### All Partial Differential Equations Resources

## Example Questions

### Example Question #1 : Initial & Boundary Value Problems

Solve the Boundary Value Problem (BVP).

**Possible Answers:**

**Correct answer:**

To solve this Boundary Value Problem (BVP) recall that the general solution for this type of derivative is,

Therefore, the equation becomes

From here, apply the boundary conditions to solve for the constants and

Thus resulting in the solution,

### Example Question #2 : Initial & Boundary Value Problems

Solve the Boundary Value Problem (BVP).

**Possible Answers:**

**Correct answer:**

To solve this Boundary Value Problem (BVP) recall that the general solution for this type of derivative is,

Therefore, the equation becomes

From here, apply the boundary conditions to solve for the constants and

Thus resulting in the solution,

### Example Question #1 : Partial Differential Equations

Determine if the statement is true or false:

The wave equation has at most one solution.

**Possible Answers:**

False

True

**Correct answer:**

True

This statement is true by the Uniqueness Theorem.

is twice differential equation in terms of and .

Now, consider the energy integral

After performing integration by parts results in the following,

From here, the initial conditions and boundary conditions are applied.

Therefore,

which proves the wave equations has only one solution and thus is unique.

### Example Question #1 : Introductions To Pd Es

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

**Possible Answers:**

**Correct answer:**

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

The conservation equation is,

Now, recall the divergence theorem which states,

Thus, by substituting

for results in,

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

,

and performing some algebraic operations results in,

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

### Example Question #2 : Partial Differential Equations

What is the order of the following partial differential equation.

**Possible Answers:**

Quasi Linear

Linear

Second Order

First Order

Third Order

**Correct answer:**

Second Order

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,

The partial derivatives are:

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

### Example Question #3 : Partial Differential Equations

What is the order of the following partial differential equation.

**Possible Answers:**

Second Order

Linear

Third Order

Quasi Linear

First Order

**Correct answer:**

Third Order

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,

The partial derivatives are:

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

### Example Question #3 : Introductions To Pd Es

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

**Possible Answers:**

**Correct answer:**

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.

is known as the heat equation.

is known as the wave equation.

is known as the Laplace equation.

is known as the Poisson equation.

is known as the biharmonic wave equation.

Therefore, the correct answer for the biharmonic wave equation is

### Example Question #4 : Partial Differential Equations

What is the order of the following partial differential equation.

**Possible Answers:**

Homogeneous

Second Order

Non-homogenous

First Order

Third Order

**Correct answer:**

Third Order

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,

The partial derivatives are:

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

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