts of the integrands in the last two integrals in the above are even in k, whereas the imaginary parts are odd. We are thus led to (66), completing the proof.
□ For (67), we have Mathematical Problems in Engineering 9 where we have used the fact that the imaginary part of the integrand in the above is odd in k while the real part is even. An analytic expression is obtained for the above integral, that is, by noting from (mathematical integral) table that is proves (67). □ For the proof of (68), we calculate F − 1 L − 1 (f 3 ), being written as e above integral becomes real, due to the fact that the real part of the integrand is even in k, on the other hand, the imaginary part is odd. is leads to (68), completing the proof. □ To show that (69) is true, recall the Fourier inverse (24) together with (61), i.e., where the imaginary part in the above vanishes because the imaginary part of the integrand (in the second line in the above) is odd in k, but the real part of the integrand is even. □

Integral Formulation Involving u Ext .
Recalling that the u Ext in Definition 2 is recovered formally through the identities because of (57), we readily arrive at the following theorem from the concerning lemmas without proof, but with a short note that the sum of the last line of (66) and (69) becomes where (40) is utilized.

Theorem 1. Let u Ext be satisfied by the Boussinesq equation (12) subject to (13)-(15); i.e., u Ext is a solution for the (extended) initial-boundary value problem in Definition 2. en, the following integral relation holds
which is considered as an (basic) integral formalism for u Ext because φ involves u Ext via (18).

Integral Equation for u
In this section, we will derive a nonlinear integral equation for u, being equivalent to the original initial-boundary value problem of (1) with (4)- (6). e derivation of the integral equation is accomplished directly by the use of the integral formalism for u Ext established in the preceding section.

Preliminaries.
Here, we need to focus on examining the triple integral, denoted by I, in (78) after substituting (18), i.e., (80) e integration domain of the above integral I can be reduced by recalling that u Ext is the odd extension of u with respect to x as was pointed out in Lemma 1, making the I depend upon u(instead of u Ext ).

Lemma 9. e integral I in (79) can be regarded as a functional of u(instead of u Ext ) through the identity
where Proof. We write I in (79) as the sum of two integrals:

Mathematical Problems in Engineering
based on the fact that u Ext is the odd extension of u with respect to x by Lemma 1.
Next, let us take a look at the first integral, denoted by J, in the above and use the change of variables σ � − ξ. We then have where we utilize the chain rule and the fact that σ is a dummy variable. Substituting (84) into (83) yields with a note on is completes the proof.

Constructing Integral Equation.
Using the result presented in the preceding section, we finally arrive at a nonlinear integral equation identical with the original initial-boundary value problem, i.e., the Boussinesq equation (1) subject to (4)- (6). We start with integration by parts.
Lemma 10. Given a kernel K in (82), the following integral, denoted by I 1 , is calculated as Proof. By integration by parts with respect to spatial variable x, e above is shortened to because the far field condition (6) implies that u vanishes as ξ ⟶ ∞ and K(x, t − τ, k; ξ � 0) � 0 by (82). Next, we interchange the order of integration in (90) due to Fubini's theorem as en, we apply integration by parts with respect to time, leading to because u vanishes at t � 0 from initial condition (4), is completes the proof. □ Lemma 11. e following triple integral, termed I 2 , involving u t is evaluated as in which the kernel K is given in (82).
Proof. We begin by rewriting the triple integral I 2 as due to Fubini's theorem, where integration by parts with respect to time is used with interchange of the order of Mathematical Problems in Engineering integration. Here, notice that u equals zero when t � 0 from initial condition (4) and K(x, t − t, k; ξ) � 0 from (82). On the other hand, the differentiation of (82) with respect to τ gives (96) is results in as required, which completes the proof.
□ Remark 3. Noting that the integral I in (81) decomposes into the sum from (88) and (94), (81) may arrange to by Lemmas 10 and 11. e following theorem is immediately directed from eorem 1 plus Remarks 2 and 3.
Proof. We first note that the last two integral terms in (100) vanish, as α, β ⟶ 0. It thus suffices to show that in the first integral in (100), which is immediate because by (36) and (39). And then, the integrand in the fourth term in (100) has an asymptotic form: as α, β ⟶ 0. is completes the proof. □ Remark 4. e kernels in integral equation (100) have singularities of (simple) poles at All the singularities become off the real line R with a proper choice of α, β > 0, which tend further to as α, β ⟶ 0.

Application of Integral Formalism
In this section, we shall apply successive approximations to integral equation (100) constructed in the previous section. is produces an iterative strategy, whereby we investigate iterative numerical solutions of a solitary wave. Furthermore, we will examine how the two pseudoparameters α, β introduced artificially in this paper effect the convergence of the iterative solutions. For this reason, the two introduced parameters can be viewed as control ones for the iterative strategy.

Successive Approximations.
Numerical solutions for the original initial-boundary value problem of the Boussinesq equation (1): subject to (4)-(6) may be found by applying successive approximations of the Banach fixed point theorem to (100). e theorem guarantees the existence and uniqueness of fixed points of certain self-operators of metric spaces, providing a constructive method, namely, the successive approximations, to find those fixed points [6][7][8][9][10][11]; in the present study, we choose the same space as the usual metric space [6][7][8][9][10][11] for the use of the theorem.
We bear in mind that (100) can be thought of a fixed form and we will seek its fixed point (solution) u through a recurrence relation with the pseudoparameters α, β for n � 0, 1, 2, . . ., as expressed as with (zero) initial guess 0.2 m and g � 9.80665 m/s 2 . e space and time domains are selected as 0 < x < 25 m (or 0 < x ′ < 216.51) and 0 < t < 18 sec (or 0 < t ′ < 218.31), respectively, with the increments, Δx � 0.10 m and Δt � 0.05 sec; we choose x 0 � − 12 m(or x 0 ′ � − 103.92). Figure 2 shows typical convergence behaviors for the iterative wave solutions with two amplitudes, i.e., 0.00115 m (or a ′ � 0.01) and 0.00231 m (or a ′ � 0.02), where two pseudoparameters are taken as α � 0.0700 rad/sec and β � 0.0490 rad/sec 2 , which are (alternatively) nondimensionalized as α ′ � 10 − 2 and β ′ � 10 − 3 , respectively, according to the (dimensionless) scaling e first iteration gives a numerical wave solution valid for the initial stage of time evolution and an almost converged solution seems to be reached at n � 10, which clearly shows a translating (moving) solitary wave.
We next inspect quantitative errors for the iterative wave solutions. Figure 3 plots the nature of convergence of (108) for various β ′ 's, when α ′ is specified as α ′ � 10 − 2 , 10 − 1 (which corresponds to α � 0.0700, 0.7002[rad/sec], respectively). Here, errors are estimated by L ∞ : where L ∞ represents L-infinity norm taken over by the space variable x only with reference time t ref being kept fixed; t ref � 18 sec (i.e., end time). We see that the L ∞ decreases as the number of iterations increases. For α ′ � 10 − 2 and 10 − 1 , convergence characteristics depend on the pseudoparameter β ′ ; to be specific, the smaller β ′ would contribute to a higher convergence rate, with a specified α ′ . is tendency is similar to that of Jang [6]. Finally, Table 1 tabulates the numerical values of errors plotted in Figure 3.

Discussions and Concluding Remarks
is work involves an initial value problem for the Boussinesq equation (1) for depth-average velocity together with an additional constraint of boundary condition, namely, an initial-boundary value problem for (1). In this paper, we have considered a nontrivial boundary condition, while being in contrast with other studies, where the null (or trivial) boundary condition has been usually imposed [6,14,15]. Physically, the nontrivial boundary condition considered here may be related with the wavemaker problem in fluid mechanics [16].
One of the main purposes of this paper is to construct an integral equation formalism, equivalent to the initial value problem for (1). Here, the derived formalism is required to be regular and different from the usual existing integral equation ones (e.g., see [17]). For the construction, as a counterpart of the original initial-boundary value problem (defined on the positive real axis), we have first built an auxiliary initial-boundary value problem (defined on the negative real axis). We also have introduced two pseudoparameters on artificial dissipation to formulate a pseudodissipative formulation for the initial-boundary value problem for (1) [6]. is results in formulating a two-parameter family of the regular nonlinear integral equations, which is different from conventional integral-equation formalisms.  By the method of successive approximations, the regular integral equations derived above have immediately led to an iterative formula for solving the initial-boundary value problem. Further, the formula has enabled us to simulate the propagating solitary wave as illustrated in the numerical experiment. e (numerically) simulated results are compared to the exact solution and the agreement is shown to be excellent.