Charpit’s Method or General Method to Solve Non-Linear Partial Differential Equation

Charpit’s Method

Charpit’s Method or General Method to Solve Non-Linear Partial Differential Equation

Charpit’s Method: It is a well-known mathematical technique known as “Method of Characteristics”. The method is widely used for solving first-order or quasi-linear PDE’s.

Explanation of Method:

Let the given differential equation be:

Different Types of Solutions in PDE

\[ f(x, y, z, p, q) = 0 \quad \text{(i)} \quad \text{where} \quad z = z(x, y) \]

We know \( dz = \frac{\partial z}{\partial x}dx + \frac{\partial z}{\partial y}dy \). [Therefore, z=z(x,y)] \( dz = p \, dx + q \, dy \quad \text{(ii)} \).

Now we shall find another relation \( F(x, y, z, p, q) = 0 \quad \text{(iii)} \) so that on solving equations (i) and (iii) for \( p \) and \( q \) and putting these values in (ii), equation (ii) becomes integrable, and this integral gives the complete solution of (i) (integral).

For finding \( F \), differentiate (i) with respect to \( x \) and (iii) also with respect to \( x \), we get:

\[ \frac{\partial f}{\partial x} + \frac{\partial f}{\partial z}\frac{\partial z}{\partial x} + \frac{\partial f}{\partial p}\frac{\partial p}{\partial x} + \frac{\partial f}{\partial q}\frac{\partial q}{\partial x} = 0 \quad \text{(iv)} \]

\[ \frac{\partial F}{\partial x} + \frac{\partial F}{\partial z}\frac{\partial z}{\partial x} + \frac{\partial F}{\partial p}\frac{\partial p}{\partial x} + \frac{\partial F}{\partial q}\frac{\partial q}{\partial x} = 0 \quad \text{(v)} \]

Similarly, differentiate (i) with respect to \( y \) and (iii) also with respect to \( y \), we get:

\[ \frac{\partial f}{\partial y} + \frac{\partial f}{\partial z}\frac{\partial z}{\partial y} + \frac{\partial f}{\partial p}\frac{\partial p}{\partial y} + \frac{\partial f}{\partial q}\frac{\partial q}{\partial y} = 0 \quad \text{(vi)} \]

\[ \frac{\partial F}{\partial y} + \frac{\partial F}{\partial z}\frac{\partial z}{\partial y} + \frac{\partial F}{\partial p}\frac{\partial p}{\partial y} + \frac{\partial F}{\partial q}\frac{\partial q}{\partial y} = 0 \quad \text{(vii)} \]

To eliminate \( \frac{\partial p}{\partial x} \) from (iv) and (v)

...(vii)

Apply (del(F)/del(p))(iv) and (del(f)/del(p))(v) we get

\[ \left(\frac{\partial F}{\partial p}\frac{\partial f}{\partial x} - \frac{\partial F}{\partial x}\frac{\partial f}{\partial p}\right) + \frac{\partial z}{\partial x}\left(\frac{\partial F}{\partial p}\frac{\partial f}{\partial z} - \frac{\partial F}{\partial z}\frac{\partial f}{\partial p}\right) + \frac{\partial q}{\partial x}\left(\frac{\partial F}{\partial p}\frac{\partial f}{\partial q} - \frac{\partial F}{\partial q}\frac{\partial f}{\partial p}\right) = 0 \]

To eliminate \( \frac{\partial q}{\partial y} \) from (vi) and (vii)

Apply (del(F)/del(q))(vi) and (del(f)/del(q))(vii) we get

\[ \left(\frac{\partial F}{\partial p}\frac{\partial f}{\partial y} - \frac{\partial F}{\partial y}\frac{\partial f}{\partial q}\right) + \frac{\partial z}{\partial y}\left(\frac{\partial F}{\partial q}\frac{\partial f}{\partial z} - \frac{\partial F}{\partial z}\frac{\partial f}{\partial p}\right) + \frac{\partial p}{\partial y}\left(\frac{\partial F}{\partial q}\frac{\partial f}{\partial p} - \frac{\partial F}{\partial p}\frac{\partial f}{\partial q}\right) = 0 \]

Adding (viii) and (ix) and using \( \frac{\partial q}{\partial x} = \frac{\partial}{\partial x}(\frac{\partial z}{\partial y}) = \frac{\partial^{2}z}{\partial x\partial y} = \frac{\partial^{2}z}{\partial x\partial y} = \frac{\partial^{2}z}{\partial y\partial x} = \frac{\partial}{\partial y}(\frac{\partial z}{\partial x}) = \frac{\partial p}{\partial y} \)

\[ \begin{align*} &\left(\frac{\partial F}{\partial p}\frac{\partial f}{\partial x} - \frac{\partial F}{\partial x}\frac{\partial f}{\partial p}\right) + \\ &\left(\frac{\partial F}{\partial q}\frac{\partial f}{\partial y} - \frac{\partial F}{\partial y}\frac{\partial f}{\partial q}\right) + \\ &\frac{\partial z}{\partial x}\left(\frac{\partial F}{\partial p}\frac{\partial f}{\partial z} - \frac{\partial F}{\partial z}\frac{\partial f}{\partial p}\right) + \\ &\frac{\partial z}{\partial y}\left(\frac{\partial F}{\partial q}\frac{\partial f}{\partial z} - \frac{\partial F}{\partial z}\frac{\partial f}{\partial q}\right) + \\ &\frac{\partial q}{\partial x}\left(\frac{\partial F}{\partial p}\frac{\partial f}{\partial q} - \frac{\partial F}{\partial q}\frac{\partial f}{\partial p} + \frac{\partial F}{\partial q}\frac{\partial f}{\partial p} - \frac{\partial F}{\partial p} - \frac{\partial f}{\partial q}\right) = 0 \quad \text{(viii)} \end{align*} \]

\[ \Rightarrow \frac{\partial F}{\partial p}\left(\frac{\partial f}{\partial x} + \frac{\partial z}{\partial x}\frac{\partial f}{\partial z}\right) + \frac{\partial F}{\partial q}\left(\frac{\partial f}{\partial y} + \frac{\partial z}{\partial y}\frac{\partial f}{\partial z}\right) + \left(-\frac{\partial z}{\partial x}\frac{\partial f}{\partial p} - \frac{\partial z}{\partial y}\frac{\partial f}{\partial q}\right)\frac{\partial F}{\partial z} + \left(-\frac{\partial f}{\partial p}\right)\frac{\partial F}{\partial x} + \left(-\frac{\partial f}{\partial q}\right)\frac{\partial F}{\partial y} = 0 \quad \text{(x)} \]

This is a Lagrange's Linear equation of order one with \( x, y, z, p, q \) as independent and \( F \) as the dependent function. The auxiliary equations of (x) are:

\[ \frac{dp}{\frac{\partial f}{\partial x} + p\frac{\partial f}{\partial z}} = \frac{dq}{\frac{\partial f}{\partial y} + q\frac{\partial f}{\partial z}} = \frac{dz}{-p\frac{\partial f}{\partial p} - q\frac{\partial f}{\partial q}} = \frac{dx}{-\frac{\partial f}{\partial p}} = \frac{dy}{-\frac{\partial f}{\partial q}} = \frac{dF}{0} \quad \text{(xi)} \]

Now any solution (integral) of equation (xi) will satisfy Equation (x) consider the simplest relation involving at least one of \( p \) or \( q \) for \( F = 0 \).

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