U T Solutions

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We saw some of the following properties in the Table of Laplace Transforms.

Recall `u(t)` is the unit-step function.

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Solutions

1. ℒ`{u(t)}=1/s`

2. ℒ`{u(t-a)}=e^(-as)/s`

3. Time Displacement Theorem:

Ut solutions plc

Yesterday's America Solution! YOU are invited to take the responsibility to control your authority following the 6 key hierarchical principals. Natural Solutions provides exceptional automotive repair services as well as CNG conversions. Our experts are knowledgeable and informative, experienced and highly qualified. Whether you need scheduled maintenance or a major fix, we’ll get your vehicle in top shape! U t +uu x =0, u(x,0)= h(x). (5.1) The characteristic equations are dx dt = z, dy dt =1, dz dt =0, and Γ may be parametrized by (s,0,h(s)). X = h(s)t+s, y = t, z = h(s). U(x,y)=h(x−uy) (5.2) The characteristic projection in the xt-plane1 passing through the point (s,0) is the line x = h(s)t+s along which u has the constant value u. The value of t = 0 is usually taken as a convenient time to switch on or off the given voltage. The switching process can be described mathematically by the function called the Unit Step Function (otherwise known as the Heaviside function after Oliver Heaviside).

If `F(s)=` ℒ`{f(t)}` then ℒ`{u(t-a)*g(t-a)}=e^(-as)G(s)`

[You can see what the left hand side of this expression means in the section Products Involving Unit Step Functions.]

Examples

Sketch the following functions and obtain their Laplace transforms:

(a) `f(t)={ {: (0,t < a), (A, a < t < b), (0, t > b) :}`

Assume the constants a, b, and A are positive, with a < b.

Answer

The function has value A between t = a and t = b only.

Graph of `f(t)=A*[u(t-a)-u(t-b)]`.

We write the function using the rectangular pulse formula.

`f(t)=A*[u(t-a)-u(t-b)]`

We use `Lap{u(t-a)}=(e^(-as))/s`

We also use the linearity property since there are 2 items in our function.

`Lap{f(t)}=A[(e^(-as))/s-(e^(-bs))/s]`

(b) `f(t)={ {: (0,t < a), (e^(t-a), a < t < b), (0, t > b) :}`

Assume the constants a and b are positive, with a < b.

Answer

Our function is `f(t)=e^(t-a)`. This is an exponential function (see Graphs of Exponential Functions).

When `t = a`, the graph has value `e^(a-a)= e^0= 1`.

Graph of `f(t)=e^(t-a)*{u(t-a)-u(t-b)}`.

The function has the form:

`f(t)=e^(t-a)*{u(t-a)-u(t-b)}`

We will use the Time Displacement Theorem:

`Lap{u(t-a)*g(t-a)}=e^(-as)G(s)`

Now, in this example, `G(s)=` `Lap{e^t}=1/(s-1)`

`Lap{e^(t-a)*[u(t-a)-u(t-b)]}`

`=` `Lap{e^(t-a)*u(t-a)-e^(t-a)*u(t-b)}`

We now make use of a trick, by noting `(t-a) = (b-a ) + (t-b)` and re-writing `e^(t-a)` as `e^(b-a)e^(t-b)`:

`= Lap{e^(t-a)*u(t-a)` `{:-e^(b-a)e^(t-b)*u(t-b)}`

[We have introduced eb−a, a constant, for convenience.]

`=` `Lap{e^(t-a)*u(t-a)}-` `e^(b-a)Lap{e^(t-b)*u(t-b)}`

[Each part is now in the form `u(t − c) · g(t − c)`, so we can apply the Time Displacement Theorem.]

`=e^(-as)xx1/(s-1)` `-e^(b-a)xxe^(-bs)xx1/(s-1)`

`=(e^(-as))/(s-1)-(e^(b-a-bs))/(s-1)`

`=(e^(-as)-e^(b-a-bs))/(s-1)`

(c) `f(t)={ {: (0,t < 0), (sin t, 0 < t < pi), (0, t > pi) :}`

Answer

Here is the graph of our function.

Graph of `f(t) = sin t * [u(t) − u(t − π)]`.

U t solutions llc

The function can be described using Unit Step Functions, since the signal is turned on at `t = 0` and turned off at `t=pi`, as follows:

`f(t) = sin t * [u(t) − u(t − π)]`

Now for the Laplace Transform:

Solutions

Ut Solutions Plc

`Lap{sin t * [u(t)-u(t-pi)]}` `=` `Lap{sin t * u(t)}- ` `Lap{sin t * u(t - pi)}`

Now, we need to express the second term all in terms of `(t - pi)`.

From trigonometry, we have:

`sin(t − π) = -sin t`

So we can write:

`Lap{sin t * u(t)}- ` `Lap{sin t * u(t - pi)}`

`= ` `Lap{sin t * u(t)}+ ` `Lap{sin(t - pi)* u(t - pi)}`

`=1/(s^2+1)+(e^(-pis))1/(s^2+1)`

`=(1+e^(-pis))/(s^2+1)`

Results 1 - 5 of 5

The Problem Of Blow-Up In Nonlinear Parabolic Equations

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Abstract - Cited by 66 (12 self) - Add to MetaCart

Existence And Blow-Up For Higher-Order Semilinear Parabolic Equations: Majorizing Order-Preserving Operators

'... . As a basic example, we establish that in the Cauchy problem for the 2m-th order semilinear parabolic equation u t = ( ) m u + juj p ; x 2 R N ; t > 0; u(x; 0) = u 0 (x); x 2 R N ; where m > 1, p > 1, with bounded integrable initial data u 0 , the critical Fujita exponent is pF = ...'
Abstract - Cited by 29 (11 self) - Add to MetaCart
. As a basic example, we establish that in the Cauchy problem for the 2m-th order semilinear parabolic equation u t = ( ) m u + juj p ; x 2 R N ; t &gt; 0; u(x; 0) = u 0 (x); x 2 R N ; where m &gt; 1, p &gt; 1, with bounded integrable initial data u 0 , the critical Fujita exponent is pF = 1 + 2m=N , so that for p &gt; pF there exists a class of small global solutions and for p 2 (1; pF ] blow-up can occur for arbitrarily small initial data. The analysis of the asymptotics of both classes of global and blow-up solutions is based on comparison with similarity solutions of the majorizing order-preserving equation, which is shown to exist for any m &gt; 1. Generalizations of this idea to dierential and pseudodierential evolution equations and relations to positivity sets for higher-order equations are discussed. 1. Introduction: majorizing order-preserving operators and equations This paper deals with a class of higher-order semilinear parabolic dierential and nonlinear integral evolution...
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Comparison Results and Steady States for the Fujita Equation with Fractional Laplacian

'... We study a semilinear PDE generalizing the Fujita equation whose evolution operator is the sum of a fractional power of the Laplacian and a convex non-linearity. Using the Feynman-Kac representation we prove criteria for asymptotic extinction versus finite time blow up of positive solutions based ...'
Abstract - Cited by 18 (3 self) - Add to MetaCart
We study a semilinear PDE generalizing the Fujita equation whose evolution operator is the sum of a fractional power of the Laplacian and a convex non-linearity. Using the Feynman-Kac representation we prove criteria for asymptotic extinction versus finite time blow up of positive solutions based on comparison with global solutions. For a critical power non-linearity we obtain a two-parameter family of radially symmetric stationary solutions. By extending

Blow-Up, Critical Exponents And Asymptotic Spectra For Nonlinear Hyperbolic Equations

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We prove nonexistence results for the Cauchy problem for the abstract hyperbolic equation in a Banach space X , u tt = f 0 (u); t &gt; 0; u(0) = u 0 ; u t (0) = u 1 ; where f : X ! R is a C¹-function. Several applications to the second and higher-order hyperbolic equations with local and nonlocal nonlinearities are presented. We also describe an approach to Kato&apos;s and John&apos;s critical exponents for the semilinear equations u t = u+b(x; t)juj p , p &gt; 1, which are responsible for phenomena of stability, unstability, blow-up and asymptotic behaviour. We construct countable spectra of different asymptotic patterns of self-similar and non self-similar types for global and blow-up solutions for the autonomous equation u tt = u + juj p 1 u in different parameter ranges.
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I T Solutions

Blow-Up Estimates for Higher-Order Semilinear Parabolic Equations

'... We prove L estimates on the blow-up behaviour of solutions of a 2m-th order semilinear parabolic equation u t = ( ) u + q(u); x 2 R ; t > 0; m > 1; with a general even function q(u) 0 with a superlinear growth for juj 1. Our comparison approach and estimates apply to general int ...'

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We prove L estimates on the blow-up behaviour of solutions of a 2m-th order semilinear parabolic equation u t = ( ) u + q(u); x 2 R ; t &gt; 0; m &gt; 1; with a general even function q(u) 0 with a superlinear growth for juj 1. Our comparison approach and estimates apply to general integral evolution equations. We also study the following problem: nd a continuous function q(u) with a superlinear growth as u !1 such that the parabolic equation exhibits regional blow-up in a domain of nite non-zero measure. We show that such a regional blow-up can occur for q(u) = uj ln jujj . We present a formal asymptotic theory explaining that the stable (generic) blow-up behaviour as t ! T is described by the self-similar solution U (x; t) = expf(T t) (x)g; : R ! C ; of the complex Hamilton-Jacobi equation U t = ( 1) 1 2m (rU rU)