# Confusion in understanding the definition of a linear and quasi-linear PDE.

I was recently studying Partial Differential Equations (PDE).

While going through the basics, I stumbled across the definition of a linear PDE and quasi-linear PDE. The definition went as follows:

• A partial differential equation is said to be linear if it is linear in the unknown function and all its derivatives with coefficients depending only on the independent variables;

• A partial differential equation is said to be quasi-linear if it is linear in the highest-ordered derivative of the unknown function.

First of all, I know that a linear equation having $$n-$$ variables $$x_1,x_2,\cdots,x_n$$ is defined as $$a_1x_1+a_2x_2+\cdots+a_nx_n=c$$ where $$a_i$$ ($$1\leq i\leq n$$) and $$c$$ are real constants.

Now, coming back to the given definitions, I am confused about what the definitions imply?

In the case of a linear PDE, it's said that a PDE is linear, if it is linear in the unknown function and all its derivatives with coefficients depending only on the independent variables.

The phrase "linear in the unknown function" seems analogous to the phrase "linear in $$n-$$ independent variables."

When we say,

A linear equation in $$n-$$ independent variables say, $$x_1,x_2,\cdots,x_n$$ then it means the equation that we are talking about is of the form $$a_1x_1+a_2x_2+\cdots+a_nx_n=c$$ where $$a_i$$ ($$1\leq i\leq n$$) and $$c$$ are real constants.

So, following a similar reasoning, the phrase, "An equation linear in the unknown function" seems to have a meaning as follows:

The equation linear in the unknown function say, $$u$$ means the equation that we're referring to has the form $$au=c,$$ for some real constants $$a,c.$$

Similarly, the phrase "linear in all its derivatives with coefficients depending only on the independent variables" with reference to the given definition of a linear PDE seems to mean the following:

We have a function $$u$$ that depends upon say, independent variables $$x_1,x_2,\cdots,x_n.$$

Hence, a function linear in all its (i.e $$u$$'s) derivatives with coefficients depending only on the independent variables (of $$u$$) means that the function we're referring to has the form $$a_1u_{x_1}+a_2u_{x_2}+\cdots+a_nu_{x_n}=c$$ where $$a_i=a_i(x_1,x_2,\cdots,x_n)$$ ($$1\leq i\leq n$$), i.e $$a_i$$'s are functions of $$x_1,x_2,\cdots,x_n$$ and $$c$$ is real constant.

## Final Meaning of a Linear PDE

So, the total phrase,

" An equation is linear in the unknown function ($$u$$) and all its derivatives ($$u_{x_1},u_{x_2},\cdots ,u_{x_n}$$) with coefficients depending only on the independent variables( $$x_1,x_2,\cdots,x_n$$)" means the equation that we are talking about has the form

$$au+a_1u_{x_1}+a_2u_{x_2}+\cdots+a_nu_{x_n}=c,$$ where $$a,c$$ are real constants and $$a_i=a_i(x_1,x_2,\cdots,x_n)$$ ($$1\leq i\leq n$$), i.e $$a_i$$'s are functions of $$x_1,x_2,\cdots,x_n.$$

Are the above interpretations correct?

Now, the meaning of a quasi-linear PDE confuses me. I am unable to decipher its meaning.

I think there might be tons of posts seeking clarification on these definitions on this site. I was able to find some of them myself. But unfortunately, none of them seems to address the type of issues I am facing and I was unable to find any posts that addresses exactly the issues I've written down here. So, any help regarding this will be greatly appreciated.

I tried my best to be clear in stating the issues explicitly. If there's any confusion with it then I am more than willing to cooperate.

I remember from my general PDE course a linear differential equation is of the form $$$$\label{2} \sum_{|\alpha| \leq k} a_\alpha(x) D^\alpha u = f(x).$$$$ A partial differential equation that is not of the previous form will be called non-linear.

A PDE, of order $$k$$, is semi-linear if it has the form $$$$a(x) D^k u + a_0 (D^{k-1} u, \ldots, Du, u, x) = 0,$$$$ that is, if it can be written as the sum of a "linear" function of the derivative of order $$k$$ plus any function of the rest of the derivatives. This means that semi-linear equations are those whose coefficients of the terms that accompany the derivative of order $$k$$ only depend on $$x$$, NOT depending on $$u$$ or its derivatives.

A linear PDE is quasilinear if it is not semilinear and can be written as follows $$$$a(D^{k-1}u, \ldots, Du, u, x) D^k u + a_0(D^{k-1}u, \ldots, Du, u, x) = 0.$$$$ This means that quasilinear equations are those where the higher order terms depend on $$x, u, \ldots, D^{k-1}u$$ but not on $$D^k u$$.

These definitions come in Evan´s book

Any ODE of n-th order of $$\begin{array}{l} \left(y^{\left((0),\dots,(n)\right)} \ = \ \text{NestList}(f\to f',\ y, \ n) \ = \ (y,\ y',\dots, \ y^{(n)}\right) \quad \mapsto \quad F(x,y,y',\dots, y^{(n)}\\ F=0\end{array}$$ is said to be explicit, if it can be solved for the highest derivative

$$y^{(n)}= \begin{cases} \left( F^{(-1,n+1)}(x,y,\dots y^{n-1}) \right)_1 , \quad \text{cond}_1\left(\ (x,y,\dots y^{n-1})\ \right)\\ \vdots\end{cases}$$ with the $$F^{(-1,n+1)}$$ denoting the inverse function of $$F$$ wrt to the last argument, all other arguments treated as constants.

If there is only one such a case globally for the total domain of the independent variable $$x$$, it is said to be quasilinear.

If F is not linear in $$y^{(n)}$$ but uniquely solvable, there exists a monotone transformation of all $$y^{(n)}$$ such that $$F$$ is linear $$y^{(n)}$$.

If this form is linear in all $$\left(y^{(k)}\ | \ 0\le k \le n\right)$$, too, it's said to be linear.

By $$y^{(k)}(x) \mapsto y_k(x)$$ $$F=0$$ expands to a truly linear system

$$y_k' = \sum_i a_{ik}(x) \ y_i + c_i$$.

Linear system are handled by the methods of linear algebra with the complication, all coefficients being functions of the independent variable $$x$$.

Linear systems are categorized as homogenous and inhomogenous systems, invoking the complete corpus of linear algebra, chapters solvability and constructing affine spaces of solutions, finding kernels and eigensystems.

Because the system is a local, linear system of equations in the tangent space of a $$n+1$$- dimensional manifold, at each point the system defines a tangent directional vector field.

All these facts become competely transparent by studying the numerical methods e.g. by direct discretization and applying and Finite Elements Methods (FEM) in all $$n+1$$ variables, $$y_k' dx \to y_k(x+1)-y_k(x)$$. Or by integral transforms and application of Fast Fourier Transform (FFT) methods to discretizations.

The last method applied to (imagined) models (of 'reality') in physics, engineering, communication and finance can be said to become the foundation of the post-modern society