# An Efficient Algorithm for Solving Hyperbolic Partial Differential Equations with a Nonlocal Conservation Condition

Authors: H. Bin Jebreen
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this paper, a numerical scheme based on the Galerkin method is extended for solving one-dimensional hyperbolic partial differential equations with a nonlocal conservation condition. To achieve this goal, we apply the interpolating scaling functions. The most important advantages of these bases are orthonormality, interpolation, and having flexible vanishing moments. In other words, to increase the accuracy of the approximation, we can individually or simultaneously increase both the degree of polynomials (multiplicityr) and the level of refinement (refinement level J). The convergence analysis is investigated, and numerical examples guarantee it. To show the ability of the proposed method, we compare it with existing methods, and it can be confirmed that our results are better than them.


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## Body

## 1. Introduction

The present paper is organized to find the approximate solution of the one-dimensional hyperbolic partial differential equations (HPDEs) with a nonlocal conservation condition [1].(1)wttx,t−wxxx,t=px,t,x∈0,1,0≤t≤T,with initial conditions(2)wx,0=qx,x∈0,1,wtx,0=gx,x∈0,1,Dirichlet boundary condition(3)w0,t=ht,0≤t≤T,and also the integral condition(4)∫01wz,tdx=kt,0≤t≤T.Here,q,p,g,h, and k are known functions. Without loss of generality, assume that T=1.Mathematical modeling of many physical phenomena leads to HPDEs with boundary and initial conditions and also integral condition replacing the standard boundary condition [2]. This problem appears when studying the dynamics of groundwater, plasma physics, thermoelasticity, and heat conduction [3–5]. The existence and uniqueness of a generalized solution of such an equation were investigated by Mesloub and Bouziani [6]. In this paper, all the known functions of equation (1) and its conditions are assumed to be such that the problem has a unique solution. Numerous studies proposed a variety of numerical and analytical solutions to such equations. Dehghan and Lakestani [1] introduced a numerical scheme based on the cubic B-spline scaling functions to solve (1). In [1], the operational matrices of the derivative and integral were used to reduce the problem to a system of algebraic equations. Several finite difference procedures based on the three-level implicit and explicit schemes are proposed for solving the problem in [5]. In [3], Ang proposed an integrodifferential formulation to find the approximate solution of (1). Ashyralyev and Aggez [7] utilized the stable difference schemes to solve multidimensional hyperbolic equations with nonlocal boundary value. The collocation method based on Jacobi polynomials and Gauss–Lobatto quadrature integration was used in [8]. Doha et al. applied shifted Jacobi spectral-Galerkin method to solve the problem [9]. An efficient computational method is proposed for solving hyperbolic partial differential equations based on Chebyshev and Legendre wavelets [10]. In [11], Turkyilmazoglu solved parabolic partial differential equations with nonlocal initial and boundary conditions using a fast and accurate analytic method based on double power series.Wavelets and their applications are a very powerful tool for solving a variety of equations such as ordinary and partial differential equations and are especially one of the best methods to solve the integral equations with the ability of them at reducing the problem to a sparse linear system. Meanwhile, the wavelet Galerkin method is used widely for solving equations [2, 12–15]. In this paper, we used interpolating scaling function constructed in [16] following [17], and these bases have been used to solved equations such as ordinary and partial differential equations and also integral equations [18–20]. These bases allow us to have high vanishing moments, compact support, and properties such as orthogonality and interpolation [16].This paper is organized as follows. Section2 is devoted to the brief introduction to the interpolating scaling function. In Section 3, Galerkin method based on interpolating scaling functions is used to solve the desired equation, and also, the convergence analysis is investigated. Section 4 is devoted to some numerical examples to show the ability and accuracy of the method.


## 2. Interpolating Scaling Functions

We introduce the subspaceVJr as a space of piecewise polynomial bases of degree less than multiplicity parameter r≥0 that is spanned by(5)VJr≔Spanϕj,bk≔ϕk2jx−k,b∈ℬj,k∈ℛ⊂L2Ω,r≥0,j∈ℤ+∪0,where two sets ℬ and ℛ are defined by(6)ℬ≔0,…,2J−1,ℛ≔0,1,…,r−1,and ϕkk∈ℛ are the primal interpolating scaling bases introduced by Alpert et al. [17]. Given nodes τkk∈ℛ, which are the roots of the Legendre polynomial of degree r, the interpolating scaling bases are defined as(7)ϕkx=2ωkLk2x−1,x∈0,1,0,o.w,where Lktk∈ℛ are the Lagrange interpolating polynomials at the point τkk∈ℛ and ωkk∈ℛ are the Gauss–Legendre quadrature weights [2, 17]. These bases form orthonormal bases on Ω with respect to the L2-inner product.LetJ∈ℤ+∪0. We consider the uniform finite discretizations Ω≔0,1=∪b∈ℬXJ,b, where the subintervals XJ,b≔xb,xb+1 are determined by the point xb≔b/2J. Every function p∈L2Ω can be represented in the form(8)p≈PJrp=∑b∈ℬJ∑k∈ℛpJ,bkϕJ,bk,where PJr is the orthogonal projection that maps L2Ω onto the subspace VJr. To find the coefficients pJ,bk that are determined by p,ϕJ,bk=∫XJ,bfxϕJ,bkxdx, we shall compute the integrals. We apply the r-point Gauss–Legendre quadrature by a suitable choice of the weights ωk and nodes τk for k∈ℛ to avoid these integrals [13, 17] via(9)pJ,bk≈2−J/2ωk2p2−Jτk+12+b,b∈ℬJ,k∈ℛ.Convergence analysis of the projectionPJrp is investigated for the r-times continuously differentiable function p∈ℂrΩ:(10)PJrp−p≤2−Jr24rr!supx∈0,1prx.For the full proof of this approximation and further details, we refer the readers to [16]. Thus, we can conclude that PJrp converges to p with the rate of convergence O2−Jr.Assume that the vector functionΦJr≔Φr,J,0,…,Φr,J,2J−1T with Φr,J,b≔ϕJ,b0,…,ϕJ,br−1 includes the scaling functions and is called the multiscaling function. Approximation (8) may be rewritten using the vector P that includes entries Pbr+k+1≔pJ,bk as follows:(11)PJrp=PTΦJr,where P is an N-dimensional vector (N≔r2J). The building blocks of these bases’ construction can be applied to approximate a higher-dimensional function. To this end, one can introduce the two-dimensional subspace VJr,2≔VJr×VJr⊂L2Ω×Ω that is spanned by(12)ϕJ,bkϕJ,b′k′:b,b′∈ℬJ,k,k′∈ℛ.Thus, by this assumption, to derive an approximation of the functionp∈L2Ω×Ω by the projection operator PJr, we get(13)ps,t≈PJrps,t=ΦJrTsPΦJrt,where components of the square matrix P of order N are obtained by(14)Prb+k+1,rb′+k′+1≈2−Jωk′2ωk2p2−Jτ^k+b,2−Jτ^k′+b′,where τ^k=τk+1/2. Consider the 2r-th partial derivatives of p:Ω×Ω⟶ℝ are continuous. Utilizing this assumption, the error of this approximation can be bounded as follows:(15)PJrp−p≤ℳmax21−rJ4rr!2+21−Jr4rr!,where ℳmax is a constant [13].(16)ℳmax=maxsupξ∈0,1∂r∂xrpξ,y,supη∈0,1∂r∂yrpx,η,supξ′,η′∈0,1∂2r∂xr∂yrpξ′,η′.


## 3. Galerkin Method

To derive Galerkin discretization based on interpolating scaling functions of one-dimensional HPDEs (1), the approximate solution can be written by projection operator PJr, i.e.,(17)wx,t≈PJrwx,t=ΦJrTxWΦJrt,where components of the square matrix W of order N are unknown coefficients that should be found and the superscript T means transpose. Taking the second derivative with respect to x and t from both sides of equation (17), one can write(18)wttx,t≈PJrwttx,t=ΦJrTxWDϕ2ΦJrt,wxxx,t≈PJrwxxx,t=ΦJrTxDϕT2WΦJrt,where Dϕ is the operational matrix of the derivative for interpolating scaling functions that are introduced in [17]. By substituting (18) in (1) and projecting the function p to the space VJr using projection PJr, we obtain(19)ΦJrTxWDϕ2ΦJrt−ΦJrTxDϕT2WΦJrt=ΦJrTxPΦJrt.Now, we introduce the residual function(20)rJr≔ΦJrTxWDϕ2ΦJrt−ΦJrTxDϕT2WΦJrt−ΦJrTxPΦJrt.Note thatPJrv=0 (v∈VJr) if ∀1≤i≤Nv,ΦJri=0. Multiplying (20) by ΨJrTt and ΨJrTx from the left and right, respectively, and integrating over IJ,b, b∈ℬJ, we obtain a linear system as follows:(21)Λ≔WDϕ2−DϕT2W−P=0,where we employ orthonormality of interpolating scaling functions and the local support of these bases. Equation (21) gives N2 equations in which 4N of these equations are linearly dependent. These equations are substituted using the conditions governing the problem. One can find 4N extra equations utilizing the projection PJr for known functions q,g,h, and k as follows:(22)wx,0=qx,⟶ΦJrTxWΦJr0=ΦJrxTQ,wtx,0=gx,⟶ΦJrTxWDϕΦJr0=ΦJrxTG,w0,t=ht,⟶ΦJrT0WΦJrt=HTΦJrt,∫01wx,t=kx,⟶ΦJrT1IϕTWΦJrt=KTΦJrx,where Iϕ is the operational matrix of the integral for interpolating scaling functions introduced in [14]. Because the interpolating scaling functions are linearly independent, we have, from (22), 4N independent equations:(23)Γ1≔WΦJr0−Q=0,Γ2≔WDϕΦJr0−G=0,Γ3≔ΦJrT0W−HT=0,Γ4≔ΦJrT1IϕTW−KT=0.ReplacingΓ1 and Γ2 in the first and second columns of Λ and Γ3 and Γ3 in the first and last rows of Λ, one can obtain the N2 equations with N2 unknowns. To find the unknowns and, equivalently, the approximate solution to the problem, this system must be solved by a convenient method.Theorem 1.
Letf∈CrΩ×0,1. Further suppose that wx,t, the solution of equation (1), is r-times derivative. The Galerkin method presented in this section applied to equation (1) is convergent.Proof.
To prove the Galerkin method convergence presented in this section, subtracting (1) from PJrwtt−wxx−p=0, we obtain(24)eJr=wtt−PJrwtt−wxx−PJrwxx−p+PJrp=eJtt−eJxx−p+PJrp,where eJ≔w−PJrw. Using the triangle inequality and taking L2-norm from both sides yield(25)eJr≤eJtt+eJxx+p−PJrp.
Now, suppose that(26)eJx,t≈PJreJx,t=ΦJrTxEΦJrt,where E is the (N×N) matrix, and thus, one can write(27)eJr≤EDϕ2+Dϕ2TE+p−PJrp≤2EDϕ2+p−PJrp,where we applied the orthonormality of bases. The operational matrix of the derivative for interpolating scaling functions is invertible [17], and its norm is finite. Let Dϕ2=η; then, using (15), we can find(28)eJr≤κ21−rJ4rr!2+21−Jr4rr!,where κ is a constant.


## 4. Numerical Results

To illustrate the validity, efficiency, and accuracy of the proposed method, we give some numerical examples. TheL∞ errors in the given tables and figures are(29)eJ≔wx,t−wJrx,t,where wJrx,t and wx,t are the approximate and exact solutions at the point x,t, respectively.Example 1.
Let us consider hyperbolic partial differential equation (1) with(30)qx≔cosπx,gx≔0,x∈Ω,ht≔cosπt,kt≔0,t∈0,1,and also px,t≔0. The exact solution of this problem is given in [1] as(31)wx,t≔12cosπx+t+cosπx−t.
Table1 shows the comparison of our results with those obtained by Dehghan and Lakestani [1]. The result confirms that the presented method is very convenient and effective. Taking r=5 and J=4, the approximate solution and L∞ error are shown in Figure 1. The effects of the multiplicity parameter r and refinement level J on the L∞ error are shown in Figure 2.Table 1
Comparison of theL∞ error for Example 1.
tPresented methodDehghan and Lakestani [1]J=2J=3J=4J=2J=3J=40.28.6e−44.2e−61.6e−79.4e−31.9e−42.4e−50.41.0e−31.1e−63.4e−86.0e−39.3e−51.1e−50.68.4e−45.6e−74.2e−87.5e−39.3e−51.1e−50.84.3e−43.9e−61.0e−77.7e−31.9e−42.4e−51.03.2e−46.9e−75.9e−86.5e−38.1e−64.3e−7Figure 1
Plot of the approximate solution (a) andL∞ errors (b) taking r=5 and J=4 for Example 1.
(a)(b)Figure 2
Effects of parametersr and J on the L∞ error for Example 1.
(a)(b)Example 2.
Consider hyperbolic partial differential equation (1) with(32)qx≔sinπx,gx≔−12sinπx,x∈Ω,ht≔0,kt≔2πe−t/2,t∈0,1,and px,t≔1/4+π2e−t/2sinπx. The exact solution of this problem is given in [1] as(33)wx,t≔e−t/2sinπx.
The approximate solution of Example1 has been plotted in Figure 3. In Table 2, results are compared with the cubic B-spline scaling function method [1] in terms of L∞ errors at different times. The effects of the refinement level J and multiplicity parameter r on L∞ errors are given in Figure 4.Figure 3
Plot of the approximate solution (a) andL∞ errors (b) taking r=7 and J=3 for Example 2.
(a)(b)Table 2
Comparison of theL∞ error for Example 2.
tPresented methodDehghan and Lakestani [1]J=2J=3J=4J=2J=3J=40.21.6e−57.8e−85.6e−101.9e−32.7e−42.1e−50.45.3e−69.5e−81.1e−95.3e−34.0e−51.0e−60.62.1e−67.6e−94.4e−95.0e−31.2e−41.9e−50.81.3e−51.1e−77.3e−102.4e−36.7e−41.5e−61.04.3e−69.3e−94.4e−91.5e−21.2e−49.7e−5Figure 4
Effects of parametersr and J on the L∞ error for Example 2.
(a)(b)Example 3.
To show the ability and accuracy of the proposed method, let us consider the following HPDEs [21]:(34)wttx,t=wxxx,t,with initial conditions(35)wx,0=0,wtx,0=πcosπx,x∈0,1,boundary condition(36)w0,t=sinπt,and also the integral condition(37)∫01wx,tdx=0.
The exact solution iscosπxsinπt.
The absolute value of errors obtained by the presented method is compared with others in Table3. Because interpolating scaling functions are flexible in choosing the multiplicity parameter r and refinement level J, it is obvious that our method gives better results than others. In Table 4, we report the absolute value of errors for different values of x at t=1 taking different values for r and J. Finally, we illustrate the approximate solution and absolute error in Figure 5.Table 3
Comparison of the absolute value of errors att=1/2 for Example 3.
xPresented methodTurkyilmazoglu [21]Dehghan and Lakestani [1]r=9r=9r=120.21.5e−96.4e−82.0e−50.41.6e−92.1e−83.8e−50.61.9e−92.1e−83.8e−50.81.9e−96.4e−82.0e−5Table 4
The absolute value of errors att=1 for Example 3.
xr=7r=9J=1J=2J=3J=1J=2J=30.21.6e−41.2e−61.2e−84.4e−62.2e−92.5e−90.44.4e−55.9e−74.3e−94.3e−69.0e−103.1e−90.62.1e−45.9e−74.7e−95.3e−68.9e−104.5e−90.87.4e−51.3e−61.2e−81.2e−52.4e−94.3e−91.01.4e−35.4e−77.0e−105.0e−59.1e−101.1e−10Figure 5
Plot of the approximate solution (a) and absolute value of errors (b) takingr=9 and J=2 for Example 3.
(a)(b)


## 5. Conclusion

Interpolating scaling functions are applied to obtain the approximate solution of one-dimensional hyperbolic partial differential equations with a nonlocal conservation condition. The convergence analysis is investigated, and the numerical results are compared with existing methods. It shows that the proposed method is much more flexible than others, and the results show that the proposed method has better accuracy than other methods.



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