analytic solution

简明释义

1. 解析解; 2. 分析解;

英英释义

例句

1.In physics, we often seek an analytic solution 解析解 to understand the behavior of systems.

在物理学中，我们常常寻求一个analytic solution 解析解来理解系统的行为。

2.The mathematician was thrilled to find an analytic solution 解析解 for the complex equation.

数学家很高兴找到这个复杂方程的analytic solution 解析解。

3.Finding an analytic solution 解析解 is often more satisfying than a numerical approximation.

找到一个analytic solution 解析解往往比数值近似更令人满意。

4.The software can only provide numerical results, but the researchers aimed for an analytic solution 解析解.

该软件只能提供数值结果，但研究人员希望得到一个analytic solution 解析解。

5.The engineer was proud to present an analytic solution 解析解 that simplified the design process.

工程师自豪地展示了一个简化设计过程的analytic solution 解析解。

作文

In the field of mathematics and science, the term analytic solution refers to a solution that can be expressed in a closed form, typically using well-defined mathematical expressions. This is in contrast to numerical solutions, which approximate the answer through computational methods. The importance of analytic solution lies in its ability to provide exact answers to complex problems, thereby offering deeper insights into the behavior of mathematical models. For instance, consider the simple equation of a line, y = mx + b. The analytic solution provides the exact values of y for any given x, which can be graphically represented to illustrate the relationship between these variables.However, not all mathematical problems can be solved analytically. In many cases, especially with non-linear differential equations or systems with multiple variables, finding an analytic solution may be impossible. In such scenarios, researchers often resort to numerical methods to approximate the solution. While these methods are valuable, they do not offer the same level of precision as analytic solutions. One of the most significant advantages of having an analytic solution is that it allows for a comprehensive understanding of the system being studied. For example, in physics, the motion of a projectile can be described by an analytic solution derived from Newton's laws of motion. By determining the projectile's trajectory through equations, scientists can predict its path, maximum height, and time of flight without needing extensive calculations for each scenario.Furthermore, analytic solutions are often easier to manipulate mathematically. They allow for the application of various mathematical techniques, including differentiation and integration, which are essential for further analysis. For instance, when studying dynamic systems, an analytic solution can lead to insights about stability and equilibrium points, which are crucial for understanding the long-term behavior of the system.Despite the advantages of analytic solutions, their limitations must be acknowledged. Many real-world problems involve complexities that cannot be captured by simple equations. In such cases, mathematicians and scientists must rely on approximations and numerical simulations to understand the system's behavior. This interplay between analytic solutions and numerical methods highlights the richness of mathematical inquiry and the necessity of both approaches in solving complex problems.In conclusion, the concept of analytic solution is fundamental in mathematics and science as it provides exact answers to various problems. While not all problems can be solved analytically, the insights gained from analytic solutions are invaluable for advancing knowledge in numerous fields. As technology continues to evolve, the integration of analytic solutions and numerical methods will remain essential for tackling the increasingly complex challenges faced by researchers today.

在数学和科学领域，术语解析解指的是可以用封闭形式表达的解，通常使用明确定义的数学表达式。这与数值解形成对比，后者通过计算方法近似答案。解析解的重要性在于它能够为复杂问题提供精确答案，从而深入洞察数学模型的行为。例如，考虑简单的直线方程y = mx + b。解析解为任何给定的x提供y的确切值，这可以通过图形表示来说明这些变量之间的关系。然而，并非所有数学问题都可以解析求解。在许多情况下，尤其是对于非线性微分方程或具有多个变量的系统，寻找解析解可能是不可能的。在这种情况下，研究人员通常依赖数值方法来近似解。虽然这些方法很有价值，但它们并不提供与解析解相同的精确度。拥有解析解的一个显著优势是，它允许对所研究的系统进行全面理解。例如，在物理学中，抛体的运动可以通过从牛顿运动定律推导出的解析解来描述。通过方程确定抛体的轨迹，科学家可以预测其路径、最大高度和飞行时间，而无需对每种情况进行广泛的计算。此外，解析解往往更容易在数学上操作。它们允许应用各种数学技术，包括微分和积分，这对进一步分析至关重要。例如，在研究动态系统时，解析解可以提供有关稳定性和平衡点的见解，这对理解系统的长期行为至关重要。尽管解析解具有优势，但必须承认其局限性。许多现实世界的问题涉及的复杂性无法通过简单方程捕捉。在这种情况下，数学家和科学家必须依赖近似和数值模拟来理解系统的行为。这种解析解与数值方法之间的相互作用突显了数学探究的丰富性，以及在解决复杂问题时两种方法的必要性。总之，解析解的概念在数学和科学中是基础，因为它为各种问题提供了确切的答案。虽然并非所有问题都可以解析求解，但从解析解中获得的见解对于推动多个领域的知识发展是无价的。随着科技的不断发展，解析解与数值方法的结合将继续在应对当今研究人员面临的日益复杂的挑战中发挥重要作用。