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 (multiplicity ) and the level of refinement (refinement level ). 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.
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].with initial conditions
Dirichlet boundary conditionand also the integral condition
Here, , and are known functions. Without loss of generality, assume that .
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. Section 2 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 subspace as a space of piecewise polynomial bases of degree less than multiplicity parameter that is spanned bywhere two sets and are defined byand are the primal interpolating scaling bases introduced by Alpert et al. [17]. Given nodes , which are the roots of the Legendre polynomial of degree , the interpolating scaling bases are defined aswhere are the Lagrange interpolating polynomials at the point and are the Gauss–Legendre quadrature weights [2, 17]. These bases form orthonormal bases on with respect to the -inner product.
Let . We consider the uniform finite discretizations , where the subintervals are determined by the point . Every function can be represented in the formwhere is the orthogonal projection that maps onto the subspace . To find the coefficients that are determined by , we shall compute the integrals. We apply the -point Gauss–Legendre quadrature by a suitable choice of the weights and nodes for to avoid these integrals [13, 17] via
Convergence analysis of the projection is investigated for the -times continuously differentiable function :
For the full proof of this approximation and further details, we refer the readers to [16]. Thus, we can conclude that converges to with the rate of convergence .
Assume that the vector function with includes the scaling functions and is called the multiscaling function. Approximation (8) may be rewritten using the vector that includes entries as follows:where is an -dimensional vector (). 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 that is spanned by
Thus, by this assumption, to derive an approximation of the function by the projection operator , we getwhere components of the square matrix of order are obtained bywhere . Consider the -th partial derivatives of are continuous. Utilizing this assumption, the error of this approximation can be bounded as follows:where is a constant [13].
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 , i.e.,where components of the square matrix of order are unknown coefficients that should be found and the superscript means transpose. Taking the second derivative with respect to and from both sides of equation (17), one can writewhere 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 to the space using projection , we obtain
Now, we introduce the residual function
Note that () if . Multiplying (20) by and from the left and right, respectively, and integrating over , , we obtain a linear system as follows:where we employ orthonormality of interpolating scaling functions and the local support of these bases. Equation (21) gives equations in which of these equations are linearly dependent. These equations are substituted using the conditions governing the problem. One can find extra equations utilizing the projection for known functions , and as follows:where 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), independent equations:
Replacing and in the first and second columns of and and in the first and last rows of , one can obtain the equations with unknowns. To find the unknowns and, equivalently, the approximate solution to the problem, this system must be solved by a convenient method.
Theorem 1. Let . Further suppose that , the solution of equation (1), is -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 , we obtainwhere . Using the triangle inequality and taking -norm from both sides yieldNow, suppose thatwhere is the () matrix, and thus, one can writewhere 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 ; then, using (15), we can findwhere is a constant.
4. Numerical Results
To illustrate the validity, efficiency, and accuracy of the proposed method, we give some numerical examples. The errors in the given tables and figures arewhere and are the approximate and exact solutions at the point , respectively.
Example 1. Let us consider hyperbolic partial differential equation (1) withand also . The exact solution of this problem is given in [1] asTable 1 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 and , the approximate solution and error are shown in Figure 1. The effects of the multiplicity parameter and refinement level on the error are shown in Figure 2.
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Example 2. Consider hyperbolic partial differential equation (1) withand . The exact solution of this problem is given in [1] asThe approximate solution of Example 1 has been plotted in Figure 3. In Table 2, results are compared with the cubic B-spline scaling function method [1] in terms of errors at different times. The effects of the refinement level and multiplicity parameter on errors are given in Figure 4.
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Example 3. To show the ability and accuracy of the proposed method, let us consider the following HPDEs [21]:with initial conditionsboundary conditionand also the integral conditionThe exact solution is .
The absolute value of errors obtained by the presented method is compared with others in Table 3. Because interpolating scaling functions are flexible in choosing the multiplicity parameter and refinement level , 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 at taking different values for and . Finally, we illustrate the approximate solution and absolute error in Figure 5.
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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.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
This project was supported by the Researchers Supporting Project number (RSP-2020/210), King Saud University, Riyadh, Saudi Arabia.