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Chapter 2: Problem 121

Why do we often utilize simplifying assumptions when we derive differential equations?

Short Answer

Expert verified
In conclusion, explain why simplifying assumptions are often utilized when deriving differential equations. Simplifying assumptions are often utilized when deriving differential equations because they reduce complexity, enhance mathematical tractability, separate important relationships from noise, and allow for easier interpretation of results. By simplifying the model or problem, researchers can analyze and understand the real-world phenomenon more effectively while still maintaining an accurate representation. These assumptions are crucial for successful problem-solving and interpretation in various fields, such as population growth modeling, vibration analysis, and temperature modeling.

Step by step solution

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1. Introduction to Simplifying Assumptions

Simplifying assumptions are statements that make a particular mathematical model or problem easier to understand and work with. These assumptions usually involve reducing the number of variables or simplifying the relationships between variables. This results in a less complicated problem that remains an accurate representation of the real-world phenomenon.
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2. Reasons for Utilizing Simplifying Assumptions

There are several reasons why simplifying assumptions are frequently made while deriving differential equations: a) Complexity reduction: Most real-world problems involve countless interacting variables and factors, making it difficult to model them realistically. Simplifying assumptions help reduce the complexity of these models, making them more manageable to work with. b) Mathematical tractability: Many real-world problems are mathematically intractable, meaning they cannot be solved using known analytical methods. Simplifying assumptions can make these problems tractable, thereby allowing us to analyze them using familiar mathematical tools and techniques. c) Reduction of noise: Real-world data often contains a considerable amount of noise and random fluctuations. Simplifying assumptions can help separate the essential underlying patterns from the noise and fluctuations, aiding in identifying important relationships and dependencies. d) Easier interpretation: Simplified models typically have fewer variables and relationships, which tends to make the interpretation of their results and implications more straightforward for researchers and practitioners.
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3. Examples of Simplifying Assumptions in Deriving Differential Equations

Some examples of simplifying assumptions when deriving differential equations include: a) Population growth models: The logistic growth model assumes that the growth rate of a population is directly proportional to the current population size and the remaining capacity for growth. This assumption greatly simplifies the dynamics of population growth, allowing for easier analysis and interpretation. b) Vibrating string: When analyzing the vibration of a string, it's often assumed that the string's tension remains constant and that the motion of the string is restricted to a single plane. These simplifying assumptions make it possible to derive the wave equation governing the string's motion, which can be solved analytically. c) Newton's law of cooling: When modeling the cooling of an object, it is often assumed that the object's heat capacity remains constant, and that the object is cooling within an environment with constant temperature. These simplifying assumptions make it possible to derive the differential equation for Newton's law of cooling, which can then be solved analytically for the temperature of the object as a function of time. Overall, simplifying assumptions play a crucial role in formulating and solving differential equations. They help to reduce the complexity of the problem, while still maintaining an accurate representation of the real-world phenomenon, making it easier to analyze and understand.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Simplifying Assumptions
When diving into the world of differential equations, the term 'simplifying assumptions' often comes to the forefront. These are deliberate choices made by mathematicians and scientists to make a problem more manageable. By reducing the number of variables or assuming linear relationships, for instance, one can transform an otherwise intricate situation into a more solvable equation.

Simplifying assumptions are not about cutting corners; instead, they're strategic decisions aimed at preserving the essence of a phenomenon while discarding unnecessary complexity. For example, when we look at population growth, it's common to assume a constant birth rate. This assumption simplifies the otherwise daunting task of accounting for every individual birth and death, making the overall pattern of growth clearer and the mathematics behind it more feasible to work with.

It is important to apply these assumptions critically; over-simplification can lead to models that misrepresent reality. Nevertheless, when used judiciously, simplifying assumptions are invaluable tools in developing usable mathematical models and leading to significant advances in our understanding of the world.
Mathematical Tractability
The notion of 'mathematical tractability' is at the heart of any discourse on differential equations. It essentially refers to our ability to deal with equations and problems using the analytical tools and methods at our disposal.

Complex real-world phenomena often lead to equations that are mathematically intractable鈥攊n other words, they can鈥檛 be neatly solved with standard methods or even at all. By utilizing simplifying assumptions, we can often transform these unwieldy problems into ones that are tractable. This opens the door to analysis, allows for more straightforward computational methods, and can enable a clearer understanding of the underlying patterns and principles.

The bottom line is that mathematical tractability is akin to making a problem accessible, something we can interact with logically and systematically. Without it, we'd be left with a collection of equations as enigmatic as the phenomena they intend to describe.
Complexity Reduction
Reducing complexity is a critical step in deriving differential equations. Real-world systems are often full of interactions and complications that can be bewildering in their entirety. Complexity reduction allows us to peel back these layers to reveal a more fundamental structure that we can analyze with the mathematical tools we have.

By stripping down a system to its core components, we not only gain computational ease but also an enhanced understanding. For instance, when we assume a constant rate of reaction in chemical kinetics, we avoid the entanglement of fluctuating temperatures and varying reactant concentrations. This leads to more streamlined equations, which can be more readily solved and interpreted.

In essence, complexity reduction is about finding the balance between a model鈥檚 simplicity and its fidelity to reality. The goal is to retain a model's ability to predict and explain while eschewing the inessential details that would make the problem intractable. This skill is one of the most powerful in a scientist or engineer's toolkit, as it paves the way for insights and solutions that would otherwise remain obscured by an overwhelming web of variables and interactions.

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Most popular questions from this chapter

In subsea oil and natural gas production, hydrocarbon fluids may leave the reservoir with a temperature of \(70^{\circ} \mathrm{C}\) and flow in subsea surrounding of \(5^{\circ} \mathrm{C}\). As a result of the temperature difference between the reservoir and the subsea surrounding, the knowledge of heat transfer is critical to prevent gas hydrate and wax deposition blockages. Consider a subsea pipeline with inner diameter of \(0.5 \mathrm{~m}\) and wall thickness of \(8 \mathrm{~mm}\) is used for transporting liquid hydrocarbon at an average temperature of \(70^{\circ} \mathrm{C}\), and the average convection heat transfer coefficient on the inner pipeline surface is estimated to be \(250 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The subsea surrounding has a temperature of \(5^{\circ} \mathrm{C}\) and the average convection heat transfer coefficient on the outer pipeline surface is estimated to be \(150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). If the pipeline is made of material with thermal conductivity of \(60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), by using the heat conduction equation (a) obtain the temperature variation in the pipeline wall, \((b)\) determine the inner surface temperature of the pipeline, \((c)\) obtain the mathematical expression for the rate of heat loss from the liquid hydrocarbon in the pipeline, and \((d)\) determine the heat flux through the outer pipeline surface.

What is the difference between an ordinary differential equation and a partial differential equation?

A circular metal pipe has a wall thickness of \(10 \mathrm{~mm}\) and an inner diameter of \(10 \mathrm{~cm}\). The pipe's outer surface is subjected to a uniform heat flux of \(5 \mathrm{~kW} / \mathrm{m}^{2}\) and has a temperature of \(500^{\circ} \mathrm{C}\). The metal pipe has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where \(k_{0}=7.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\beta=0.0012 \mathrm{~K}^{-1}\), and \(T\) is in \(\mathrm{K}\). Determine the inner surface temperature of the pipe.

A spherical vessel is filled with chemicals undergoing an exothermic reaction. The reaction provides a uniform heat flux on the inner surface of the vessel. The inner diameter of the vessel is \(5 \mathrm{~m}\) and its inner surface temperature is at \(120^{\circ} \mathrm{C}\). The wall of the vessel has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where \(k_{0}=1.01 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\beta=0.0018 \mathrm{~K}^{-1}\), and \(T\) is in \(\mathrm{K}\). The vessel is situated in a surrounding with an ambient temperature of \(15^{\circ} \mathrm{C}\), the vessel's outer surface experiences convection heat transfer with a coefficient of \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). To prevent thermal burn on skin tissues, the outer surface temperature of the vessel should be kept below \(50^{\circ} \mathrm{C}\). Determine the minimum wall thickness of the vessel so that the outer surface temperature is \(50^{\circ} \mathrm{C}\) or lower.

Heat is generated uniformly at a rate of \(4.2 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}\) in a spherical ball \((k=45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) of diameter \(24 \mathrm{~cm}\). The ball is exposed to iced-water at \(0^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(1200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Determine the temperatures at the center and the surface of the ball.
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