All Partial Differential Equations Resources
Example Questions
Example Question #1 : Initial & Boundary Value Problems
Solve the Boundary Value Problem (BVP).
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).
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.
False
True
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?
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.
Quasi Linear
Linear
Second Order
First Order
Third Order
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.
Second Order
Linear
Third Order
Quasi Linear
First Order
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?
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.
Homogeneous
Second Order
Non-homogenous
First Order
Third Order
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 #1 : Partial Differential Equations
Determine if the statement is true or false:
If the -axis is the axis of symmetry and a surface is revolving around it and is an arbitrary function, then the partial differential equation associated with that said surface, satisfies the equation:
False
True
True
To determine the truth of this statement, assume the following.
is some function
From here, differential with respect to and .
Next eliminate as it is an arbitrary function.
This leads to the result,
Therefore, the statement,
If the -axis is the axis of symmetry and a surface is revolving around it and is an arbitrary function, then the partial differential equation associated with that said surface, satisfies the equation:
is true.
Example Question #2 : Partial Differential Equations
Which of the following describes the physical phenomena that is the wave equation?
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 wave equation is
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