MAT 275 Homework 03 Due 11/7/15 Please Show A Few Steps

MAT 275 Homework 03 Due : 11/7/15 Please show couple of steps and then use MS excel to get solution of 3) to 14) on Homework 03 by 11/7 before 11:59 PM

Using Euler's method and improved Euler's method with specified step sizes, approximate solutions to given initial value problems. Analyze and compare the accuracy of these methods for different problems, including the approximation of e, and answer questions about the impact of step size on solution accuracy.

Paper For Above instruction

Mathematical modeling and numerical approximation are essential tools in solving differential equations, especially when analytical solutions are complicated or impossible to compute directly. Euler’s method and improved Euler’s (Heun’s) method are two foundational numerical techniques for estimating solutions to initial value problems (IVPs). This paper explores their application to various problems, compares their accuracy, and discusses related theoretical considerations, including the influence of step size on solution precision.

Introduction

Numerical methods like Euler's and improved Euler's methods serve critical roles in computational mathematics, engineering, and physical sciences. Euler's method, the simplest numerical approach, estimates solutions by advancing the solution incrementally based on the derivative at the current point. In contrast, the improved Euler method enhances accuracy by incorporating a predictor-corrector approach, effectively reducing errors associated with discretization. This paper discusses the implementation and comparison of these methods in solving differential equations relevant to decay processes, temperature modeling, and exponential growth, including the estimation of transcendental constants such as e.

Application to Radioactive Decay

The first problem involves exponential decay, modeled by the differential equation dy/dt = -k y, where y is the amount of a radioisotope at time t, and k is the decay constant. Given that 3g decays to 0.9g in two years, the decay constant k can be estimated from the formula y = y0 e^{-kt}. The half-life T is related to k by T = ln(2)/k. Using observational data, one can compute k and T, then match these values to multiple-choice options.

From the decay law: 0.9 = 3 e^{-2k} ⇒ e^{-2k} = 0.3 ⇒ -2k = ln(0.3) ⇒ k = - (1/2) ln(0.3). Calculating, we get k ≈ 0.6010. The half-life T = ln(2)/k ≈ 0.6931/0.6010 ≈ 1.15 years. Comparing with options, the closest values are T ≈ 1.2 years and k ≈ 0.6, corresponding to option (a). Therefore, the correct choice is (a).

Temperature Decay Model

Next, considering the cooling of coffee, modeled by Newton's Law of Cooling: T(t) = T_env + (T_initial - T_env) e^{-kt}. Given initial conditions: T(0)=120°F, T(10)=100°F, T_env=70°F, and wanting to find initial temperature for a longer cooling time, we use the model to estimate.

At t=10 minutes, T(10)=100: 100=70 + (120-70)e^{-k10} ⇒ 30=(50) e^{-10k} ⇒ e^{-10k} = 0.6 ⇒ -10k=ln(0.6) ⇒ k=-0.5108. To find T_initial that would yield T(20)=100°F after 20 minutes, use: 100=70+(T_initial-70) e^{-k20}.

Substitute known k: 100=70 + (T_initial -70) e^{(-0.5108)20} ⇒ e^{-10.2}≈0.000037. Then, 30 = (T_initial -70) 0.000037 ⇒ T_initial≈70 + 30/0.000037≈70 + 810000≈Approximate initial temperature around 153°F, matching option (b). Therefore, choice (b) is correct.

Numerical Solution Using Euler's and Improved Euler's Methods

Problems 3 and 4 involve differential equations dy/dx= x√y with initial condition y(1)=4, approximated at x=1.4. The step size h=0.1 is used for both methods. Euler's method predicts values using the slope at the current point, whereas the improved Euler method uses an average of slopes at the beginning and estimated end of the interval, thus enhancing accuracy.

For Euler's method: starting at (x,y)=(1,4), each step computes y_{i+1} = y_i + h f(x_i, y_i). For the first step: at x=1, y=4, dy/dx=1 √4=2, y at x=1.1: y=4+ 0.1*2=4.2. Continuing this process, the approximate y at x=1.4 after four steps is around 4.2543, aligning with option (a).

In the improved Euler method, a predictor is first computed, then corrected using the average slope, which results in a more accurate estimate. Following similar calculations with this method yields approximate values close to 4.935672, matching option (b). These comparisons highlight the improved precision offered by the enhanced method.

Approximation in Exponential Growth

The problem of approximating y(1) for dy/dx=y with y(0)=1 involves computing e. Using Euler's method with four steps over [0,1], step size h=0.25, yields an approximate value of about 2.56578, corresponding to option (b). Employing improved Euler's method enhances the approximation, producing a value near 2.71408 (option (c)). These approximations illustrate how numerical techniques approach the exponential function.

Estimating e Numerically

The calculation of e via its differential equation y' = y, with y(0)=1, effectively computes e^x at x=1. Using Euler's method with 10 steps (h=0.1) yields an approximation close to 1.94872 (option (b)), while improved Euler's method with the same number of steps produces an approximation near 2.71408 (option (c)). These results attest to the increased accuracy of the improved Euler method in estimating exponential functions over the basic Euler approach.

Significance of Step Size in Numerical Methods

Choosing the step size h influences the accuracy: smaller step sizes generally yield more precise results, as errors decrease with h. Increasing step size, however, tends to increase the truncation error in numerical methods like Euler's, typically decreasing accuracy. The statement that increasing step size increases accuracy is false; smaller Step sizes are essential for higher precision.

Conclusion

Euler's and improved Euler’s methods are robust tools for approximating solutions to differential equations, with improved Euler’s method generally providing higher accuracy, especially over larger intervals or when higher precision is needed. The choice of step size directly impacts the accuracy, emphasizing the importance of selecting appropriately small steps for sensitive calculations. Numerical analysis confirms that reducing step size enhances the fidelity of approximations of exponential functions and decay processes, reflecting practical considerations in computational modeling.

References

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