Differential Equations - Derivatives from Conservation Laws

What is the order of the following partial differential equation.

$u_{xx}+2u_{xy}+\frac{7}{5}u_{yy}=10y$

Second Order

First Order

Third Order

Linear

Quasi Linear

Explanation

Just like with ordinary differential equations, partial differential equations can be characterized by their order.

The order of an equation is defined by the highest ordered partial derivatives in the equations.

Looking at the equation in question,

$u_{xx}+2u_{xy}+\frac{7}{5}u_{yy}=10y$

The partial derivatives are:

$\u_{xx} \u_{xy} \u_{yy}$

Notice that each partial derivative contains two variables, thus this equation is a second order partial differential equation.

What is the conservation law written as a partial differential equation?

$\frac{\partial U}{\partial t}+\bigtriangledown F=S$

$\frac{\partial U}{\partial t}+\bigtriangledown S=F$

$\frac{\partial F}{\partial t}+\bigtriangledown U=S$

$\int_{\partial \wp } U dx+\int_{\partial \wp} F(U)n\ ds$

$\int_{\partial \wp } U dx+\int_{\partial \wp} F(U)n\ ds=\int_\wp S(U,t)\ dx$

Explanation

The conservation law written as a partial differential equation is found by applying the divergence theorem to the conservation equation.

The conservation equation is,

$\frac{d}{dt}\int_\wp U\ dx +\int_{\partial \wp} F(U)n\ ds=\int_\wp S(U,t)\ dx$

Now, recall the divergence theorem which states,

$\int_{\partial \wp}Fn\ ds=\int_{\wp}\bigtriangledown F\ dx$

Thus, by substituting

$\int_{\partial \wp} Fn\ ds$ for $\int_{\partial \wp} F(U)n\ ds$ results in,

$\frac{d}{dt}\int_\wp U\ dx+\int_{\partial \wp}Fn\ ds=\int_\wp S\ dx$

From here, rewriting this equation to bring the derivative inside the integral along with substituting

$\int_{\partial \wp} Fn\ ds=\int_\wp \bigtriangledown F\ ds$ ,

and performing some algebraic operations results in,

$\int_\wp \left(\frac{\partial U}{\partial t}+\bigtriangledown F-S \right )dx=0$

After integrating over the domain $\wp$ the partial differential equation that is found is,

$\frac{\partial U}{\partial t}+\bigtriangledown F=S$

What is the order of the following partial differential equation.

$u_{xyz}+u_{xy}=e^y$

Third Order

Second Order

First Order

Homogeneous

Non-homogenous

Explanation

Just like with ordinary differential equations, partial differential equations can be characterized by their order.

The order of an equation is defined by the highest ordered partial derivatives in the equations.

Looking at the equation in question,

$u_{xyz}+u_{xy}=e^y$

The partial derivatives are:

$\u_{xyz} \u_{xy}$

Notice that one of them partial derivative contains three variables, thus this equation is a third order partial differential equation.

Which of the following describes the physical phenomena that is the biharmonic wave equation?

$u_{tt}+c^2\bigtriangledown^4u=0$

$u_t-k(u_{xx}+u_{yy}+u_{zz})=0$

$u_{tt}-c^2(u_{xx}+u_{yy}+u_{zz})=0$

$u_{xx}+u_{yy}+u_{zz}=0$

$\bigtriangledown^2u=f(x,y,z)$

Explanation

When dealing with partial differential equations, there are phenomenons in the physical world that have specific equations related to them in the mathematical world.

Looking at the possible answer selections below, identify the physical phenomena each represents.

$u_t-k(u_{xx}+u_{yy}+u_{zz})=0$ is known as the heat equation.

$u_{tt}-c^2(u_{xx}+u_{yy}+u_{zz})=0$ is known as the wave equation.

$u_{xx}+u_{yy}+u_{zz}=0$ is known as the Laplace equation.

$\bigtriangledown^2u=f(x,y,z)$ is known as the Poisson equation.

$u_{tt}+c^2\bigtriangledown^4u=0$ is known as the biharmonic wave equation.

Therefore, the correct answer for the biharmonic wave equation is

$u_{tt}+c^2\bigtriangledown^4u=0$

What is the order of the following partial differential equation.

$u_{xyz}+u_{xy}+\frac{10}{3}=10y$

Third Order

Second Order

First Order

Linear

Quasi Linear

Explanation

Just like with ordinary differential equations, partial differential equations can be characterized by their order.

The order of an equation is defined by the highest ordered partial derivatives in the equations.

Looking at the equation in question,

$u_{xyz}+u_{xy}+\frac{10}{3}=10y$

The partial derivatives are:

$\u_{xyz} \u_{xy}$

Notice that one of them partial derivative contains three variables, thus this equation is a third order partial differential equation.

Derivatives from Conservation Laws

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