具有模糊边界值的线性微分方程
Linear Differential Equations with Fuzzy Boundary Values
AI总结：
针对具有模糊边界值的线性微分方程，提出一种基于线性变换性质的方法求解模糊解，并证明解的唯一性及三角模糊数边界下解仍为三角模糊数。
摘要：
在本研究中，我们考虑具有模糊边界值的线性微分方程。我们将问题的解表示为一组清晰实函数的模糊集。解集中的每个实函数满足微分方程，其边界值属于由相应模糊数确定的区间。边界值在相应模糊集中的最小可能性被定义为该实函数在模糊解中的可能性。为了找到模糊解，我们提出了一种基于线性变换性质的方法。我们证明，如果相应的清晰问题有唯一解，那么模糊问题也有唯一解。我们还证明，如果边界值是三角模糊数，则任意时刻的解也是三角模糊数。我们发现，通过我们的方法确定的模糊解与通过扩展原理从清晰问题的解中获得的解相同。我们给出两个例子来描述所提出的方法。
英文摘要：
In this study, we consider a linear differential equation with fuzzy boundary values. We express the solution of the problem in terms of a fuzzy set of crisp real functions. Each real function from the solution set satisfies differential equation, and its boundary values belong to intervals, determined by the corresponding fuzzy numbers. The least possibility among possibilities of boundary values in corresponding fuzzy sets is defined as the possibility of the real function in the fuzzy solution. In order to find the fuzzy solution we propose a method based on the properties of linear transformations. We show that, if the corresponding crisp problem has a unique solution then the fuzzy problem has unique solution too. We also prove that if the boundary values are triangular fuzzy numbers, then the value of the solution at any time is also a triangular fuzzy number. We find that the fuzzy solution determined by our method is the same as the one that is obtained from solution of crisp problem by the application of the extension principle. We present two examples describing the proposed method.
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