Abstract
In this paper, we study the performance of Boundary Value Methods (BVMs) on second-order PDEs. The PDEs are transformed into a system of second-order ordinary differential equations (ODEs) using the Lanczos-Chebyshev reduction technique. The conditions under which the BVMs converge and the computational complexities of the algorithms are discussed. Numerical illustrations are given to show the simplicity and high accuracy of the approach.
1. Introduction
The study of methods which approximates the solution of partial differential equations is an important issue in numerical analysis. This is because most modeled problems in science and engineering results in PDEs such as the problem of the elastic torsion of prismatic bars, axisymmetric ideal flow around sphere, the heat equation, the wave equations and so on. In this paper, we are interested in the approximate solution of the initial or boundary value problems of second-order PDEs where is a fixed elliptic or hyperbolic operator of order , , and is a space function defined on a domain .
The reduction of (1) to a lower dimensional problem results, in most cases, to either a system of second-order ordinary differential equation (ODE) or an algebraic system. The reduction technique described in this paper is based on the Lanczos -method [1, 2]. In this approach, polynomials are used as the trial functions with an important advantage that the “economized” solutions which can be readily integrated or differentiated are obtained [1, 2].
In the last few years, Boundary Value Methods (BVMs) have been used for the solution of the first-order initial and boundary value problems and their convergence and linear stability properties have been fully discussed in [3–8]. These BVMs are also used to solve higher order initial and boundary value problems by first reducing the higher order differential equations into an equivalent first-order system which increases the computational cost and time. Biala [9] and Biala et al. [10–12] developed BVMs for the direct solution of systems of the general second-order ODEs and PDEs with initial or boundary conditions. The boundary value technique simultaneously generates approximate solution to the exact solution on the entire interval of integration. This approach has the advantage of producing smaller global errors (at the end of the range of integration) than those produced by the step-by-step methods due to the presence of accumulated errors at the each step in the step-by-step method. The novel property of this paper is in the ease with which the second-order PDEs are transformed into a system of ODEs.
The paper is organized as follows: In Section 2, we derive a continuous approximation of the exact solution from which the class of BVMs are developed. The convergence and computational complexities of the methods are discussed in Section 3. In Section 4, the Lanczos-Chebyshev reduction technique is introduced. Several numerical illustrations, given in Section 5, are given to show the efficacy of this approach. Finally, we give a concluding remark in Section 6.
2. Boundary Value Methods
The BVMs given in [9] are a class of methods for the general second-order ODEs based on the Linear Multistep Formulas. Most formulas [4] used for second-order ODEs are implemented by reducing them into a system of first-order ODEs. The BVMs in [9] were introduced in order to lessen the computational cost and time and also to utilize additional information associated with specific differential equations such as the oscillatory nature of solutions. One main feature of the BVMs in [9] is that they can be used in the same way for solving both initial and boundary value problems with some slight change in the code used. Therefore such methods are the best candidates for solving second-order PDEs after reduction into an equivalent second-order system.
In what follows, we consider the general system of second-order boundary value problems subject to the mixed boundary conditionswhere are continuous functions, , and is the dimension of the system. The BVMs for solving (2) take the general form with the derivative formulas where is the constant stepsize and , , , , , and are chosen so that (4) and (5) have order and , respectively.
2.1. Derivation of the BVMs
In this section, we shall use the interpolation and collocation approach [9–15] to construct a -step continuous LMM (CLMM) which will be used to produce formulas for solving (2). The CLMM has the general form
Evaluating (6) at we obtain the formulas of the form whose derivative formula is To obtain (6), we seek an approximation to the exact solution of the form where and are coefficients to be uniquely determined so that the method has order and and are the number of interpolation and collocation points, respectively. We impose that the interpolating function (9) satisfies the following conditions: which lead to a system of equations which is solved using a Computer Algebra System (CAS) such as Mathematica to obtain , .
The CLMM is developed by substituting the values of into (9). After some algebraic manipulations, the CLMM is expressed in form (6). The -step CLMM is used to generate the main methods (7) and derivative formulas of form (8) which are combined to solve (2). For example, the BVMs of order 6 used are as follows.
BVM order 6:with the derivative formulasAlso, the BVMs of order 8 are as follows.
BVM order 8:with the derivatives
Remark 1. We note here that, for higher orders (higher degree polynomials of (9)), the accuracy of the scheme becomes lower and tends to integrate the scheme less accurately. This is due to the truncation errors in the numerical solution.
3. Convergence of the BVMs
In this section, we shall establish the convergence of the BVMs derived in the previous section. We emphasize that we evaluate (6) at to obtain and also evaluate at to obtainWe note that the formulas in (15) are of while the derivative formulas are of . Equations (15) and (16) can be compactly written in matrix form by introducing the following matrix notations. Let be a matrix defined by where are matrices given as is an null matrix, and is an identity matrix.
Similarly, let be a matrix defined by where are matrices given as , are null matrices.
We further define the following vectors:The exact form of the system formed by (15) and (16) is given by where is the truncation error vector of the formulas in (15) and (16). The approximate form of the system is given by where is the approximate solution of vector .
Subtracting (22) from (23) and letting and using the mean value theorem, we have the error system where is the Jacobian matrix and its entries , , , and are defined as Let be a matrix of dimension so that (24) becomes and for sufficiently small , is a monotone matrix and thus nonsingular (Jator and Li [14]). Therefore which shows that the methods are convergent and the global error is of order .
3.1. Computational Complexities
In this section, we discuss the use of (15) and (16), which we henceforth call BVM (BVM), for , where is a multiple of . We emphasize that the methods in the BVM are all main methods since they are weighted the same and their use lead to a single matrix equation which can be solved for the unknowns. For example, for BVM6 (order 6), we make use of each of the methods above in steps of 4; that is, . This results in a system of equations in unknowns which can be easily solved for the unknowns. Below is an algorithm for the use of the methods.
The methods are implemented as BVMs by efficiently using the following steps.
Step 1. Use the BVM for to obtain in the interval and for is obtained in the interval . Similarly, for we obtain , where in the intervals, , respectively.
Step 2. The unified block given by the system obtained in Step 1 results in a system of equations in unknowns which can be easily solved.
Step 3. The values of the solution and the first derivatives of (1) are generated by the sequence , , , obtained as the solution in Step 2.
4. Lanczos-Chebyshev Reduction Technique
We consider the second-order problem of the form with the boundary condition where is the boundary of the region . The Lanczos -method involves the replacement of one of the two functions in the trial solution by an approximate polynomial of the form and is approximated by and must be bounded by the lines in the -direction. The problem is slightly perturbed to obtain where are arbitrary functions and are the order shifted Chebyshev polynomials in the range . Equating the powers of in (32) together with the boundary conditions (33) gives equations in unknowns . The operator may also contain polynomial functions. The inclusion of (33) is to ensure the exact satisfaction of the exact solutions at and . The arbitrary functions are eliminated to give a set of second-order ordinary differential equations. Hence, can be uniquely determined.
5. Numerical Illustrations
In this section, we consider five numerical examples using the BVMs of order 6 to solve the ODE system arising from the semidiscretization of the PDEs. All computations were carried out using a written code in Mathematica 11.0. In all examples, a uniform stepsize was used and the maximum absolute errors were computed as where is the numerical approximation of the exact solution at the mesh point .
Example 1. We consider the following one-dimensional elliptic PDE:
We seek an approximation of the form to obtain and the perturbed form with where Equating the powers of in (38), we obtain From (39) and (41), we eliminate to obtain second-order differential equations in unknowns . To use the BVMs to solve the resulting second-order system, we require boundary conditions which is obtained from ; that is, . Table 1 shows the computational results; Figures 1–3 show the exact, approximate, and error function of the elliptic problem, respectively.
Example 2. We consider the one-dimensional wave equation with variable coefficients
The perturbed form of the problem is with Equating powers of , we obtain Eliminating from (44) and (45), we obtain second-order differential equations in unknowns , . The initial conditions to be used with this system of second-order ODE are Table 2 shows the maximum errors on the line for different . Also Figures 4–6 depict the exact, approximate, and error function of the wave equation.
Example 3. Consider the one-dimensional linear inhomogeneous Klein-Gordon equation
With the procedures of the previous examples, we obtain the maximum errors as shown in Table 3. Figures 7–9 also show the exact, approximate, and error function of the Klein-Gordon equation.
Example 4. Consider the Helmholtz equation
Table 4 shows the numerical results for the Helmholtz equation. We observe that, for this example, as increases the method gives more accurate results. This is due to the accurate matching of the derivative conditions as increases. Figures 10–12 show the plot of the exact, approximate, and error function for this example.
Example 5. Lastly, we consider the nonlinear equation
Table 5 and Figures 13–15 show the numerical results for the nonlinear equation.
6. Conclusion
The Boundary Value Methods have been used to approximate second-order PDEs. This has been achieved by using the Lanczos-Chebyshev reduction technique to transform the PDEs into an equivalent second-order system. The high accuracy and simplicity of the approach is evidenced by the preceding examples.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this article.