Falkner Hybrid Block Methods for Second-Order IVPs:
A Novel Approach to Enhancing Accuracy and Stability Properties

Ordinary differential equations (ODEs) are prevalent in various scientific disciplines, allowing for the modeling of temporal and spatial changes in a wide range of scenarios. Practical applications of ODEs include predicting the movement of electricity, analyzing the oscillatory motion of objects like pendulums, and explaining principles of thermodynamics. In medicine, ODEs are used to estimate disease progression visually. The general $n$th-order ODE can be written as,

Solving (1) often involves converting it into a system of first-order equations and using the appropriate method to solve the system, (see, [8], [18], [19], [22], and [25]). [9] noted that this process can be time-consuming, especially when developing a computer program. In addition to the main program, separate programs for initial values and functions from the equation system are typically required. This complexity may discourage newcomers from exploring promising numerical methods due to a lack of knowledge and confidence in crafting programs to validate their results.

The second-order IVPs in ODEs that we aim to approximate on a given interval in this research stem from (1) when $n$ = 2, resulting in the following IVPs,

where $y\in R^N$ and $f:R\times R\times R^N\to R^N$ are continuous vector-valued functions. The theoretical solutions to (1.2) are usually highly oscillatory. Second-order IVPs are fundamental in modeling various phenomena, such as oscillations, vibrations, and electrical circuits. However, their numerical integration requires careful consideration of accuracy, stability, and efficiency. The earlier proposed schemes for the direct solution of second-order ODEs (2) in literature can be found in the articles of the authors: [1], [2], [3], [4], [5], [6], [7], [9], [10], [11]-[13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [28], [29], and [30]. In this article we formulate and derive a family of Falkner hybrid block methods with off-step points, using interpolation and collocation techniques. The proposed scheme modifies the Falkner block method in [29].

The paper is organized as follows: In Section 2, we reviewed the Falkner block method in [29]. Section 3 deals with the formulation and the derivation of the Falkner hybrid block method with an off-step point of order $k$ + 2. In Section 4, we give the main properties of the proposed method. Section 5 is devoted to the numerical experiments of the method. We compared the accuracy of our methods with other methods proposed in the literature. Finally, Section 6 is dedicated to conclusion.

Falkner method (FM) was introduced by V. M Falkner in 1936. This scheme was for the numerical solution of second-order ODEs. The general form of a couple $k$-step Falkner method is,

where, $y_{n+k}=y(x_{n+k})$, and $y_{n+k}^{\prime }=y^{\prime }(x_{n+k})$ are approximations to the solutions and its derivative at $x_n=x_0+nh$, and $f(x_{n+j},y_{n+j})$ = $y_{n+j}^{\prime \prime }$ is the function emanating from the second order ODEs. The h denotes the step size, $k$ represents the step number and ${\beta }_j$ is a constant parameter. The algorithm in (3) is: (i). explicit, if ${\beta }_k=0$ and (ii) implicit, if ${\beta }_k\ne 0$. A subclass of the formula in (3) is the explicit Falkner method in [14] and the two implicit Falkner schemes in [20].

The Falkner method is another form of hybrid method, it combines different numerical methods, to solve second-order boundary value problems (see [17] and [29]). The block form of the Falkner method in (3) is,

where $A,B,$ and $C$ are matrices of coefficients of dimension 2$k$ by ($k$+1), and

The theoretical solution to (5) is the vectors,

The associated local truncation error of the methods in (5) is,

where $A,B,C$ are matrices of coefficients of dimension 2$k$ by ($k$+1). The block method in (4) and the associated linear difference operator in (7) is said to be of order $p$ if after expanding (6) by Taylor series about $x_n$ and inserting the resulting expressions into (7) we obtain

$\parallel \cdot \parallel$ may be the maximum norm we adopted for convenience.

To determine whether a numerical formula will yield realistic results as step size $h$ becomes large, there is a need for more insight into the types of stability properties the method processes with zero stability being one of them. In most numerical schemes proposed for solving second-order ODEs, the stability properties are generally investigated by considering the linear test equation in [25],

Recall that the second-order ODEs in (2) contain the first derivative component, but the [25] test equation, does not contain the first derivative component $y^{\prime }(x)$, thus, a generalized test equation that includes the first derivative component, $y^{\prime }(x)$, as in ODEs (2) is,

which has bounded solutions for $\mu \ge 0$ that tend to zero as $x\to \mathrm{\infty }$ and this shall be the test equation for the stability analysis in this article. For further reading, see [29].

An example of the two-step Falkner method in [29] is,

The methods in (11) have order $p$ = 3 respectively. The block form of the schemes in (11) is,

Applying (12) to (10) after eradicating the first derivatives yields the stability polynomial,

Plotting the roots of the stability polynomial in (13) via boundary locus sense gives the interval of absolute stability of the scheme in (12) as (0, 1.73205).

In this section, we introduce Falkner hybrid methods for the numerical solution of ODEs in (2). The proposed method modifies the Falkner method discussed in [29]. The general form of the proposed FHM is

where . The ${\alpha }_{k0}(c_i)$, ${\alpha }_{k1}(c_i)$, ${\{{\beta }_{kj}(c_i)\}}_{j=0}^k$, ${\beta }_v(c_i)$, ${\varphi }_{k0}(c_i)$, ${\{{\overline{\beta }}_{kj}\}}_{j=0}^k(c_i)$ and ${\overline{\beta }}_v(c_i)$ are the continuous coefficients of the method, $c$ is the abscissa vector, $c={(c_1,c_2,\mathrm{\dots },c_{r-1},c_r)}^T$, and $h$ is the step size. The output points are $y_{n+c_i}$ and $y_{n+c_i}^{\prime }$. Also, in (14), $y_{n+c_i}^{\prime }$ and $y_{n+k-1}^{\prime }$ are the first derivative component while ${\{y^{\prime \prime }(x_{n+j})\}}_{j=0}^k={\{f(x_{n+j},y_{n+j},y_{n+j}^{\prime })\}}_{j=0}^k$ and $y^{\prime \prime }(x_{n+v})=f(x_{n+v},y_{n+v},y_{n+v}^{\prime })$ is the second derivative function. The $v$ in formula (14) is the hybrid point (i.e., off-step point). if $c_i=v$ in (14) a new hybrid LMM is defined. The block form of the Falkner hybrid methods in (14) is,

where $c={(c_0,c_1,\mathrm{\dots },c_{r-1},c_r)}^T$, is the abscissa vector, $A,B,$ and $C$ are the coefficients matrices of dimension 2$k$+2 by ($k$+2), and the vectors,

The theoretical solution to (15) is the vectors,

The associated local truncation error of the methods in (15) is,

The block hybrid multistep method in (15) and the associated linear difference operator in (18) are said to be of order $p$ if after expanding (17) by Taylor series about $x_n$ and inserting the resulting expressions into (18) we obtain

where $\overline{C}$ is the error constant of the method in (15) and $\parallel \cdot \parallel$ is the maximum norm we adopted for convenience.

This subsection introduces the derivation of the Falkner hybrid method in (14). To do this we use the following polynomial interpolant,

Differentiating (20) with respect to $x$ yields

Again, differentiating (21) with respect to $x$ gives,

Collocating (21) at $x=x_{n+j}$, $j$ = 0 (1) $k$, and $x$ = $x_{n+v}$, and interpolating (22) at $x$ = $x_{n+k-1}$, results in the linear system of equations,

where

We obtained the value of $a_j^{\prime }s$ by the Gaussian elimination method. Substituting the resulting values of $a_j^{\prime }s$, () with $x=x_n+c_{i}h$, (without loss of generality we set $x_n=0$ so that $x=c_{i}h$) into (19) yields the continuous formulas in (14a).

Inserting the value of $c={(c_0,c_1,\mathrm{\dots },c_{r-1},c_r)}^T$ into the continuous formulas gives the discrete form of (14a). Similarly, to obtain the second method (14b), collocate (22) at $x=x_{n+j}$, , and $x=x_{n+v}$ and interpolate (21) at $x=x_{n+k-1}$ gives the linear system of equations

where

Following the above procedures we obtain the formula in (14b).

Our interest herein is to determine the performance of the proposed scheme via a fixed step-size approach. The newly proposed block method (15) under consideration is implicit. Hence, the set of non-linear equations arising from the method when applied to IVPs in ODEs (1) is resolved using the Newton-Raphson iterative method,

where, $c={(c_0,c_1,\mathrm{\dots },c_r)}^T$, $c_{r-1}=v$, $c_r=k$, $c\in (k-1,k)$ and, the function,

where, $A$, $B$, and $C$ are the coefficients matrices of dimension $2k+2$ by $(k+2)$ form our proposed method and the vectors are,

The $J\left(Y_{n+c}^{[s]}\right)$ in (12) is the Jacobian matrix given by

The starting value, $Y_{n+c}^{[0]}$ for the Newton method (26) is obtained from an explicit Runge-Kutta Nystrom method. To demonstrate the application of our proposed Falkner hybrid block methods using constant step size (fixed step size) we solve the following second-order IVPs given in Example 4-Example 8.