A new approach to the Benjamin-Bona-Mahony equation via ultraspherical wavelets collocation method

Linear and nonlinear phenomena are essential in physics, applied mathematics, and engineering problems. Optical fibers, fluid mechanics, plasma physics, electricity, fluid dynamics, wave circulation phenomena, elastic media, movement of heat, and other fields are related to wave phenomena. PDEs, such as Advection, Burgers, Boussinesq, Fisher equations, and many others, are of endless interest in nonlinear optics and quantum mechanics. Occasionally, in addition to helping authors fully understand the process they are describing, solving nonlinear equations enlightens them on facts that are not readily apparent through everyday observation. Additionally, finding precise answers to these problems is somewhat unexplored. However, numerical analysis [1] has become increasingly popular for linear and nonlinear partial differential equations, including BBM equations, in recent years. We look into two physical scenarios with an initial-boundary value problem of the BBM equations type. In 1972, Benjamin, Bona, and Mahony [2] enhanced the equation has the following form: (1) yt(x,t)+yx(x,t)+y(x,t)yx(x,t)−yxxt(x,t)=0, {y_t}(x,t) + {y_x}(x,t) + y(x,t){y_x}(x,t) - {y_{xxt}}(x,t) = 0, with different constraints, y(x,0)=f(x), 0≤x≤1,y(0,t)=g(t), y(1,t)=h(t), t≥0. \matrix{ {y(x,0) = f(x),\quad 0 \le x \le 1,} \cr {y(0,t) = g(t),\;y(1,t) = h(t),\quad t \ge 0.} \cr } The BBM equation, an alternate smoothness model for the KdV equation [3], is given in equation (1).

A lot of mathematicians have also recently introduced several new techniques for solving differential equations, such as the Hermite wavelet technique [4], the Bäcklund transformation method [5], the Fibonacci wavelet scheme for solving hyperbolic PDE, dispersive PDE, the Rosenau-Hyman equation [6,7,8], the first integral method for MBBM equation [9], Hirota's bilinear method [10], Bernoulli wavelet scheme for nonlinear Murray equation [11], BBMB equations by Exp-Function method [12], (2+1) dimensional Sobolev equation via wavelet technique [13], the Haar wavelet method for the BBM equations [14], ultraspherical wavelet scheme for spectral solutions of Riccati equations [15], ultraspherical wavelet technique for solving 2nth-order boundary value problems [16], ultraspherical operational matrices of derivatives [17], clique polynomial and Adomian decomposition method for solving differential equations [18], and Laguerre wavelets scheme for solving delay differential equations [19], Bernoulli wavelet technique for solving biological models [20], explicit solution of atmosphere-soil-land plant carbon cycle system [21], study on Kudryashov-Sinelshchikov dynamical equation [22], study on Caudrey-Dodd-Gibbon-Sawada-Kotera partial differential equation [23], structure of the analytic solutions for Schrödinger equation [24], solutions for Konopelchenko-Dubrovsky equation [25], solutions of Kadomtsev-Petviashvili-Benjamin-Bona-Mahony equation [26], solutions of the Korteweg-de Vries-Zakharov-Kuznetsov equation [27].

Wavelets are utilized in numerous locations in numerical analysis as ideal instruments to provide a proper mathematical approach to scientific phenomena, often modelled by linear or nonlinear differential equations. The ultraspherical wavelets approach is one of such methods due to its unique properties called orthogonality, which compactly supports and holds multiresolution analysis. Also, these wavelets are continuous in respective domains; hence they play a vital role in applied mathematics. Our literature survey found only a few work-related to ultraspherical wavelets led by Youssri et al. [15,16,17]. In literature, no one has solved BBM equations by ultraspherical wavelets; this impels us to propose a new technique called the ultraspherical wavelets method (UWM).

The main goal of this paper is to describe and explain a novel numerical method UWM for obtaining a numerical solution to the nonlinear PDEs. The results are compared with wavelet and non-wavelet techniques [14]. According to the current literature survey, UWM has not been used to solve BBM equations.

The structure of this article is organized as follows. Section 2 is dedicated to the properties of ultraspherical wavelets. Section 3 gives the projected technique's convergence analysis. Section 4 is dedicated to the ultraspherical wavelets method. The numerical experiment, outcomes, and error analysis of the examples are covered in Section 5. In Section 6, the suggested work's conclusion is covered.

The family of functions known as wavelets is descended from a single function called the mother wavelet through dilatation and translation. When the translation parameter b and the dilation parameter a are continuously variable, we have a family of continuous wavelets as ψa,b(x)=|a|−12ψx−ba, a,b∈ℝ, a≠0. {\psi _{a,b}}(x) = |a{|^{{{ - 1} \over 2}}}\psi \left( {{{x - b} \over a}} \right),\;a,b \in {\mathbb {R}},\;a \ne 0. If the parameters a and b are restricted to discrete values as a=a0−k, b=nb0a0−k, a0>1, b0>0, a = a_0^{ - k},\;b = n{b_0}a_0^{ - k},\;{a_0} > 1,\;{b_0} > 0, the discrete wavelets family that we have is as follows ψk,n(x)=|a0|12ψ(a0kx−nb0). {\psi _{k,n}}(x) = |{a_0}{|^{{1 \over 2}}}\psi (a_0^kx - n{b_0}). Where L²(R)'s wavelet basis is formed by ψ_k,n. The orthonormal basis ψ_k,n(x) is particularly formed when a₀ = 2 and b₀ = 1. The ultraspherical wavelets ψn,mα(x)=ψ(k,n,m,α,x) \psi _{n,m}^\alpha (x) = \psi (k,n,m,\alpha ,x) involve five arguments where k is supposed to be any positive integer, n = 1,2,3,⋯,2^k−1, m be the degree of the ultraspherical polynomials, x be the normalized time, and α is the known ultraspherical parameter. The ultraspherical wavelets be defined on the interval [0,1] as in [16], (2) ψn,mα(x)=2k+12μm,α cmα(2k+1x−2n−1),n2k≤x<n+12k,0,otherwise, \psi _{n,m}^\alpha (x) = \left\{ {\matrix{ {{2^{{{k + 1} \over 2}}}{\mu _{m,\alpha }}\;c_m^\alpha ({2^{k + 1}}x - 2n - 1),} \hfill & {{n \over {{2^k}}} \le x < {{n + 1} \over {{2^k}}},} \hfill \cr {0,} \hfill & {{\rm{otherwise}},} \hfill \cr } } \right. where, μm,α=2αΓ(α)m!(m+α)2πΓ(m+2α) {\mu _{m,\alpha }} = {2^\alpha }\Gamma (\alpha )\sqrt {{{m!(m + \alpha )} \over {2\pi \Gamma (m + 2\alpha )}}} and m = 0,1,2,⋯,M − 1. Here cmα c_m^\alpha is the ultraspherical polynomial having degree m and satisfies the recursive formula as follows, c0α=1 c_0^\alpha = 1 and c1α=x c_1^\alpha = x , [17], cm+1α(x)=2(m+α)xcmα(x)−mcm−1α(x)m+2α, m=1,2,3,⋯. c_{m + 1}^\alpha (x) = {{2(m + \alpha )xc_m^\alpha (x) - mc_{m - 1}^\alpha (x)} \over {m + 2\alpha }},\quad \quad m = 1,2,3, \cdots .

The generalized BBM equation of the form (3) αyt(x,t)+βyx(x,t)+χy(x,t)yx(x,t)−δyxxt(x,t)=μ(x,t), \alpha {y_t}(x,t) + \beta {y_x}(x,t) + \chi y(x,t){y_x}(x,t) - \delta {y_{xxt}}(x,t) = \mu (x,t), with initial-boundary conditions, y(x,0)=f(x), 0≤x≤1, y(x,0) = f(x),\quad \quad 0 \le x \le 1, and y(0,t)=g0(t), y(1,t)=g1(t), t≥0, y(0,t) = {g_0}(t),\quad \quad y(1,t) = {g_1}(t),\quad \quad t \ge 0, where α,β,δ,ξ are constants, and f (x),g₀(t),g₁(t),μ(x,t) are continuous real functions.

Consider, (4) yxxt(x,t)=∑n=1∞∑m=0∞an,mψn,m(x), {y_{xxt}}(x,t) = \sum\limits_{n = 1}^\infty \sum\limits_{m = 0}^\infty {a_{n,m}}{\psi _{n,m}}(x), truncating the (4), we get (5) yxxt(x,t)≈∑n=12k−1∑m=0M−1an,mψn,m(x), {y_{xxt}}(x,t) \approx \sum\limits_{n = 1}^{{2^{k - 1}}} \sum\limits_{m = 0}^{M - 1} {a_{n,m}}{\psi _{n,m}}(x), where ultraspherical coefficients a_n,m have to be determined. Integrate (5) with respective t from 0 to t. (6) yxx(x,t)=yxx(x,0)+t∑n=12k−1∑m=0M−1an,mψn,m(x). {y_{xx}}(x,t) = {y_{xx}}(x,0) + t\sum\limits_{n = 1}^{{2^{k - 1}}} \sum\limits_{m = 0}^{M - 1} {a_{n,m}}{\psi _{n,m}}(x). Now integrate (6) concerning x from 0 to x. (7) yx(x,t)=yx(x,0)+yx(x,0)−yx(0,0)+∫0xt∑n=12k−1∑m=0M−1an,mψn,m(x). {y_x}(x,t) = {y_x}(x,0) + {y_x}(x,0) - {y_x}(0,0) + \int_0^x t\sum\limits_{n = 1}^{{2^{k - 1}}} \sum\limits_{m = 0}^{M - 1} {a_{n,m}}{\psi _{n,m}}(x). Now integrate (7) concerning x from 0 to x. (8) y(x,t)=y(0,t)+x(yx(0,t)−yx(0,0))+y(x,0)−y(0,0)+t∑n=12k−1∑m=0M−1an,m∫0x∫0xψn,m(x). y(x,t) = y(0,t) + x({y_x}(0,t) - {y_x}(0,0)) + y(x,0) - y(0,0) + t\sum\limits_{n = 1}^{{2^{k - 1}}} \sum\limits_{m = 0}^{M - 1} {a_{n,m}}\int_0^x \int_0^x {\psi _{n,m}}(x). Put x = 1 in (8), and by given conditions, we get (9) yx(0,t)−yx(0,0)=g1(t)−g0(t)+f(0)−f(1)−t∑n=12k−1∑m=0M−1an,m∫0x∫0xψn,m(x)dxdx|x=1. {y_x}(0,t) - {y_x}(0,0) = {g_1}(t) - {g_0}(t) + f(0) - f(1) - t\left( {\sum\limits_{n = 1}^{{2^{k - 1}}} \sum\limits_{m = 0}^{M - 1} {a_{n,m}}\int_0^x \int_0^x {\psi _{n,m}}(x)dxdx} \right){|_{x = 1}}. Substitute (9) in (7) and (8), we get (10) yx(x,t)=yx(x,0)+g1(t)−g0(t)+f(0)−f(1)−t∑n=12k−1∑m=0M−1an,m∫0x∫0xψn,m(x)dxdx|x=1+t∑n=12k−1∑m=0M−1an,m∫0xψn,m(x)dx, \matrix{ \hfill {{y_x}(x,t) = {y_x}(x,0) + \left[ {{g_1}(t) - {g_0}(t) + f(0) - f(1) - t\left( {\sum\limits_{n = 1}^{{2^{k - 1}}} \sum\limits_{m = 0}^{M - 1} {a_{n,m}}\int_0^x \int_0^x {\psi _{n,m}}(x)dxdx} \right){|_{x = 1}}} \right]} \cr \hfill { + t\sum\limits_{n = 1}^{{2^{k - 1}}} \sum\limits_{m = 0}^{M - 1} {a_{n,m}}\int_0^x {\psi _{n,m}}(x)dx,} \cr } and (11) y(x,t)=y(0,t)+xg1(t)−g0(t)+f(0)−f(1)−t∑n=12k−1∑m=0M−1an,m∫0x∫0xψn,m(x)dxdx|x=1+y(x,0)−y(0,0)+t∑n=12k−1∑m=0M−1an,m∫0x∫0xψn,m(x)dxdx. \matrix{ \hfill {y(x,t) = y(0,t) + x\left[ {{g_1}(t) - {g_0}(t) + f(0) - f(1) - t\left( {\sum\limits_{n = 1}^{{2^{k - 1}}} \sum\limits_{m = 0}^{M - 1} {a_{n,m}}\int_0^x \int_0^x {\psi _{n,m}}(x)dxdx} \right){|_{x = 1}}} \right]} \cr \hfill { + y(x,0) - y(0,0) + t\sum\limits_{n = 1}^{{2^{k - 1}}} \sum\limits_{m = 0}^{M - 1} {a_{n,m}}\int_0^x \int_0^x {\psi _{n,m}}(x)dxdx.} \cr } Now differentiate (11) concerning t, and we get (12) yt(x,t)=yt(0,t)+xg1(t)−g0(t)−∑n=12k−1∑m=0M−1an,m∫0x∫0xψn,m(x)dxdx|x=1+∑n=12k−1∑m=0M−1an,m∫0x∫0xψn,m(x)dxdx. \matrix{ \hfill {{y_t}(x,t) = {y_t}(0,t) + x\left[ {{g_1}(t) - {g_0}(t) - \left( {\sum\limits_{n = 1}^{{2^{k - 1}}} \sum\limits_{m = 0}^{M - 1} {a_{n,m}}\int_0^x \int_0^x {\psi _{n,m}}(x)dxdx} \right){|_{x = 1}}} \right]} \cr \hfill { + \sum\limits_{n = 1}^{{2^{k - 1}}} \sum\limits_{m = 0}^{M - 1} {a_{n,m}}\int_0^x \int_0^x {\psi _{n,m}}(x)dxdx.} \cr } Substituting (5), (10), (11), and (12) in (3) by assuming t=1M t = {1 \over M} and using the collocation points xi=2i−12M {x_i} = {{2i - 1} \over {2M}} where i = 1,2,⋯,M the obtained equation (3) can be converted into a system of algebraic equations. The system is solved by Newton's iterative method. We get the ultraspherical coefficients a_n,m, then subsitute these an,m's a_{n,m}^\prime s in (11) will contribute to the ultraspherical wavelets-based approximate solution of a chosen equation. AE = |y_e(x,t) − y_a(x,t)|, where y_a(x,t) and y_e(x,t) approximate and exact solutions will determine the absolute error (AE).

We adapt the UWM mentioned in section 4 to various BBM equations in this part.

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The present study used the ultraspherical wavelet method to find the numerical solution of BBM equations. The proposed approach is easily computer implementable and supported with illustrative examples. We also observed that this method is computationally efficient and gives better results. The present approach was compared with Haar wavelet and finite difference methods through tables and graphs. We have drawn 1-D and 2-D graphs of UWM, HWM, and FDM. From the figures and tables, we observed that the present technique has more accuracy than HWM and FDM, which are available in literature [14]. By raising the M, the precision of the solution is improved. As a result, the suggested approach is highly efficient and straightforward for solving linear and nonlinear BBM equations. Additionally, this approach can be extended to higher-order equations by making minor adjustments to the suggested procedure.