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→Superposition principle
===Superposition principle===
===Superposition principle===
叠加原理
{{further| Superposition principle }}
{{further| Superposition principle }}
The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example 2 sin x}}. The same principle can be observed in PDEs where the solutions may be real or complex and additive. superposition
The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example 2 sin x}}. The same principle can be observed in PDEs where the solutions may be real or complex and additive. superposition
'''<font color = "#ff8000">叠加原理 Superposition Principle<\font>适用于任何线性系统，包括偏微分方程的线性系统。这个概念的一个常见的可视化是两个同相位的波相互作用结合在一起会产生更大的振幅，例如 {{math|sin ''x'' + sin ''x'' {{=}} 2 sin ''x''}}。在偏微分方程中也可以观察到同样的原理，其中的解可能是真实的或复杂的和可加的。叠加
If {{math|''u''<sub>1</sub>}} and {{math|''u''<sub>2</sub>}} are solutions of linear PDE in some function space {{mvar|R}}, then {{math|''u'' {{=}} ''c''<sub>1</sub>''u''<sub>1</sub> + ''c''<sub>2</sub>''u''<sub>2</sub>}} with any constants {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}} are also a solution of that PDE in the same function space.
If {{math|''u''<sub>1</sub>}} and {{math|''u''<sub>2</sub>}} are solutions of linear PDE in some function space {{mvar|R}}, then {{math|''u'' {{=}} ''c''<sub>1</sub>''u''<sub>1</sub> + ''c''<sub>2</sub>''u''<sub>2</sub>}} with any constants {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}} are also a solution of that PDE in the same function space.
If and are solutions of linear PDE in some function space , then c<sub>1</sub>u<sub>1</sub> + c<sub>2</sub>u<sub>2</sub>}} with any constants and are also a solution of that PDE in the same function space.
If and are solutions of linear PDE in some function space , then c<sub>1</sub>u<sub>1</sub> + c<sub>2</sub>u<sub>2</sub>}} with any constants and are also a solution of that PDE in the same function space.
若线性偏微分方程在某个函数空间{{mvar|R}}中有解{{math|''u''<sub>1</sub>}} 和 {{math|''u''<sub>2</sub>}}，则{{math|''u'' {{=}} ''c''<sub>1</sub>''u''<sub>1</sub> + ''c''<sub>2</sub>''u''<sub>2</sub>}，其中{{math|''c''<sub>1</sub>}} 和 {{math|''c''<sub>2</sub>}}是任意常数，也是该偏微分方程在同一函数空间中的解。
===Methods for non-linear equations===
===Methods for non-linear equations===