We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Partial Differential Equations : Existence & Uniqueness

Study concepts, example questions & explanations for Partial Differential Equations

Create An Account

All Partial Differential Equations Resources

11 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Introductions To Pd Es

Determine if the statement is true or false:

The wave equation has at most one solution.

Possible Answers:

True

False

Correct answer:

True

Explanation:

This statement is true by the Uniqueness Theorem.

$\\u_{tt}=c^2u_{xx}, 0<x<1, t>0 \\\text{Initial Conditions: }u(x,0)=f(x), u_t(x,0)=g(x), 0\leq x\leq l \\\text{Boundary Conditions: }u(0,t)=0, u(l,t)=0, t\geq 0$

u(x,t) is twice differential equation in terms of and .

Now, consider the energy integral

$E(t)=\frac{1}{2}\int_0^l(c^2u^2_x+u^2_t)\ dx$

After performing integration by parts results in the following,

$\int_0^lc^2u_xu_{xt}\ dx=[c^2u_xu_t]_0^l-\int_0^lc^2u_tu_{xx}\ dx$

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

$\\\frac{dE}{dt}=\int_0^l u_t(u_{tt}-c^2u_{xx})\ dx \\\\\frac{dE}{dt}=0 \\\\E(t)=C$

Therefore,

u_1(x,t)=u_2(x,t)=u(x,t)

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

Report an Error

View Tutors

Omar
Certified Tutor

Arizona State University, Bachelors, Civil Engineering.

View Tutors

Dawn
Certified Tutor

University of Oregon, Bachelor in Arts, Biochemistry and Molecular Biology. Johns Hopkins University, Doctor of Philosophy, E...

View Tutors

Jacob
Certified Tutor

University of Pennsylvania, Bachelor of Science, Computational Biology.

All Partial Differential Equations Resources

11 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Math Tutors in New York City, SAT Tutors in Philadelphia, GRE Tutors in Seattle, Algebra Tutors in Denver, ISEE Tutors in Phoenix, LSAT Tutors in Denver, Physics Tutors in Denver, SSAT Tutors in Chicago, ISEE Tutors in Philadelphia, Math Tutors in Seattle

Popular Courses & Classes

MCAT Courses & Classes in Philadelphia, LSAT Courses & Classes in San Diego, SSAT Courses & Classes in Miami, ISEE Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in Los Angeles, LSAT Courses & Classes in Denver, GRE Courses & Classes in San Diego, ACT Courses & Classes in Phoenix, MCAT Courses & Classes in Boston, SSAT Courses & Classes in San Diego

Popular Test Prep

LSAT Test Prep in Philadelphia, SAT Test Prep in Chicago, SSAT Test Prep in San Francisco-Bay Area, MCAT Test Prep in Miami, MCAT Test Prep in Atlanta, LSAT Test Prep in Boston, ISEE Test Prep in Los Angeles, ISEE Test Prep in New York City, SAT Test Prep in Los Angeles, MCAT Test Prep in Washington DC

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Partial Differential Equations : Existence & Uniqueness

Study concepts, example questions & explanations for Partial Differential Equations

All Partial Differential Equations Resources

Example Questions

Example Question #1 : Introductions To Pd Es

All Partial Differential Equations Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: