Numerical solutions to initial value problems (IVPs),

play a critical role in scientific and engineering applications, as exact analytical solutions are often unattainable [1]. These problems typically involve solving differential equations that model various real-world systems, ranging from fluid dynamics to population growth models. Over the years, a variety of classical methods have been developed to provide approximate solutions to IVPs, including the Euler method, Heun’s method, and the widely-used Runge-Kutta family of methods. Each of these approaches aims to generate numerical solutions by approximating the integral of the function over small time intervals, relying on diferent strategies for calculating the slope of the function.

The Euler method, one of the simplest techniques, estimates the solution using the slope at the initial point. While it provides a straightforward approximation, its first-order accuracy often results in significant errors for stif or complex problems [2]. Heun’s method, sometimes referred to as the improved Euler method, ofers a refinement by averaging the slope at the beginning and the end of the interval, thereby increasing the order of accuracy of two. Runge-Kutta methods, on the other hand, provide a general framework that further improves the accuracy and stability by computing intermediate slopes within the interval. These classical methods form the foundation for many numerical solvers in use today.

Despite their widespread use, these traditional methods such as Euler’s method, Heun’s method, and Runge-Kutta method, are not without their limitations, and over time, mathematicians have proposed modifications to enhance their accuracy, stability, and eficiency. For instance, adaptive step-size control, higher-order methods, combining transcendental functions on finite diference, and enhanced stability conditions have been incorporated into existing techniques to handle more complex or stif diferential equations as can be studied in [3, 4, 5, 6, 7, 8, 9, 1]. These improvements have proven particularly useful for problems where the classical methods exhibit poor convergence or sufer from excessive computational cost.

In this paper, we propose a novel modification to Heun’s method by altering how the average slope of the tangents is computed. Instead of using the arithmetic mean, we replace it with the mean of harmonic mean and the root mean. The harmonic mean, which emphasizes smaller values [10], and the root mean square, which provides a balanced approach between the arithmetic and geometric means [11], both ofer potential improvements in accuracy and stability for specific classes of problems.

The motivation for choosing this combination arises from the nature of stif initial value problems in diferential equations, where maintaining numerical stability while controlling local truncation error is crucial. As known, the harmonic mean emphasize the smaller of two values, which suggests it may reduce the efect of large tangent slopes and hence enhance numerical stability of stif problems. Similarly, the root mean square gives more weight to larger values of slope magnitude, which suggests a potential for better approximation in regions of rapid variation. Motivated by these complementary properties, we propose combining them and develop a new method for solving initial value problems, in particular stif problem as the model.

The rest of the paper is structured as follows. In Section 2, we provide the derivation of our proposed method, the stability and consistency analysis. Section 3 we present a series of numerical experiments comparing the performance of the modified methods against traditional approaches. Finally, Section 4 ofers a conclusion and discusses potential future directions for research, including further refinements and possible applications in more complex diferential equation models.

In this section, we test our proposed method to solve three IVPs with diferent step-size In order to see the performance of the method, we compare its results with several numerical methods, i.e. EuM, HeM, HM, and AHM.

Example 3.1. Consider the IVP

The exact solution is . The solution is given in the following tables.

From Table 1, it can be seen that HM shows the highest error across all values of t, indicating that it performs the worst for this step-size. EuM and AHM progressively reduce error, with HeM consistently showing better results than HM and AHM. On the other hand, HRM demonstrates its superiority, with significantly lower error compared to the other methods. At a smaller step-size as can be viewed in Table 2, the errors of all methods decrease significantly where HRM continues to provide consistent accuracy, followed by AHM, EuM and finally HM. As step-size decreases, the errors of all methods decreases while HRM and HeM consistently perform well.

Example 3.2. Given IVP

The exact solution is . The following tables present the numerical solutions for and

Table 3 presents the results in which HRM again shows a consistent accuracy across the entire time range. HM performs poorly with the largest errors among all methods while EuM, HeM, and AHM performs moderately, with AHM closer to HRM in accuracy. At smaller the step-size, as can be seen in Table 5, the errors across all methods reduce and HRM maintains the consistency despite not as accurate as EuM and HeM.

Example 3.3. The exact solution of IVP

is . The tables below presents the errors given by the mentioned methods for 0 ≤ t ≤ 1.

Similar to previous examples, HRM still provides the smallest errors, after HeM. At this step-size, HM shows the largest error values, followed by EuM and AHM, as seen in Table 6. Reducing the step-size for this example, results in improved accuracy of all methods, but the ranking remains consistent which is apparent in Table 7.

Overall, from the results of the numerical simulations, HRM shows the best performance in terms of accuracy. It consistently has the smallest error for both large end small step-size. HRM can also handle larger step-size without sacrificing much accuracy, unlike HM, which performs poorly with large step-size. In addition, HRM works well for diferent types of diferential equations as demonstrated across all examples. This shows the versatility of HRM and marks consistency of the method for solving IVPs that require precise solutions.

This paper introduces a novel numerical technique for solving initial value problems (IVPs) by modifying Heun’s method. The modification involves using the harmonic and root mean square to calculate the average of the slopes of the tangents in Heun’s method. Stability and consistency analyses demonstrate the reliability of the approach, while numerical simulations emphasize its accuracy in producing approximate solutions with improved precision. The proposed method shows competitive potential compared to existing methods, ofering a viable alternative for solving IVPs. Future advancements could involve applying other averaging techniques to diferent numerical methods, expanding their use for solving real-world problems.