Documentation improvements: fix grammar, typos, and clarity

README.md

@@ -1 +1,12 @@
-DiffEqDevMaterials
+# DiffEqDevMaterials
+
+Development materials, research notes, and examples for the SciML differential equations ecosystem.
+
+This repository contains:
+
+- **operators/**: Examples and tutorials for DiffEqOperators.jl, including solving PDEs (Heat equation, KdV equation, Bellman equations)
+- **newton/**: Research notes on nonlinear systems arising from stiff equations
+- **dae_adjoint/**: Derivation of semi-explicit DAE adjoint methods
+- **nordsieck/**: Materials on Nordsieck form for multistep methods
+
+These materials support the development of various SciML packages and provide deeper mathematical context for differential equation solving techniques.

dae_adjoint/README.md

@@ -1,3 +1,5 @@
-# Derivation of semi-explicit DAE adjoint
+# Derivation of Semi-Explicit DAE Adjoint
 
-The PDF is at output/main.pdf.
+This document derives the adjoint sensitivity method for semi-explicit differential-algebraic equations (DAEs).
+
+The PDF is available at `output/main.pdf`.

newton/README.md

@@ -1,3 +1,5 @@
-# A Brief Note on the Nonlinear Systems Arise from Stiff Equations
+# A Brief Note on the Nonlinear Systems that Arise from Stiff Equations
 
-The PDF is at output/main.pdf.
+This note explores the nonlinear systems that arise when solving stiff differential equations using implicit methods.
+
+The PDF is available at `output/main.pdf`.

operators/DiffEqOperators.md

@@ -2,7 +2,7 @@
 
 In this tutorial we will explore the basic functionalities of PDEOperator which is used to obtain the discretizations of PDEs of appropriate derivative and approximation order.
 
-So an operator API is as follows:-
+So an operator API is as follows:
 
     A = DerivativeOperator{T}
         (
@@ -20,21 +20,21 @@ Taking a specific example
 
     A = DerivativeOperator{Float64}(2,2,1/99,10,:Dirichlet,:Dirichlet; BC=(u[1],u[end]))
 
-this is the time independent Dirichlet BC. You can also specify a time dependent Dirichlet BC as follows:-
+this is the time-independent Dirichlet BC. You can also specify a time-dependent Dirichlet BC as follows:
 
     A = DerivativeOperator{Float64}(2,2,1/99,10,:Dirichlet,:Dirichlet; bndry_fn=(t->(u[1]*cos(t)),u[end]))
 
-We have generated an operator which produces the 2nd order approximation of the Laplacian. We can checkout the stencil as follows:-
+We have generated an operator which produces the 2nd order approximation of the Laplacian. We can check out the stencil as follows:
 
     julia> A.stencil_coefs
     3-element SVector{3,Float64}:
       1.0
      -2.0
       1.0
 
-We can get the linear operator as a matrix as follows:-
+We can get the linear operator as a matrix as follows:
 
-    julia> full(A)
+    julia> Matrix(A)
     10×10 Array{Float64,2}:
      -2.0   1.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0
       1.0  -2.0   1.0   0.0   0.0   0.0   0.0   0.0   0.0   0.0
@@ -50,7 +50,7 @@ We can get the linear operator as a matrix as follows:-
 Note that we **don't** need to define the `BC` only for `:D0` and `:periodic` boundary conditions so you can ignore it.
 
 
-Now coming to the main functionality of DiffEqOperators ie. taking finite difference discretizations of functions.
+Now coming to the main functionality of DiffEqOperators, i.e., taking finite difference discretizations of functions.
 
     julia> x = collect(0 : 1/99 : 1);
     julia> u0 = x.^2 -x;
@@ -76,8 +76,8 @@ The derivative values at the boundaries are in accordance with the `Dirichlet` b
 
 You can also take derivatives of matrices using `A*M` or `M*A` where the order of multiplication decides the axis along which we want to take derivatives.
 
-    julia> xarr = linspace(0,1,51)
-    julia> yarr = linspace(0,1,101)
+    julia> xarr = range(0, 1, length=51)
+    julia> yarr = range(0, 1, length=101)
     julia> dx = xarr[2]-xarr[1]
     julia> dy = yarr[2]-yarr[1]
     julia> F = [x^2+y for x = xarr, y = yarr]

operators/HeatEquation.md

@@ -1,6 +1,6 @@
 # Solving the Heat Equation using DiffEqOperators
 
-In this tutorial we will solve the famous heat equation using the explicit discretization on a 2D `space x time` grid. The heat equation is:-
+In this tutorial we will solve the famous heat equation using the explicit discretization on a 2D `space x time` grid. The heat equation is:
 
 $$\frac{\partial u}{\partial t} - \frac{{\partial}^2 u}{\partial x^2} = 0$$
 
@@ -11,7 +11,7 @@ For this example we consider a Dirichlet boundary condition with the initial dis
         julia> u0 = -(x - 0.5).^2 + 1/12;
         julia> A = DerivativeOperator{Float64}(2,2,2pi/511,512,:Dirichlet,:Dirichlet;BC=(u0[1],u0[end]));
 
-Now solving equation as an ODE we have:-
+Now solving the equation as an ODE we have:
 
         julia> prob1 = ODEProblem(A, u0, (0.,10.));
         julia> sol1 = solve(prob1, dense=false, tstops=0:0.01:10);

operators/KdV.md

@@ -1,9 +1,9 @@
-# Solving the Heat Equation using DiffEqOperators
+# Solving the KdV Equation using DiffEqOperators
 
 In this tutorial we will try to solve the famous **KdV equation** which describes the motion of waves on shallow water surfaces.
-The equation is commonly written as 
+The equation is commonly written as
 
-        $$\frac{\partial u}{\partial t} + \frac{{\partial}^3 u}{\partial t^3} - 6*u*\frac{\partial u}{\partial t} = 0$.
+        $$\frac{\partial u}{\partial t} + \frac{{\partial}^3 u}{\partial x^3} - 6u\frac{\partial u}{\partial x} = 0$$.
 
 Lets consider the cosine wave as the initial waveform and evolve it using the equation with a [periodic boundary condition](https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.15.240). This example is taken from [here](https://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation).
 
@@ -23,7 +23,7 @@ Now defining our DiffEqOperators
     C = DerivativeOperator{Float64}(3,2,1/99,199,:periodic,:periodic);
 ```
 
-Now call the ODE solver as follows:-
+Now call the ODE solver as follows:
 
     prob1 = ODEProblem(KdV, u0, (0.,5.));
     sol1 = solve(prob1, dense=false, tstops=0:0.01:10);