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In general, a partial differential equation (PDE) of order n takes the form: \(\displaystyle \begin {array} [c]{c} F\left ( x_{1},\ldots ,x_{m},w,\dfrac {\partial w(x_{1},\ldots x_{m})}{\partial x_{1}},\ldots ,\dfrac {\partial w(x_{1},\ldots x_{m})}{\partial x_{m}} ,\dfrac {\partial ^{2}w(x_{1},\ldots x_{m})}{\partial x_{1}^{2}},\ldots ,\right . \\ \qquad \left . ,\ldots ,\dfrac {\partial ^{n}w(x_{1},\ldots x_{m})}{\partial x_{1}^{n}},\dfrac {\partial ^{n}w(x_{1},\ldots ,x_{m})}{\partial x_{1}^{n-1}\partial x_{2}},\ldots ,\dfrac {\partial ^{n}w(x_{1},\ldots x_{m} )}{\partial x_{m}^{n}}.\right ) =0, \end {array} \) \(w(x_{1},\ldots x_{m})\) being the unknown function.

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Introduction to Partial Differential Equations

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  • 发表日期：

    2025年

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  • Michele Nichelatti

摘要

In general, a partial differential equation (PDE) of order n takes the form: \(\displaystyle \begin {array} [c]{c} F\left ( x_{1},\ldots ,x_{m},w,\dfrac {\partial w(x_{1},\ldots x_{m})}{\partial x_{1}},\ldots ,\dfrac {\partial w(x_{1},\ldots x_{m})}{\partial x_{m}} ,\dfrac {\partial ^{2}w(x_{1},\ldots x_{m})}{\partial x_{1}^{2}},\ldots ,\right . \\ \qquad \left . ,\ldots ,\dfrac {\partial ^{n}w(x_{1},\ldots x_{m})}{\partial x_{1}^{n}},\dfrac {\partial ^{n}w(x_{1},\ldots ,x_{m})}{\partial x_{1}^{n-1}\partial x_{2}},\ldots ,\dfrac {\partial ^{n}w(x_{1},\ldots x_{m} )}{\partial x_{m}^{n}}.\right ) =0, \end {array} \) \(w(x_{1},\ldots x_{m})\) being the unknown function.