0
votes
0answers
33 views

How to solve a system of first-order partial differential equations?

I have a system of first-order partial differential equations. \begin{align} & -\frac{a}{a_1} \frac{\partial P_{12}}{\partial a} - \frac{a b}{a_1 b_1} \frac{\partial P_{12}}{\partial b} + ...
0
votes
1answer
37 views

Solve a system of equations.

I have a system of equations: \begin{align} & x_{21} (\frac{\partial}{\partial x_{11}}f_{1111})( x_{11} , x_{21}, y_{11} , y_{21} ) + \frac{y_{21}}{x_{11}^2} (\frac{\partial}{\partial ...
4
votes
3answers
81 views

Pair of PDEs to be solved together

I have the following pair of equations to be solved together to find the functions $H_{x}$ and $H_{y}$, which are the components of a vector $\bar{H}(x,y)=H_{x}(x,y)\hat{x}+H_{y}(x,y)\hat{y}$ in ...
1
vote
0answers
21 views

System of ODEs and DAE system

Let us consider the following system of ODEs: $$ y' = f(y,z),\quad z' = g(y,z),\quad y(0) = y_0,\;z(0)=z_0 $$ and the following one: $$ y' = f(y,z),\quad 0 = g(y,z), \quad y(0) = y_0. $$ $f$ and $g$ ...
0
votes
0answers
37 views

Methods of characteristic for system of first order linear hyperbolic partial differential equations: reference and examples

I would like to understand a few points on the methods of characteristics used to resolve a system of coupled, linear first order partial differential equation (of the hyperbolic type). Some example ...
0
votes
1answer
38 views

Are there standard approaches, to solving a system of nonlinear PDE?

If we have a system of PDE's, where each PDE is different i.e. for $u:U\subset \Bbb R^2\to\Bbb R^3$, $u(x,y)=(a(x,y),b(x,y),c(x,y))$, which needs to satisfy $ \left\{ \begin{array}{ll} ...
2
votes
1answer
61 views

Method of characteristics for a system of pdes

I can do parts a) and b) as follows $\begin{pmatrix} 1&0&0 \\ 0&1&0 \\ 0&0&1\end{pmatrix}\frac{\partial}{\partial{}x}\begin{pmatrix} u \\ v \\ w\end{pmatrix}+\begin{pmatrix} ...
5
votes
0answers
98 views

How to solve a particular initial-boundary value problem

I have the following initial-boundary value problem $$\begin{cases}\dfrac{\partial^2 u_1}{\partial x^2}=A_{11}\dfrac{\partial u_1}{\partial t}+A_{12}\dfrac{\partial u_2}{\partial ...
1
vote
0answers
54 views

Decoupling system of two partial differential equations

If I have the following systems of PDE $$ u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)} = 0,\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)} = 0, $$ where ...
1
vote
1answer
43 views

Changing variables for a partial differential equation

If I have the following systems of PDE \begin{align} u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)}=0\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)}=0, ...
0
votes
0answers
34 views

Simple Question- Is this a Sturm Liouville regular problem?

I have the following differential system: \begin{align} &(1-t^2)x'' -2tx' +\lambda x= 0 \\&x(0)=0 \\&x(l)+x'(l)=0 \end{align} I have to decide if it is an homogeneous regular Sturm ...
1
vote
2answers
39 views

Given a solution flow to find periodic solutions

Given the system of differential equations $x' = 2x + y^3$ and $y' = -y$ i found the flow $$\phi_t(x,y) = ((x_0 + 1/5y_0^3)e^{2t} - 1/5 y_0^3e^{-3t}, y_0 e^{-t})$$. I am wondering are there any ...
1
vote
1answer
33 views

How to solve these two second-order coupled PDE?

I have two second-order equations governing the behaviour of two spatial function which are coupled: $$ 0 = A f(x,y) + B \frac{\partial^2 f(x,y)}{\partial x^2} + C \frac{\partial^2 f(x,y)}{\partial ...
0
votes
1answer
50 views

system of partial differential equation and boundary condition.

Let $\Omega$ be a regular domain, for example be a rectangle. Is it true that solve system of PDE's like this: $$u+\Delta w=0, ~~~~~~~~~~~~~~~~~~~~~~w=b1,~\frac{\partial w}{\partial ...
0
votes
0answers
53 views

How to solve three second-order coupled PDE?

I need to solve these three second-order coupled partial differential equations: \begin{align} \left( A + B\frac{\partial^2}{\partial x^2} + C \left( \frac{\partial^2}{\partial y^2} + ...
0
votes
0answers
66 views

Series solution for coupled PDEs

So I have this system for $f(z, v, t)$ and $\Psi(z, t)$, $$ \frac{\partial f}{\partial t} + v \frac{\partial f}{\partial z} - g(v) \frac{\partial \Psi}{\partial z} = 0 \tag{1} $$ $$ \frac{\partial^2 ...
0
votes
0answers
31 views

Equating two systems of PDEs

I'm trying to relate two sets of a pair of PDEs, but massively struggling! They should be equivalent up to a linear transformation. Any help would be wonderful! First set: $$\frac{\partial ...
0
votes
0answers
40 views

Existence of global solution to a system of ordinary differential equations

In Evans's Partial Differential Equations, second edition, pp 401, in establishing the existence of solution to wave equation, the author uses the Galerkin method and constructs a sequence of ...
1
vote
0answers
30 views

Let $r_t+ru_x+ur_x=0$ and $u_t+uu_x=0$ for $u(x,0)=f(x)$ and $r(x,0)=g(x)$.

Let $r_t+ru_x+ur_x=0$ and $u_t+uu_x=0$ for $u(x,0)=f(x)$ and $r(x,0)=g(x)$. I know I have to solve for implicit $u$ first and then $r$, but I don't know if I'm using the right method here. What I've ...
7
votes
2answers
112 views

Very simple partial differential equation

I am solving $$ \frac {\partial f}{\partial x} = \frac y{x^2 + y^2} \\ \frac {\partial f}{\partial y} = \frac {-x}{x^2 +y^2} $$ As $y$ was held constant when the partial derivative with respect to ...
5
votes
1answer
81 views

For a system of PDEs, how many equations are needed generally for the system to have unique solution?

For an algebraic system of equations or a system of ordinary differential equations the following rule holds:(right?) the total number of unknown variables must be equal to the number of equations ...
1
vote
0answers
26 views

System of linear differential equations eigenproblem

In the Smith and Young 2001 paper on the Barotropic tide they have the governing equations... \begin{array}{rcl} u_t-f_0v+p_x & = & 0 \\v_t+f_0u+p_y & = & 0 \\p_z &= &b ...
1
vote
1answer
58 views

On a system of PDE

I would like to know what is the set of solutions to the following PDE. I think it consists of just constants, but I need help to prove. Let $f_1(p_1,p_2)$ and $f_2(p_1,p_2)$ be two functions. The ...
3
votes
2answers
78 views

Constructing a Distributional Solution to the Inhomogeneous C.R. Equations

The question is to find a fundamental solution to the system of equations in $\mathbb{R}^{2}$ \begin{array}{l} u_{x}-v_{y}=f\\ u_{y}+v_{x}=g\end{array} and to express the answer as a $2\times2$ ...
3
votes
1answer
442 views

How to solve coupled linear 1st order PDE

It is fairly straight forward to solve linear 1st order PDEs by the method of characteristics. For example, if $\partial_tf+a\partial_xf=bf$ , we have that $\dfrac{df}{dt}=bf$ on the characteristic ...
1
vote
1answer
261 views

Numerical techniques for solving systems of first-order semi-linear hyperbolic PDEs in two variables

I need to solve such systems of PDEs numerically and I'm wondering what the "standard" method (if there is one) is. The systems I'm interested in have the form: $\frac{\partial F_i}{\partial t} + ...