Partial Differential Equations through Examples and Exercises 1st. Edition by Endre Pap, Arpad Takači, Djurdjica Takači | 9789401063494 | 9401063494 | Holooly

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. ...

Verified Answer:

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. ...

Verified Answer:

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. ...

Verified Answer:

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. ...

Verified Answer:

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 ...

Verified Answer:

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). ...

Verified Answer:

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) ...

Verified Answer:

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. ...

Verified Answer:

a) Let

y \in \mathcal {D} ^{\prime}( \mathb...

Question: 12.9

a) Let f be a contin uous function on R \ { a }, which is also continuously differentiable on the intervals (-∞, a] and [a, +∞). Prove that Df = Tf′ + |f|a δa, where D f denotes the distributional derivative of the function f, while Tf′ is the regular distribution defined by the classical derivative ...

Verified Answer:

Clearly it is enough to prove part a). To that end...

Question: 12.8

Let us denote ej(x) = exp(2jπιx) (j ∈ Z), D^α the derivation operator in the sense of D′(R) and assume that for the sequence {cj}j ∈ Z of complex numbers there exist a positive constant A and a natural number k such that ...

Verified Answer:

a) Let us us start from the sequence of functions ...

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