Abstract
A powerful technique based on the sincGalerkin method is presented for obtaining numerical solutions of secondorder nonlinear Dirichlettype boundary value problems (BVPs). The method is based on approximating functions and their derivatives by using the Whittaker cardinal function. Without any numerical integration, the differential equation is reduced to a system of algebraic equations via new accurate explicit approximations of the inner products; therefore, the evaluation is based on solving a matrix system. The solution is obtained by constructing the nonlinear (or linear) matrix system using Maple and the accuracy is compared with the Newton method. The main aspect of the technique presented here is that the obtained solution is valid for various boundary conditions in both linear and nonlinear equations and it is not affected by any singularities that can occur in variable coefficients or a nonlinear part of the equation. This is a powerful side of the method when being compared to other models.
Keywords:
Maple; sincGalerkin approximation; sinc basis function; nonlinear matrix system; Newton method1 Introduction
We present here the sincGalerkin approximation technique using Maple to solve systems of nonlinear BVPs such as
where NL is the nonlinear part of Eq. (1.1) which can take any form of nonlinearity, and we
investigate the approximate solution on some closed interval
We start by casting a given linear or nonlinear BVP into a sincGalerkin form accurate
to the order
We start with some literature on the sincGalerkin methods. The sinc methods were introduced in [2] and expanded in [1] by Frank Stenger. Sinc functions were first analyzed in [3] and [4]. An extensive research of sinc methods for twopoint boundary value problems can be found in [5,6]. In [7,8] parabolic and hyperbolic problems are discussed in detail. Some kind of singular elliptic problems are solved in [9], and the symmetric sincGalerkin method is introduced in [10]. The sinc domain decomposition is presented in [1114]. Also, iterative methods for symmetric sincGalerkin systems are given in [1517]. Sinc methods are studied thoroughly in [18]. Applications of sinc methods can also be found in [1921]. The article [22] summarizes the results that are obtained by sinc numerical methods of computation. In [14] a numerical solution of the Volterra integrodifferential equation by means of the sinc collocation method is considered. The paper [1] illustrates the application of a sincGalerkin method to the approximate solution of linear and nonlinear secondorder ordinary differential equations and to the approximate solution of some linear elliptic and parabolic partial differential equations in the plane. The fully sincGalerkin method is developed for a family of complexvalued partial differential equations with timedependent boundary conditions [19]. In [23] some novel procedures of using sinc methods to compute the solutions of three types of medical problems are illustrated. In [24], the sincbased algorithm is used to solve a nonlinear set of partial differential equations. A new sincGalerkin method is developed for approximating the solution of convection diffusion equations with mixed boundary conditions on halfinfinite intervals in [25]. The work which is presented in [26] deals with the sincGalerkin method for solving nonlinear fourthorder differential equations with homogeneous and nonhomogeneous boundary conditions. In [27], the sinc methods are used to solve secondorder ordinary differential equations with homogeneous Dirichlettype boundary conditions. In the paper given in [28], the sincGalerkin method is applied to solving Troesch’s problem. The properties of the sinc procedure are utilized to reduce the computation of Troesch’s equation to nonlinear equations with unknown coefficients.
2 Sinc basis functions
Let C denote the set of all complex numbers, and for all
For
For various values of k, the sinc basis function
Figure 1. The basis functions
Figure 2. Central sinc basis function
If a function
is called the Whittaker cardinal expansion of f whenever this series converges. The infinite strip Ds of the complex w plane, where
In general, approximations can be constructed for infinite, semiinfinite, and finite intervals. Define the function
which is a conformal mapping from
This is shown in Figure 3.
Figure 3. The relationship between the eyeshaped domain
For the sincGalerkin method, the basis functions are derived from the composite translated sinc functions:
for
For the evenly spaced nodes
A list of conformal mappings may be found in Table 2.1 [6].
Figure 4. Three adjacent members
Definition 2.1 Let
and
We can use Table 1 to choose convenient conformal map according to boundary conditions.
Table 1. Conformal mappings and nodes for several subintervals ofR
Definition 2.2 Let
where
and on the boundary of
The proof of following theorems can be found in [1].
Theorem 2.1Let Γ be
where
For the sincGalerkin method, the infinite quadrature rule must be truncated to a finite sum. The following theorem indicates the conditions under which an exponential convergence results.
Theorem 2.2If there exist positive constantsα, βandCsuch that
then the error bound for the quadrature rule (2.14) is given by
The infinite sum in (2.14) is truncated with the use of (2.16) to arrive at (2.17).
Making the selections
where
We used Theorems 2.1 and 2.2 to approximate the integrals that arise in the formulation of the discrete systems corresponding to the secondorder boundary value problem.
Theorem 2.3Letϕbe a conformal onetoone map of the simply connected domain
3 Convergence analysis
Consider the following problem:
with Dirichlettype boundary condition
where P, Q, R, and F are analytic on D. We consider sinc approximation by the formula
The unknown coefficients
Let
where w is the weight function. For the secondorder problems, it is convenient to take [1]
For Eq. (3.1), we use the notations (2.21)(2.23) together with the inner product given in (3.5) [1] to get the following approximation formulas:
where
Using (3.5), (3.8)(3.11), we obtain a nonlinear system of equations for
The nonlinear system with
Let
With these notations, the discrete system in (3.5) takes the form:
Theorem 3.1Letc,
Now we have a nonlinear system with
4 Examples
In this section, three examples are given to illustrate the performance of the sincGalerkin
method by solving nonlinear Dirichlettype boundary value problems. Each of these
problems have been chosen to simulate how the solutions change in different zero boundary
intervals. In the following examples, the discrete sinc system defined by (3.18) is
used to compute the coefficients
Example 4.1 Consider the following nonlinear Dirichlettype boundary value problem on the interval
We choose the weight function according to [1],
Figure 5. The redcolored curve displays the Newton solution and the green one is an approximate solution of Eq. (4.1).
Table 2. The numerical results for the approximate solutions obtained by sincGalerkin in comparison
with the Newton solutions of Eq. (4.1) for
Example 4.2 Let us have the following form of nonlinear Dirichlettype boundary value problem
on the interval
where
Figure 6. The redcolored curve displays the Newton solution and the green one is an approximate solution of Eq. (4.2).
Table 3. The numerical results for the approximate solutions obtained by sincGalerkin in comparison
with the exact solutions of Eq. (4.2) for
Example 4.3 In this case, we take the problem to be given on the interval
where we chose
5 Discussion
A new efficient computer application of sincGalerkin method has been presented for
nonlinear BVPs. The main advantage of our technique compared to other methods (e.g., Newton’s method) is that the solution is independent of the singularity conditions
and valid for Dirichlettype boundary conditions. The order of accuracy used in this
paper is
6 Conclusion
In this study, the sincGalerkin method has been employed to find the solutions of secondorder nonlinear Dirichlettype boundary value problems on some closed real interval and the method has been compared to the Newton method. Our main purpose is to find the solution of boundary value problems which arise from the singular problems for which the Newton method does not converge at singular points. The powerful side of our method is that it can easily compute solutions even if the equation has singularities. The Newton method can fail when computing some complicated forms of governing equations; on the other hand, our method can easily handle this situation. The examples show that the accuracy improves by increasing the number of sinc grid points N. The method presented here is simple and gives a numerical solution, which is valid for various boundary conditions. We have developed a very efficient algorithm to solve secondorder nonlinear Dirichlettype boundary value problems with sincGalerkin method in Maple Computer Algebra System. Several nonlinear BVPs have been solved by using our technique in less than 20 seconds. All computations and graphical representations have been prepared automatically by our algorithm.
Appendix: A computer application of numeric solutions for nonlinear boundary value problems (NBVPs)
We demonstrate below how to solve and simulate for a nonlinear BVP. For example, the following Maple code computes and simulates Example 4.3.
Set all parameters as default values
> restart:
For drawing approximation graphics, we must type the following line
> with(plots):
A user has to specify with (linalg) for linear algebra operations in Maple
> with(linalg):
A user can define the grid point size N for sincGalerkin approximation
> N:=48:
The boundary conditions are given as Eq. (4.3).
> a:=4:
> b:=5:
> Boundaries:=y(a)=0,y(b)=0;
P, Q and R are the variable coefficients of Eq. (1.1). In Maple for Eq. (4.3) they are defined as follows:
> P(x):=1;
> Q(x):=1;
> R(x):=1;
F is right side of Eq. (4.3)
> F(x):=cos(Pi*x^2)*x;
We can write a nonlinear part of Eq. (1.1) as follows. User can define any form of nonlinearity in this section.
> NLPart:=exp(sin(y(x)))*y(x)^2/(1+y(x));
The main form of Eq. (1.1)
> Equation:=P(x)*diff(y(x),x$2)+Q(x)*diff(y(x),x$1)+R(x)*NLPart=F(x);
If the user needs, the main equation can be written in the latex format
In order to compare our method with the Newton interpolation (for nonlinear ODE) method, we first solve Eq. (4.3) numerically as follows:
Prepare the plot of the Newton solution
> PlotNewtonSolution:=odeplot(NewtonSolution,a....b):
To define
> delta[0]:=unapply(piecewise(j=k,1,j<>k,0),j,k):
> delta[1]:=unapply(piecewise(j=k,0,j<>k,((1)^(kj))/(kj)),j,k):
> delta[2]:=unapply(piecewise(j=k,(Pi^2)/3,j<>k,2*(1)^(kj)/(kj)^2),j,k):
The parameters for sincapproximation given [1]
> d:=Pi/2:
> h:=2/sqrt(N):
The evenly spaced nodes given (2.9) and Table 1 are defined as follows:
> xk:=unapply((a+b*exp(k*h))/(1+exp(k*h)),k);
The conformal map in Table 1 for sincGalerkin method and its derivatives is computed as follows:
> phi:=unapply(log((xa)/(bx)),x);
> Dphi:=unapply(simplify(diff(phi(x),x)),x):
> D2phi:=unapply(simplify(diff(phi(x),x$2)),x):
The weight function and its derivatives are computed for using an inner product to discretization Eq. (4.3)
> w:=unapply(1/Dphi(x),x):
> Dw:=unapply(simplify(diff(w(x),x$1)),x):
> D2w:=unapply(simplify(diff(w(x),x$2)),x):
By using sincdiscretization in (3.16), the matrix system with
If we want to obtain solutions of linear BVPs, we can use the following lines. They can reduce time complexity. Here, the linear solution is given as a comment (“#”).
> #for Linear system
> #vars:=seq(c[i],i=N..N):
> #A,b:=LinearAlgebra[GenerateMatrix](evalf(MatrixSystem),[vars]):
> #c:=linsolve(A,b);
In this paper, we want to solve nonlinear problems. Then we use fsolve function given by Maple to find unknown
> c:=fsolve(evalf(MatrixSystem)):
Finally, we have unknown
We define plot of Eq. (4.3) obtained by the sincGalerkin solution
> SincGalerkinPlot:=plot({ApproximateSol(x)},x=a..b,color=green,thickness=1):
Simulation: Figure 5, Figure 6, and Figure 7 are obtained as
Enter the number of digits here
> Digits := 15:
Tables 2, 3, and 4 are obtained by the following code:
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
AS proposed main idea of the solution schema by using Sinc Method. He developed computer algorithm and worked on theoretical aspect of problem. MK searched the materials about study and compared with other techniques. MAA contributed us with his experience on Nonlinear Approximation methods. MB contributed us with his experience on Nonlinear Approximation methods, suggested us some valuable techniques.
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