偏微分方程在物理学中的完美应用——热方程,推导和示例
作者:老胡
偏微分方程是一个将具有一个以上变量的函数与其偏导数联系起来的方程。为了引入偏微分方程,我们要解决一个简单的问题:模拟薄金属棒内的温度作为位置和时间的函数。在此过程中,我们将从物理原理推导出一维热方程,并求解一些简单的条件:
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在这个方程中,温度T是位置x和时间t的函数,k、ρ和c分别是金属的热导率、密度和比热容,
称为扩散系数。
我们想要研究,随着时间的增加,热量如何在长为L的金属棒中传导的。
金属棒的一端在
处,另一端在
处。金属棒的长度远大于它的截面半径,所以我们可以把热传导看成是x和t的函数。
假设金属棒的比热容是已知的,如果我们能找到温度
的函数,我们就能知道热量是如何扩散的。
假设棒沿其长度方向是绝热的,因此它只能通过两端吸收或散发热量。这意味着温度分布只取决于以下三个因素:
●初始温度分布情况
,这叫做初始条件。
●金属棒两端的温度,
和
这些叫做边界条件。
●热量在金属棒内由一点传递到另一点的规律。热方程是这种物理定律的数学表示。
对于一组特定的初始和边界条件,求解偏微分方程的问题被称为初始边值问题
。
在本文中,我们将求解的热方程的初始边值为
。这些叫做齐次边界条件。
热方程可以从能量守恒导出:金属杆上某一点储存的热量的时间变化率等于进入该点的净热量流量。这个过程显然符合连续性方程。如果Q是各点处的热量,V是热量流动的矢量场,则:
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根据热力学第二定律,如果两个相同的物体进行热接触,其中一个比另一个热,那么热量必然以与温度差成比例的速度从较热的物体流向较冷的物体。因此,V与温度的负梯度成正比,所以
,其中k为金属的导热系数。在一维中,它简化为
,其中
是
方向的单位向量。![](data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2220%22%20width%3D%2259%22%20wrs%3Abaseline%3D%2216%22%3E%3C%21--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3EQ%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmi%3E%26%23x3C1%3B%3C%2Fmi%3E%3Cmi%3Ec%3C%2Fmi%3E%3Cmi%3ET%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A%27math17f39f8317fbdb1988ef4c628eb%27%3Bsrc%3Aurl%28data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADRjdnQgDVUNBwAAAVAAAAA6Z2x5ZoPi2VsAAAGMAAAAsmhlYWQQC2qxAAACQAAAADZoaGVhCGsXSAAAAngAAAAkaG10eE2rRkcAAAKcAAAACGxvY2EAHTwYAAACpAAAAAxtYXhwBT0FPgAAArAAAAAgbmFtZaBxlY4AAALQAAABn3Bvc3QB9wD6AAAEcAAAACBwcmVwa1uragAABJAAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACAAAAAEAAQAAQAAAD3%2F%2FwAAAD3%2F%2F%2F%2FEAAEAAAAAAAABVAMsAIABAABWACoCWAIeAQ4BLAIsAFoBgAKAAKAA1ACAAAAAAAAAACsAVQCAAKsA1QEAASsABwAAAAIAVQAAAwADqwADAAcAADMRIRElIREhVQKr%2FasCAP4AA6v8VVUDAAACAIAA6wLVAhUAAwAHAGUYAbAIELAG1LAGELAF1LAIELAB1LABELAA1LAGELAHPLAFELAEPLABELACPLAAELADPACwCBCwBtSwBhCwB9SwBxCwAdSwARCwAtSwBhCwBTywBxCwBDywARCwADywAhCwAzwxMBMhNSEdASE1gAJV%2FasCVQHAVdVVVQAAAAEAAAABAADVeM5BXw889QADBAD%2F%2F%2F%2F%2F1joTc%2F%2F%2F%2F%2F%2FWOhNzAAD%2FIASAA6sAAAAKAAIAAQAAAAAAAQAAA%2Bj%2FagAAF3AAAP%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%2F%29format%28%27truetype%27%29%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%226.5%22%20y%3D%2216%22%3EQ%3C%2Ftext%3E%3Ctext%20font-family%3D%22math17f39f8317fbdb1988ef4c628eb%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2221.5%22%20y%3D%2216%22%3E%3D%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2234.5%22%20y%3D%2216%22%3E%26%23x3C1%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2243.5%22%20y%3D%2216%22%3Ec%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2252.5%22%20y%3D%2216%22%3ET%3C%2Ftext%3E%3C%2Fsvg%3E)
,代入V和Q的表达式,得到热方程:
,
。
在我们进一步讨论之前,我们需要证明对于任何有物理意义的初始和边界条件,热方程必须存在一个唯一的解。对此的正式证明超出了本文的范围,因此我们将使用一个经验论证。
热力学定律告诉我们,无论一开始金属棒的温度分布是怎样的,系统必须经历一个过程,使金属棒达到热平衡,我们在前面讲过这个过程必须服从热方程,因此,对于有物理意义的初始和边界条件,热方程的解是存在的。此外,经典物理学的基本假设之一是,相同的实验条件必然会导致相同的结果,因此,金属棒进入热平衡的特定方式,由初始条件和边界条件所唯一规定。
这意味着,对于热方程,如果
和
是两个不同的函数且满足相同的
,那么
和
有相同的形式。此外,热方程是线性的,因此如果
和
是解,
和
是任何实数,那么
也是一个解。所以我们可以得出结论,解是相同形式的函数的线性组合。
考虑下面的函数,我们可以通过试错来推测:
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其中n是大于0的正整数。该??数满足热方程:
。
,
。
这个函数也满足边界条件,因为
。因此通解为:
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如果我们能找到系数
,使这个通解满足初始条件,问题就解决了。也就是说,我们需要找到一个
,这样:
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这叫做初始条件下的傅里叶正??级数展开式。系数
叫做傅里叶系数。
初始条件![](data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2221%22%20width%3D%2245%22%20wrs%3Abaseline%3D%2216%22%3E%3C%21--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmi%3ET%3C%2Fmi%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A%27math1f7177163c833dff4b38fc8d287%27%3Bsrc%3Aurl%28data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAADRjdnQgDVUNBwAAAVAAAAA6Z2x5ZoPi2VsAAAGMAAAAUmhlYWQQC2qxAAAB4AAAADZoaGVhCGsXSAAAAhgAAAAkaG10eE2rRkcAAAI8AAAACGxvY2EAHTwYAAACRAAAAAxtYXhwBT0FPgAAAlAAAAAgbmFtZaBxlY4AAAJwAAABn3Bvc3QB9wD6AAAEEAAAACBwcmVwa1uragAABDAAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACAAAAAEAAQAAQAAACz%2F%2FwAAACz%2F%2F%2F%2FVAAEAAAAAAAABVAMsAIABAABWACoCWAIeAQ4BLAIsAFoBgAKAAKAA1ACAAAAAAAAAACsAVQCAAKsA1QEAASsABwAAAAIAVQAAAwADqwADAAcAADMRIRElIREhVQKr%2FasCAP4AA6v8VVUDAAABAFX%2FZADVAIAACgAAMzUzFRQGByc%2BATdVgC8vGx4eAYB6PVEUKQ40MQAAAAEAAAABAADVeM5BXw889QADBAD%2F%2F%2F%2F%2F1joTc%2F%2F%2F%2F%2F%2FWOhNzAAD%2FIASAA6sAAAAKAAIAAQAAAAAAAQAAA%2Bj%2FagAAF3AAAP%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%2F%29format%28%27truetype%27%29%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A%27round_brackets18549f92a457f2409%27%3Bsrc%3Aurl%28data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMjwHLFQAAADMAAAATmNtYXDf7xCrAAABHAAAADxjdnQgBAkDLgAAAVgAAAASZ2x5ZmAOz2cAAAFsAAABJGhlYWQOKih8AAACkAAAADZoaGVhCvgVwgAAAsgAAAAkaG10eCA6AAIAAALsAAAADGxvY2EAAARLAAAC%2BAAAABBtYXhwBIgEWQAAAwgAAAAgbmFtZXHR30MAAAMoAAACOXBvc3QDogHPAAAFZAAAACBwcmVwupWEAAAABYQAAAAHAAAGcgGQAAUAAAgACAAAAAAACAAIAAAAAAAAAQIAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAo8AMGe%2F57AAAHPgGyAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACACgAKf%2F%2FAAAAKAAp%2F%2F%2F%2F2f%2FZAAEAAAAAAAAAAAFUAFYBAAAsAKgDgAAyAAcAAAACAAAAKgDVA1UAAwAHAAA1MxEjEyMRM9XVq4CAKgMr%2FQAC1QABAAD%2B0AIgBtAACQBNGAGwChCwA9SwAxCwAtSwChCwBdSwBRCwANSwAxCwBzywAhCwCDwAsAoQsAPUsAMQsAfUsAoQsAXUsAoQsADUsAMQsAI8sAcQsAg8MTAREAEzABEQASMAAZCQ%2FnABkJD%2BcALQ%2FZD%2BcAGQAnACcAGQ%2FnAAAQAA%2FtACIAbQAAkATRgBsAoQsAPUsAMQsALUsAoQsAXUsAUQsADUsAMQsAc8sAIQsAg8ALAKELAD1LADELAH1LAKELAF1LAKELAA1LADELACPLAHELAIPDEwARABIwAREAEzAAIg%2FnCQAZD%2BcJABkALQ%2FZD%2BcAGQAnACcAGQ%2FnAAAQAAAAEAAPW2NYFfDzz1AAMIAP%2F%2F%2F%2F%2FVre7u%2F%2F%2F%2F%2F9Wt7u4AAP7QA7cG0AAAAAoAAgABAAAAAAABAAAHPv5OAAAXcAAA%2F%2F4DtwABAAAAAAAAAAAAAAAAAAAAAwDVAAACIAAAAiAAAAAAAAAAAAAkAAAAowAAASQAAQAAAAMACgACAAAAAAACAIAEAAAAAAAEAABNAAAAAAAAABUBAgAAAAAAAAABAD4AAAAAAAAAAAACAA4APgAAAAAAAAADAFwATAAAAAAAAAAEAD4AqAAAAAAAAAAFABYA5gAAAAAAAAAGAB8A%2FAAAAAAAAAAIABwBGwABAAAAAAABAD4AAAABAAAAAAACAA4APgABAAAAAAADAFwATAABAAAAAAAEAD4AqAABAAAAAAAFABYA5gABAAAAAAAGAB8A%2FAABAAAAAAAIABwBGwADAAEECQABAD4AAAADAAEECQACAA4APgADAAEECQADAFwATAADAAEECQAEAD4AqAADAAEECQAFABYA5gADAAEECQAGAB8A%2FAADAAEECQAIABwBGwBSAG8AdQBuAGQAIABiAHIAYQBjAGsAZQB0AHMAIAB3AGkAdABoACAAYQBzAGMAZQBuAHQAIAAxADgANQA0AFIAZQBnAHUAbABhAHIATQBhAHQAaABzACAARgBvAHIAIABNAG8AcgBlACAAUgBvAHUAbgBkACAAYgByAGEAYwBrAGUAdABzACAAdwBpAHQAaAAgAGEAcwBjAGUAbgB0ACAAMQA4ADUANABSAG8AdQBuAGQAIABiAHIAYQBjAGsAZQB0AHMAIAB3AGkAdABoACAAYQBzAGMAZQBuAHQAIAAxADgANQA0AFYAZQByAHMAaQBvAG4AIAAyAC4AMFJvdW5kX2JyYWNrZXRzX3dpdGhfYXNjZW50XzE4NTQATQBhAHQAaABzACAARgBvAHIAIABNAG8AcgBlAAAAAAMAAAAAAAADnwHPAAAAAAAAAAAAAAAAAAAAAAAAAAC5B%2F8AAY2FAA%3D%3D%29format%28%27truetype%27%29%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%225.5%22%20y%3D%2216%22%3ET%3C%2Ftext%3E%3Ctext%20font-family%3D%22round_brackets18549f92a457f2409%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%2214.5%22%20y%3D%2216%22%3E%28%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%2220.5%22%20y%3D%2216%22%3Ex%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1f7177163c833dff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是区间
上的分段连续函数,且在边界处为零。结果证明具有这些性质的函数集合是加法和标量乘法下的向量空间。我们称这个向量空间为:
。
这个向量空间有一个内积。对于
,一个可能的内积为:
![](data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2248%22%20width%3D%22190%22%20wrs%3Abaseline%3D%2229%22%3E%3C%21--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3E%26%23x27E8%3B%3C%2Fmo%3E%3Cmi%3Ef%3C%2Fmi%3E%3Cmo%3E%2C%3C%2Fmo%3E%3Cmi%3Eg%3C%2Fmi%3E%3Cmo%3E%26%23x27E9%3B%3C%2Fmo%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfrac%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi%3EL%3C%2Fmi%3E%3C%2Fmfrac%3E%3Cmsubsup%3E%3Cmo%3E%26%23x222B%3B%3C%2Fmo%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmi%3EL%3C%2Fmi%3E%3C%2Fmsubsup%3E%3Cmi%3Ef%3C%2Fmi%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%29%3C%2Fmo%3E%3Cmo%20mathvariant%3D%22bold%22%3E%26%23xB7%3B%3C%2Fmo%3E%3Cmi%3Eg%3C%2Fmi%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%29%3C%2Fmo%3E%3Cmi%3Ed%3C%2Fmi%3E%3Cmi%3Ex%3C%2Fmi%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A%27aec8956637a99787bd197eacd77acce%27%3Bsrc%3Aurl%28data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMjv%2FLJYAAADMAAAATmNtYXDgWxEdAAABHAAAADRjdnQgAAAABwAAAVAAAAAEZ2x5ZoYrxVAAAAFUAAAAsmhlYWQOdyayAAACCAAAADZoaGVhC0UVwQAAAkAAAAAkaG10eCg8AIUAAAJkAAAACGxvY2EAAAVKAAACbAAAAAxtYXhwBIoEWwAAAngAAAAgbmFtZXSF9ZsAAAKYAAABrXBvc3QDogHPAAAESAAAACBwcmVwukanGAAABGgAAAANAAAGtAGQAAUAAAgACAAAAAAACAAIAAAAAAAAAQIAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg8AMGe%2F57AAAHPgGyAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACAAAAAEAAQAAQAAAGb%2F%2FwAAAGb%2F%2F%2F%2BbAAEAAAAAAAAABwACAFUAAAMAA6sAAwAHAAAzESERJSERIVUCq%2F2rAgD%2BAAOr%2FFVVAwAAAQAB%2FvUC9QZDABoANRgBsBsQsAjUsAgQsAHUsBsQsAzUsAwQsBjUALAbELAK1LAKELAL1LAbELAA1LAAELAY1DAxAREQIyInNRY3ESM3FzUQITIXFSYjIgYdASEVAWXfWSyxAYYDgwFkWYVZWVmFAQsDp%2FxZ%2FvUshlmyA6eHAYUBkVmGWVmyhYYAAAABAAAAAQAAmr%2FbrF8PPPUAAwgA%2F%2F%2F%2F%2F9Wt7j3%2F%2F%2F%2F%2F1a3uPQAB%2FvUEAwZDAAAACgACAAEAAAAAAAEAAAc%2B%2Fk4AABdwAAH%2F%2FAQDAAEAAAAAAAAAAAAAAAAAAAACA1IAVQLxAAEAAAAAAAAAKAAAALIAAQAAAAIAHgADAAAAAAACAIAEAAAAAAAEAAA7AAAAAAAAABUBAgAAAAAAAAABABYAAAAAAAAAAAACAA4AFgAAAAAAAAADADQAJAAAAAAAAAAEABYAWAAAAAAAAAAFABYAbgAAAAAAAAAGAAsAhAAAAAAAAAAIABwAjwABAAAAAAABABYAAAABAAAAAAACAA4AFgABAAAAAAADADQAJAABAAAAAAAEABYAWAABAAAAAAAFABYAbgABAAAAAAAGAAsAhAABAAAAAAAIABwAjwADAAEECQABABYAAAADAAEECQACAA4AFgADAAEECQADADQAJAADAAEECQAEABYAWAADAAEECQAFABYAbgADAAEECQAGAAsAhAADAAEECQAIABwAjwBNAGEAdABoACAARgBvAG4AdAAgADIAUgBlAGcAdQBsAGEAcgBNAGEAdABoAHMAIABGAG8AcgAgAE0AbwByAGUAIABNAGEAdABoACAARgBvAG4AdAAgADIATQBhAHQAaAAgAEYAbwBuAHQAIAAyAFYAZQByAHMAaQBvAG4AIAAxAC4AME1hdGhfRm9udF8yAE0AYQB0AGgAcwAgAEYAbwByACAATQBvAHIAZQAAAAADAAAAAAAAA58BzwAAAAAAAAAAAAAAAAAAAAAAAAAAuQciAACNhRgAsgAAAAAAAA%3D%3D%29format%28%27truetype%27%29%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A%27math1e3f8073e97afd084a6a0060de7%27%3Bsrc%3Aurl%28data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAAExjdnQgDVUNBwAAAWgAAAA6Z2x5ZoPi2VsAAAGkAAABbWhlYWQQC2qxAAADFAAAADZoaGVhCGsXSAAAA0wAAAAkaG10eE2rRkcAAANwAAAAFGxvY2EAHTwYAAADhAAAABhtYXhwBT0FPgAAA5wAAAAgbmFtZaBxlY4AAAO8AAABn3Bvc3QB9wD6AAAFXAAAACBwcmVwa1uragAABXwAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADgAAAAKAAgAAgACACwAPQC3Iiv%2F%2FwAAACwAPQC3Iiv%2F%2F%2F%2FV%2F8X%2FTN3ZAAEAAAAAAAAAAAAAAAABVAMsAIABAABWACoCWAIeAQ4BLAIsAFoBgAKAAKAA1ACAAAAAAAAAACsAVQCAAKsA1QEAASsABwAAAAIAVQAAAwADqwADAAcAADMRIRElIREhVQKr%2FasCAP4AA6v8VVUDAAABAFX%2FZADVAIAACgAAMzUzFRQGByc%2BATdVgC8vGx4eAYB6PVEUKQ40MQACAIAA6wLVAhUAAwAHAGUYAbAIELAG1LAGELAF1LAIELAB1LABELAA1LAGELAHPLAFELAEPLABELACPLAAELADPACwCBCwBtSwBhCwB9SwBxCwAdSwARCwAtSwBhCwBTywBxCwBDywARCwADywAhCwAzwxMBMhNSEdASE1gAJV%2FasCVQHAVdVVVQABAIABVQDrAcAAAwAbGAGwBBCxAAP0sQID9LEFA%2FQAsAQQsQMG9DAxEzMVI4BrawHAawABAEv%2FawILA0AAGQAAASYOAQMCDgEjJjQ2BiMWPgETEj4BMxYOATYB9TU1ICAgQDZAKlULNTU2ICAgQDVAKwFVCwMVFiuL%2FuD%2BwIogFUAVQBUrigEgAUCLIBVAFT8AAAAAAQAAAAEAANV4zkFfDzz1AAMEAP%2F%2F%2F%2F%2FWOhNz%2F%2F%2F%2F%2F9Y6E3MAAP8gBIADqwAAAAoAAgABAAAAAAABAAAD6P9qAAAXcAAA%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%2BgAAAAAAAAAAAAAAAAAAAAAAAAAAuQcRAACNhRgAsgAAABUUE7EAAT8%3D%29format%28%27truetype%27%29%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A%27round_brack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font-family%3D%22round_brackets18549f92a457f2409%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22128.5%22%20y%3D%2229%22%3E%29%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1e3f8073e97afd084a6a0060de7%22%20font-size%3D%2216%22%20font-weight%3D%22bold%22%20text-anchor%3D%22middle%22%20x%3D%22136.5%22%20y%3D%2229%22%3E%26%23xB7%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22146.5%22%20y%3D%2229%22%3Eg%3C%2Ftext%3E%3Ctext%20font-family%3D%22round_brackets18549f92a457f2409%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22155.5%22%20y%3D%2229%22%3E%28%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22161.5%22%20y%3D%2229%22%3Ex%3C%2Ftext%3E%3Ctext%20font-family%3D%22round_brackets18549f92a457f2409%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22168.5%22%20y%3D%2229%22%3E%29%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22175.5%22%20y%3D%2229%22%3Ed%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22184.5%22%20y%3D%2229%22%3Ex%3C%2Ftext%3E%3C%2Fsvg%3E)
我们可以通过使用单位向量的点积将其投影到轴上来找到几何向量的分量,单位向量构成了的基。同样,如果我们能为找到一个基,那么我们可以将任何
投影到基函数上,以便将f表示为基函数的线性组合。
![](http://www.yun-zhang.com/upload/20213111093156692.svg)
●求v的分量
对于整数
,函数
是标准正交的:
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因此,我们可以将任意函数f∈表示为基集中函数的线性组合:
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线性组合的系数由欧拉积分给出:
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作为演示,让我们找到一个单位锯齿脉冲的傅立叶系数:
![](data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2248%22%20width%3D%22169%22%20wrs%3Abaseline%3D%2229%22%3E%3C%21--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3Esaw%3C%2Fmo%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%29%3C%2Fmo%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmfenced%20close%3D%22%22%20open%3D%22%7B%22%3E%3Cmtable%20columnalign%3D%22left%22%3E%3Cmtr%3E%3Cmtd%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%20mathvariant%3D%22italic%22%3E%2C%3C%2Fmo%3E%3C%2Fmtd%3E%3Cmtd%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmo%3E%26%23x2264%3B%3C%2Fmo%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%26lt%3B%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3Cmtr%3E%3Cmtd%3E%3Cmn%3E0%3C%2Fmn%3E%3Cmo%3E%2C%3C%2Fmo%3E%3C%2Fmtd%3E%3Cmtd%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmtd%3E%3C%2Fmtr%3E%3C%2Fmtable%3E%3C%2Fmfenced%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A%27math1595b5ca43595c4d2275abe88ce%27%3Bsrc%3Aurl%28data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAAExjdnQgDVUNBwAAAWgAAAA6Z2x5ZoPi2VsAAAGkAAAB6WhlYWQQC2qxAAADkAAAADZoaGVhCGsXSAAAA8gAAAAkaG10eE2rRkcAAAPsAAAAFGxvY2EAHTwYAAAEAAAAABhtYXhwBT0FPgAABBgAAAAgbmFtZaBxlY4AAAQ4AAABn3Bvc3QB9wD6AAAF2AAAACBwcmVwa1uragAABfgAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADgAAAAKAAgAAgACACwAPAA9ImT%2F%2FwAAACwAPAA9ImT%2F%2F%2F%2FV%2F8b%2Fxt2gAAEAAAAAAAAAAAAAAAABVAMsAIABAABWACoCWAIeAQ4BLAIsAFoBgAKAAKAA1ACAAAAAAAAAACsAVQCAAKsA1QEAASsABwAAAAIAVQAAAwADqwADAAcAADMRIRElIREhVQKr%2FasCAP4AA6v8VVUDAAABAFX%2FZADVAIAACgAAMzUzFRQGByc%2BATdVgC8vGx4eAYB6PVEUKQ40MQACAIAAVQKAAqsABQALAGwYsEEaALEAABMQsQAG5LEAARMQsAk8sQEF9bAKPLAAELEFA%2FWxAgXtsQQD9bEDBe2wChCxCwP1sQgF7bEGA%2FWxBwXtAbAMELEBA%2FayCQAKPDw8sQUE9bICCAs8PDyxBAT1sgYHAzw8PLENA%2BYTBwUXNScBNQ0BFzeBAQEA%2F%2F8BAP8A%2FwAB%2FwGrVoCAVoABKlaAgFaAAAIAgADrAtUCFQADAAcAZRgBsAgQsAbUsAYQsAXUsAgQsAHUsAEQsADUsAYQsAc8sAUQsAQ8sAEQsAI8sAAQsAM8ALAIELAG1LAGELAH1LAHELAB1LABELAC1LAGELAFPLAHELAEPLABELAAPLACELADPDEwEyE1IR0BITWAAlX9qwJVAcBV1VVVAAMAgP%2BrAoACqwADAAcACwAvGAEAsQkAP7EIBeSxAgX0sQEE9LEGBfSxAwTksQUE9LEEBfSxBwEQPLEABhA8MDETBwE1EzUBFwE1IRWBAQH%2FAf4AAQH%2F%2FgABq1b%2FAFYBqlb%2FAFb%2BVlVVAAAAAAEAAAABAADVeM5BXw889QADBAD%2F%2F%2F%2F%2F1joTc%2F%2F%2F%2F%2F%2FWOhNzAAD%2FIASAA6sAAAAKAAIAAQAAAAAAAQAAA%2Bj%2FagAAF3AAAP%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%2F%29format%28%27truetype%27%29%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A%27brack_sm74a4ec520979a73b06ad972%27%3Bsrc%3Aurl%28data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7PH4UAAADMAAAATmNtYXA3kjw6AAABHAAAAExjdnQgAQYDiAAAAWgAAAASZ2x5ZkyYQ7YAAAF8AAABLGhlYWQLyR8fAAACqAAAADZoaGVhAq0XCAAAAuAAAAAkaG10eDEjA%2FUAAAMEAAAAFGxvY2EAAEKZAAADGAAAABhtYXhwBJsEcQAAAzAAAAAgbmFtZW7QvZAAAANQAAAB5XBvc3QArQBVAAAFOAAAACBwcmVwu5WEAAAABVgAAAAHAAACDAGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg9AMD%2FP%2F8AAABVAABAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADgAAAAKAAgAAgACI6cjqCOpI6r%2F%2FwAAI6cjqCOpI6r%2F%2F9xa3FrcWtxaAAEAAAAAAAAAAAAAAAABVABUAQAAKwCMAIAAqAAHAAAAAgAAAAAA1QEBAAMABwAAMTMRIxcjNTPV1auAgAEB1qsAAQAAAAABAAFVAAcAIxgBsQAAP7EHAv2xAwf8sAQ8ALEAAD%2BwBzyxAwH2sQQE%2FTEwMTQ2NxUOARVrlWtAa8AqKivAQAAB%2F4AAAABVAVQACQAhGAGxBQA%2FsAE8sQYC%2FbAAPACxBQA%2FsAY8sQEBP7AAPDEwEyMUBxYVMzQnNlVVgIBVgIABVFVUVlVVVlQAAQAAAAABAAFUAAcAIxgBsQAAP7EHAv2xAwf8sAQ8ALEAAT%2BwBzyxAwH2sQQE%2FTEwERQWFzUuATVrlWtAAVRqwCorKcBAAAEAAAAAAFUBVAADACAYAbAEELAAPLADPLEBAvWwAjwAsAMvsAI8sQAB9bABPBEzESNVVQFU%2FqwAAQAAAAEAAIsesexfDzz1AAMEAP%2F%2F%2F%2F%2FVre5k%2F%2F%2F%2F%2F9Wt7mT%2FgP%2F%2FAdYBWAAAAAoAAgABAAAAAAABAAABVP%2F%2FAAAXcP%2BA%2F4AB1gABAAAAAAAAAAAAAAAAAAAABQDVAAABAAAAAQD%2FgAEAAAABAAAAAAAAAAAAACEAAABlAAAArgAAAPQAAAEsAAEAAAAFAAoAAgAAAAAAAgCABAAAAAAABAAAZQAAAAAAAAAVAQIAAAAAAAAAAQAmAAAAAAAAAAAAAgAOACYAAAAAAAAAAwBEADQAAAAAAAAABAAmAHgAAAAAAAAABQAWAJ4AAAAAAAAABgATALQAAAAAAAAACAAcAMcAAQAAAAAAAQAmAAAAAQAAAAAAAgAOACYAAQAAAAAAAwBEADQAAQAAAAAABAAmAHgAAQAAAAAABQAWAJ4AAQAAAAAABgATALQAAQAAAAAACAAcAMcAAwABBAkAAQAmAAAAAwABBAkAAgAO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![](http://www.yun-zhang.com/upload/20213111093156767.svg)
显然,
,因此:
。
傅里叶系数为:
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因此,锯齿波的傅里叶级数展开为:
![](data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2253%22%20width%3D%22247%22%20wrs%3Abaseline%3D%2233%22%3E%3C%21--MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmo%3Esaw%3C%2Fmo%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%29%3C%2Fmo%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmunderover%3E%3Cmo%3E%26%23x2211%3B%3C%2Fmo%3E%3Cmrow%3E%3Cmi%3En%3C%2Fmi%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3Cmo%3E%26%23x221E%3B%3C%2Fmo%3E%3C%2Fmunderover%3E%3Cmfrac%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmrow%3E%3Cmi%3En%3C%2Fmi%3E%3Cmi%3E%26%23x3C0%3B%3C%2Fmi%3E%3C%2Fmrow%3E%3C%2Fmfrac%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3Cmsup%3E%3Cmo%3E%29%3C%2Fmo%3E%3Cmrow%3E%3Cmi%3En%3C%2Fmi%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmn%3E1%3C%2Fmn%3E%3C%2Fmrow%3E%3C%2Fmsup%3E%3Cmi%3Esin%3C%2Fmi%3E%3Cmo%3E%28%3C%2Fmo%3E%3Cmi%3En%3C%2Fmi%3E%3Cmi%3E%26%23x3C0%3B%3C%2Fmi%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%29%3C%2Fmo%3E%3C%2Fmath%3E--%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A%27math1a3768a30235512d6b1f7ab26d1%27%3Bsrc%3Aurl%28data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAAFxjdnQgDVUNBwAAAXgAAAA6Z2x5ZoPi2VsAAAG0AAADVWhlYWQQC2qxAAAFDAAAADZoaGVhCGsXSAAABUQAAAAkaG10eE2rRkcAAAVoAAAAHGxvY2EAHTwYAAAFhAAAACBtYXhwBT0FPgAABaQAAAAgbmFtZaBxlY4AAAXEAAABn3Bvc3QB9wD6AAAHZAAAACBwcmVwa1uragAAB4QAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEAEgAAAAOAAgAAgAGACsAPQPAIhEiEiIe%2F%2F8AAAArAD0DwCIRIhIiHv%2F%2F%2F9b%2FxfxD3fPd893oAAEAAAAAAAAAAAAAAAAAAAAAAVQDLACAAQAAVgAqAlgCHgEOASwCLABaAYACgACgANQAgAAAAAAAAAArAFUAgACrANUBAAErAAcAAAACAFUAAAMAA6sAAwAHAAAzESERJSERIVUCq%2F2rAgD%2BAAOr%2FFVVAwAAAQCAAFUC1QKrAAsASQEYsgwBARQTELEAA%2FaxAQT1sAo8sQMF9bAIPLEFBPWwBjyxDQPmALEAABMQsQEG5LEBARMQsAU8sQME5bELBfWwBzyxCQTlMTATIREzESEVIREjESGAAQBVAQD%2FAFX%2FAAGrAQD%2FAFb%2FAAEAAAIAgADrAtUCFQADAAcAZRgBsAgQsAbUsAYQsAXUsAgQsAHUsAEQsADUsAYQsAc8sAUQsAQ8sAEQsAI8sAAQsAM8ALAIELAG1LAGELAH1LAHELAB1LABELAC1LAGELAFPLAHELAEPLABELAAPLACELADPDEwEyE1IR0BITWAAlX9qwJVAcBV1VVVAAEAVQAAAsACQAAZAEEYAbAaELAN1LANELAH1LAHELAE1LAEELAY1ACwGhCwC9SwGhCwAtSwGhCwD9SwDxCwFNSwDxCwBjywDxCwFzwwMSUGIyI1AyMRFAYrATY1NCciFSM0MyEVIxMUAsAVQIABqikBVSoBVUBqAgGBAVVV1QEW%2FuqVQECVgJYrgFX%2B6sAAAQCAAAADKwMrAA8AShgBsBAQsQQD9rAAPLEJA%2FWwCzyxBgv1sA48sRED5rEHBv2wDTwAsQQAP7EIBf2xAwk8PLEGBe2xAAI%2FsQwF%2FbELATw8sQ4F%2FTEwExUJARUhNSMVIQkBIRUzNYABK%2F7VAqsr%2FgABK%2F7VAgArAytV%2Fv%2F%2BgFWrVgGAAQFWqwABAIABVQLVAasAAwAwGAGwBBCxAAP2sAM8sQIH9bABPLEFA%2BYAsQAAExCxAAblsQABExCwATyxAwX1sAI8EyEVIYACVf2rAatWAAMAAACEA9ACPAAjADMAQwAAATY3NjMyFxYVFAcGIyInJicmJwYHBiMiJyY1NDc2MzIXFhcWFxYXFjMyNzY1NCcmIyIHBgcmJyYjIgcGFRQXFjMyNzYB90djLC5kPzJFPVl8WhENBiFHZCwuZD8yRT1ZfFoQDQdJVig3O0wxIz0uO3RdDndWKDc7TDEjPS47dF0NAaxeIhBPPVBfQzpJDBAGJV0jEE89UF9DOkgODgZWZCArOCg0TzUncA5KZR8sOCg1TzQocA4AAAAAAQAAAAEAANV4zkFfDzz1AAMEAP%2F%2F%2F%2F%2FWOhNz%2F%2F%2F%2F%2F9Y6E3MAAP8gBIADqwAAAAoAAgABAAAAAAABAAAD6P9qAAAXcAAA%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%2F%29format%28%27truetype%27%29%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%40font-face%7Bfont-family%3A%27round_brackets18549f92a457f2409%27%3Bsrc%3Aurl%28data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMjwHLFQAAADMAAAATmNtYXDf7xCrAAABHAAAADxjdnQgBAkDLgAAAVgAAAASZ2x5ZmAOz2cAAAFsAAABJGhlYWQOKih8AAACkAAAADZoaGVhCvgVwgAAAsgAAAAkaG10eCA6AAIAAALsAAAADGxvY2EAAARLAAAC%2BAAAABBtYXhwBIgEWQAAAwgAAAAgbmFtZXHR30MAAAMoAAACOXBvc3QDogHPAAAFZAAAACBwcmVwupWEAAAABYQAAAAHAAAGcgGQAAUAAAgACAAAAAAACAAIAAAAAAAAAQIAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAo8AMGe%2F57AAAHPgGyAAAAAAACAAEAAQAAABQAAwABAAAAFAAEACgAAAAGAAQAAQACACgAKf%2F%2FAAAAKAAp%2F%2F%2F%2F2f%2FZAAEAAAAAAAAAAAFUAFYBAAAsAKgDgAAyAAcAAAACAAAAKgDVA1UAAwAHAAA1MxEjEyMRM9XVq4CAKgMr%2FQAC1QABAAD%2B0AIgBtAACQBNGAGwChCwA9SwAxCwAtSwChCwBdSwBRCwANSwAxCwBzywAhCwCDwAsAoQsAPUsAMQsAfUsAoQsAXUsAoQsADUsAMQsAI8sAcQsAg8MTAREAEzABEQASMAAZCQ%2FnABkJD%2BcALQ%2FZD%2BcAGQAnACcAGQ%2FnAAAQAA%2FtACIAbQAAkATRgBsAoQsAPUsAMQsALUsAoQsAXUsAUQsADUsAMQsAc8sAIQsAg8ALAKELAD1LADELAH1LAKELAF1LAKELAA1LAD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下面的动图展示了,随着
项的增加,傅里叶级数如何接近锯齿状(波形)的。
![770cedccbacb34c16ed358b0e3fa6143.gif 770cedccbacb34c16ed358b0e3fa6143.gif](http://www.yun-zhang.com/upload/20213111093156872.gif)
附近的误差称为吉布斯现象。吉布斯现象是一种不可避免的误差,它使不连续函数的傅里叶级数将不连续时的函数值高估约9%。吉布斯现象永远不能完全消除,但当傅里叶级数中的项数接近无穷时,误差收敛到完全局限于不连续点。例如,如果在锯齿形的傅里叶级数展开式中包含无限项,我们会发现当
时,级数将完全等于x,而在
时,级数的值约为
。
这告诉我们,求解热方程的齐次
等于使用欧拉积分来求傅里叶系数:
,
,
例:金属棒初始温度是均匀的
假设一个绝热的,一米长的金属棒扩散系数为
(不现实,为了方便作图),最初温度为
,在温度为
、
时夹紧冷却元件。初始条件和边界条件为:
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![](http://www.yun-zhang.com/upload/20213111093156905.svg)
傅里叶系数是:
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让我们来验证这个匹配初始和边界条件的傅里叶级数:
![5ad71da79a65295ffe928f1424f2f2fb.gif 5ad71da79a65295ffe928f1424f2f2fb.gif](http://www.yun-zhang.com/upload/20213111093157109.gif)
所以,解是:
,
现在让我们画出解:
![cf8c5312bf62aa301e19e10c2b55eba2.gif cf8c5312bf62aa301e19e10c2b55eba2.gif](http://www.yun-zhang.com/upload/20213111093157192.gif)
作为一个三维图:
![d01373f082025aaf05d5c4ed6d2e816c024f1a78.png d01373f082025aaf05d5c4ed6d2e816c024f1a78.png](http://www.yun-zhang.com/upload/20213111093157249.png)
例:温度“尖峰”
现在假设金属棒每个地方的初始温度都是0°C,除了中间10厘米的温度是
,这次的扩散系数是
。
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![](http://www.yun-zhang.com/upload/20213111093157283.svg)
计算傅里叶系数最简单的方法是把它们转换成更一般的形式:
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%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%22203.5%22%20y%3D%2287%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22brack_sm2882ad605b1e27be87c7468%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%22203.5%22%20y%3D%2292%22%3E%26%23x239F%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22bracketse552f5417ff4680c6b50499%22%20font-size%3D%2216%22%20text-anchor%3D%22start%22%20x%3D%22203.5%22%20y%3D%22106%22%3E%26%23x23A0%3B%3C%2Ftext%3E%3Cline%20stroke%3D%22%23000000%22%20stroke-linecap%3D%22square%22%20stroke-width%3D%221%22%20x1%3D%22165.5%22%20x2%3D%22199.5%22%20y1%3D%2287.5%22%20y2%3D%2287.5%22%2F%3E%3Ctext%20font-family%3D%22math1d212b866a3c45dc66854f4e02c%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22173.5%22%20y%3D%2281%22%3E%26%23x3C0%3B%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22183.5%22%20y%3D%2281%22%3En%3C%2Ftext%3E%3Ctext%20font-family%3D%22ae2ef524fbf3d9fe611d5a8e90fefdc%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22192.5%22%20y%3D%2281%22%3Ea%3C%2Ftext%3E%3Ctext%20font-family%3D%22Arial%22%20font-size%3D%2216%22%20font-style%3D%22italic%22%20text-anchor%3D%22middle%22%20x%3D%22182.5%22%20y%3D%22104%22%3EL%3C%2Ftext%3E%3Ctext%20font-family%3D%22math1d212b866a3c45dc66854f4e02c%22%20font-size%3D%2216%22%20text-anchor%3D%22middle%22%20x%3D%22211.5%22%20y%3D%2293%22%3E%2C%3C%2Ftext%3E%3C%2Fsvg%3E)
,
,
,
所以解是:
,
![5760064a376bfe280704fedc8eef49f9.gif 5760064a376bfe280704fedc8eef49f9.gif](http://www.yun-zhang.com/upload/20213111093157348.gif)
虽然金属棒最终会达到热平衡状态,但是5秒后温度下降的非常缓慢,所以动画值展示5秒。下面是3D图:
![1b4c510fd9f9d72a64bbaec942e9023c369bbb56.png 1b4c510fd9f9d72a64bbaec942e9023c369bbb56.png](http://www.yun-zhang.com/upload/20213111093157394.png)
例:随机热分布
这一次,这根金属棒,我们假设是铜做的,扩散系数为
,它的热量分布是随机的(以10厘米为一段),如下表:
![0824ab18972bd407a53a77fced4ab4590eb30979.png 0824ab18972bd407a53a77fced4ab4590eb30979.png](http://www.yun-zhang.com/upload/20213111093157430.png)
???这种情况下,最好是通过数值积分来求傅里叶系数而不是试图找到一个封闭式的表达式。
下面是一个解的动画:
![e74ee65b3b587cbf60477aa62c5dbddb.gif e74ee65b3b587cbf60477aa62c5dbddb.gif](http://www.yun-zhang.com/upload/20213111093157771.gif)
240秒后,温度的变化非常缓慢。最有趣的行为发生在最初的60秒:
![caef76094b36acafa88b8f40e61aa71800e99c6a.png caef76094b36acafa88b8f40e61aa71800e99c6a.png](http://www.yun-zhang.com/upload/20213111093157840.png)
结束语
这就是第一部分的内容。现在你知道了如何解最简单的情况下的热方程,你可以使用热方程来分析更有趣的问题。
想了解更多精彩内容,快来关注老胡说科学。
注:转载本文,仅为展示云章多学科编辑器在理工科复合内容的编辑排版效果,如有异议可致电删除。