- #1

Math100

- 771

- 219

- Homework Statement
- Find the Euler equation (strong form) for ## \int ((\mathrm{u}')^2+e^{\mathrm{u}}) \, dx ##.

- Relevant Equations
- Euler's equation: ## J(y)=\int_{a}^{b} F(x, y, y', y") \, dx ##

By the Euler's equation of the functional, we have

## J(\mathrm u)=\int ((\mathrm{u})^{2}+e^{\mathrm{u}}) \, dx ##.

Then ## J(\mathrm{u}+\epsilon\eta)=\int ((\mathrm{u}'+\epsilon\eta')^{2}+e^{\mathrm{u}+\epsilon\eta}) \, dx=\int ((\mathrm{u})'^{2}+2\epsilon\mathrm{u}'\eta'+\epsilon^{2}(\eta')^{2}+e^{\mathrm{u}}+\epsilon e^{\mathrm{u}}\eta) \, dx ##.

Note that ## \frac {J(\mathrm{u}+\epsilon\eta)-J(\mathrm{u})} {\epsilon}=\int (2\mathrm{u}'\eta'+\epsilon(\eta')^{2}+e^{\mathrm{u}}\eta) \, dx ##.

Consider the following limit:

## \lim_{\epsilon \rightarrow 0} \frac {J(\mathrm{u}+\epsilon\eta)-J(\mathrm{u})} {\epsilon}=\int (2\mathrm{u}'\eta'+e^{\mathrm{u}}\eta) \, dx=0 ##.

Applying the method of integration by parts, we obtain

## \int (2\mathrm{u}'\eta'+e^{\mathrm{u}}\eta) \, dx=(2\mathrm{u}'\eta)-\int (2\mathrm{u}''\eta+e^{\mathrm{u}}\eta) \, dx=0 ##.

Thus ## 2\mathrm{u}'\eta-2\mathrm{u}''\eta-e^{\mathrm{u}}\eta=0\implies 2\mathrm{u}'-2\mathrm{u}''-e^{\mathrm{u}}=0 ##.

Therefore, the Euler equation (strong form) is ## 2\mathrm{u}'-2\mathrm{u}''-e^{\mathrm{u}}=0 ##.

## J(\mathrm u)=\int ((\mathrm{u})^{2}+e^{\mathrm{u}}) \, dx ##.

Then ## J(\mathrm{u}+\epsilon\eta)=\int ((\mathrm{u}'+\epsilon\eta')^{2}+e^{\mathrm{u}+\epsilon\eta}) \, dx=\int ((\mathrm{u})'^{2}+2\epsilon\mathrm{u}'\eta'+\epsilon^{2}(\eta')^{2}+e^{\mathrm{u}}+\epsilon e^{\mathrm{u}}\eta) \, dx ##.

Note that ## \frac {J(\mathrm{u}+\epsilon\eta)-J(\mathrm{u})} {\epsilon}=\int (2\mathrm{u}'\eta'+\epsilon(\eta')^{2}+e^{\mathrm{u}}\eta) \, dx ##.

Consider the following limit:

## \lim_{\epsilon \rightarrow 0} \frac {J(\mathrm{u}+\epsilon\eta)-J(\mathrm{u})} {\epsilon}=\int (2\mathrm{u}'\eta'+e^{\mathrm{u}}\eta) \, dx=0 ##.

Applying the method of integration by parts, we obtain

## \int (2\mathrm{u}'\eta'+e^{\mathrm{u}}\eta) \, dx=(2\mathrm{u}'\eta)-\int (2\mathrm{u}''\eta+e^{\mathrm{u}}\eta) \, dx=0 ##.

Thus ## 2\mathrm{u}'\eta-2\mathrm{u}''\eta-e^{\mathrm{u}}\eta=0\implies 2\mathrm{u}'-2\mathrm{u}''-e^{\mathrm{u}}=0 ##.

Therefore, the Euler equation (strong form) is ## 2\mathrm{u}'-2\mathrm{u}''-e^{\mathrm{u}}=0 ##.

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