Abstract
A general one-step three-hybrid (off-step) points block method is proposed for solving fourth-order initial value problems of ordinary differential equations directly. A power series approximate function is employed for deriving this method. The approximate function is interpolated at while its fourth and fifth derivatives are collocated at all points , , in the interval of approximation. Several fourth-order initial value problems of ordinary differential equations are then solved to compare the performance of the proposed method with the derived methods. The analysis of the method reveals that the method is consistent and zero stable concluding that the method is also convergent. The numerical results demonstrate the superiority of the new method over the existing ones in terms of error.
1. Introduction
In this article, we consider the following general fourth-order initial value problems (IVPs) as shown: with the assumption that . Several phenomena in physical fields such as neural networks, electric circuits, and ship dynamics can be expressed in differential equations (DEs) forms (see [1–3]).
Block method is one of the efficient methods proposed in 1953 by [4] to enhance the performance of the numerical methods. In 1967, [5] employed block method to provide the essential starting values needed for the predictor schemes. Subsequently, hybrid methods were initiated by [6], which involve the evaluation of functions off-step (nonstep) points. The introduction of hybrid (off-step) points in block methods has many advantages such as the ability to change step size, utilizing data off-step points, and the most important feature according to [7] is the capability to circumvent zero stability barrier condition ([8]).
To increase the accuracy further and to solve the stiffness problem in ordinary differential equations (ODEs), [9] derived second derivative multistep methods for stiff ODEs. Recently, authors like in [10, 11] proposed high derivative methods for the same reason. The former developed block hybrid-second derivative method for stiff systems, while the latter introduced a Simpson’s type second derivative method for the solution of a first-order stiff system of IVP. In addition, [12] proposed a continuous fourth derivative method for third-order boundary value problems. Following these scholars’ footsteps a new generalized three-hybrid one-step fifth derivative method for solving fourth-order ODEs directly using the approach of interpolation and collocation will be proposed.
This article consists of five sections: Section 1 is for the introduction, while Section 2 illustrates the method derivation, where we consider three-off-step points and employing the collocation approach. The analysis of the approach is discussed in Section 3 which includes zero stability, order, consistency, and convergence. Section 4 covers the solution of some mathematical problems to show the performance of the developed method. Finally, a brief conclusion is performed in Section 5.
2. Development of the Method
Let the following power series polynomial be the approximate solution of (1)where and , the number of interpolation and collocation points. Now, differentiating (2) four and five times yieldsInterpolate (2) at , , and collocate (3) at all points , , where produces a system of equations in matrix form as below where . Matrix manipulation is then employed to solve the resulting system (4) for the unknown coefficients , . Substituting the obtained values of the coefficients back into (2) yieldswhere , is the constant step size for the partition of the interval which is given by , , and are undetermined constants listed in Appendix in the Supplementary Material (see Supplementary Material available online at https://doi.org/10.1155/2017/7637651). For simplicity we shall use , , , and .
Calculating the first, second, and third derivatives of (5) produces Evaluating (5) at the noninterpolating points with (6) and (7) at all points , , gives the following general equations in block form:where is an identity matrix of order and The entries of , , , and are listed in Appendix in the Supplementary Material section, while the vectors , , , , , are defined as follows:
3. Analysis of the Method
3.1. Zero Stability
Definition 1. The hybrid block method formula (9) is said to be zero stable if no root of the first characteristic equation has modulus greater than one; that is, , and if then the multiplicity of must not exceed four.
To prove that the roots of the first characteristic equation satisfy the previous definition which imply that As a result, the developed method is zero stable.
3.2. Order of the Method
The linear operator associated with the hybrid block methods formula (9) is defined as Expanding the above equation in Taylor series and combining like terms imply According to [7, 13] method (9) is said to be of order if The term is called the error constant and the local truncation error is given by Comparing like terms of and in (15) produces the coefficients with vector of error constants where which concludes that the order of the developed method is .
3.3. Consistency
Definition 2. A block method is said to be consistent if its order is greater than one.
Consistency property is achieved for the hybrid block method from the above analysis since the order .
3.4. Convergence
Theorem 3 (see [14]). Consistency and zero stability are sufficient conditions for a linear multistep method to be convergent.
The hybrid block method (9) is convergent since it fulfils both the consistency and zero stability conditions.
3.5. Region of Absolute Stability
Stability region of the hybrid block (9) is discussed in spirit of [7]. The test problem of the forms , , , , and , where are substituted into the main methods of block (9), is shown belowwhere while the vectors , , , , , , , and are defined as follows: substituting in (20) yieldswhere the amplification matrix is given byThe analysis shows that the matrix has eigenvalues , where the dominant eigenvalue is a function of subject to the values given by where and the values and are listed in Table 1.
The region of absolute stability is depicted in the dark area in Figure 1.
3.6. Pseudocode of the Developed Method
The following pseudocode illustrates how the developed method is implemented for solving fourth-order initial value problems.
Step 1. The values on the interval from the problem are given.
Step 2. Set the values of the hybrid points and step size .
Step 3. For , set , .
Step 4. Evaluate the approximate values , , , using the direct hybrid block method.
Step 5. Find the solution for the resultant system using the built in function fsolve in Matlab.
Step 6. Calculate the maximum error = .
4. Numerical Examples
The general three-hybrid one-step hybrid block method (9) with order in this section was employed to solve five problems from the literature. In order to show the performance of the proposed method, the problems are solved with different values of step size and hybrid points .
Problem 1 (linear fourth-order problem). Exact Solution.
Source. See [15].
Problem 2 (nonlinear fourth-order problem). Exact Solution. with
Source. See [15].
Problem 3 (application to problem from ship dynamics). In this example, the derived method is implemented to solve a physical problem from ship dynamics. As stated by [3], when a sinusoidal wave of frequency passes along a ship or off-shore structure, the resultant fluid actions vary with time. In a particular case study by [3], the fourth-order problem is defined as Exact Solution.
Problem 4 (system of linear fourth-order IVPs). Exact Solution. , , , and with
Problem 5 (system of nonlinear fourth-order IVPs). Exact Solution. , , , and , with
5. Conclusion
A hybrid one-step block method of order with generalized three-off-step points has been derived successfully. The developed method was employed to solve general fourth-order IVPs of ODEs directly. The numerical analysis performed shows that the developed method is consistent and zero stable which leads to the convergence of the method. The numerical results were then compared with the results obtained by the existing methods in terms of error. The new method is found to have better performance over the other methods (refer to Tables 2–12).
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors thank Universiti Utara Malaysia for the financial support of the publication of this article.
Supplementary Materials
The Supplementary Material consists of two sections the first is appendix I which displays the values of α_i, β_i, and γ_i. While, on the other hand, appendix II lists the elements of the matrices B^[0], B^[1], D^[0], and D^[1].