###### Partial Differential Equations through Examples and Exercises

262 SOLVED PROBLEMS

Question: 2.27

## Assume equation (2.32) can be written in the form ∂z/∂x = f (x,y,z,∂z/∂y) (2.41) (for example, in a neighbourhood of a point where it holds ∂F/∂p ≠0.) Then using the method of variation of constants, prove that all the solutions of (2.32) can be obtained from the complete solution. ...

Hint. Firstly take a = a( x, y) and b = b( x, y) i...
Question: 12.23

## (Malgrange – Ehrenpreiss) Every linear partial differential operator with constant coefficients P(D) has a fundamental solution, i.e., there exists a solution of the equation P(D)u = δ. ...

Let us define K(x)=\int_{ \mathbf{R} ^{n-1}...
Question: 12.14

## Prove the following equalities for every f ∈ D′(R) : a) δa * f = f (a ∈ R); b) D^m δ * f = D^(m) f, m ∈ N. ...

a) Let \varphi \in \mathcal {D} ( \mathbf{R...
Question: 12.20

## Let E(x) =1/σn(2-n)|x|^n-2 for n ≥ 3, n where σn = 2π^n / 2/Γ(n/2). Prove that a) E is a locally integrable function; b) E is the fundamental solution for the Laplace operator Δ. ...

a) There exist M > 0 and a < n such that [la...
Question: 12.18

## Prove that the function E : R × R → R given by E(x,t) = 1/2 H(t – |x|) = { 1/2 if t > |x|, 0 if t < |x|, (12.36) is the fundamental solution of the one-dimensional wave equation. In other words, it holds ...

Let us note first that E from (12.36) is a locally...
Question: 12.17

## Let the function f : R² → R be given for (x, y) = x + ιy by f(x+ιy ) = 1/π · 1/x+ιy. Prove that a) the function f is locally integrable on R², and so it defines a distribution on R²; b) the function f is the fundamental solution for the Cauchy-Riemann operator ∂ = 1/2 (∂/∂x + ι∂/∂y). ...

a) Since we have |f(z)|=\frac{1}{\pi} \cdot...
Question: 12.15

## Let P(x, D) be the linear differential operator P(x,D) = ∑j=0^m aj(x)D^j, (12.33) where aj ∈ C^∞ (R), j = 0, 1, …, m, and D is the distributional derivation operator. a) Prove that the solution of the following equation in D′(R) ...

a) Firstly we find the distributional derivatives ...
Question: 12.10

## Find all distributions y such that a) y’ = 0; b) y^(m) = 0, m∈N. ...

a) Let y \in \mathcal {D} ^{\prime}( \mathb...
Question: 12.9