diff --git a/operator_examples/first_target.jl b/operator_examples/first_target.jl
new file mode 100644
index 0000000..b78a19f
--- /dev/null
+++ b/operator_examples/first_target.jl
@@ -0,0 +1,141 @@
+using LinearAlgebra, DiffEqOperators
+import DiffEqBase: AbstractDiffEqLinearOperator
+import Base: *, size, convert
+
+#############################
+# Matrix-free operators
+# The operators are defined as a subtype of AbstractDiffEqLinearOperator to make use of the composition
+# functionality. In actual code we should define a separate AbstractDerivativeOperator type hierarchy.
+# Only the most essential interface for AbstractDiffEqLinearOperator is defined: multiplication,
+# size (needed for linear combination) and conversion to AbstractMatrix (needed for linsolve).
+struct GenericDerivativeOperator{LT,QT} <: AbstractDiffEqLinearOperator{Float64}
+  L::LT
+  QB::QT
+end
+size(LB::GenericDerivativeOperator) = (size(LB.L, 1), size(LB.QB, 2))
+*(LB::GenericDerivativeOperator, x::AbstractVector{Float64}) = L.L * (L.QB * x)
+convert(::Type{AbstractMatrix}, LB::GenericDerivativeOperator) = convert(AbstractMatrix, LB.L) * convert(AbstractMatrix, LB.QB)
+
+struct UniformDiffusionStencil <: AbstractDiffEqLinearOperator{Float64}
+  M::Int
+  dx::Float64
+end
+size(L::UniformDiffusionStencil) = (L.M, L.M+2)
+*(L::UniformDiffusionStencil, x::AbstractVector{Float64}) = [x[i] + x[i+2] - 2x[i+1] for i = 1:L.M] / L.dx^2
+function convert(::Type{AbstractMatrix}, L::UniformDiffusionStencil)
+  # spdiagm/diagm always creates a square matrix so we have to construct manually.
+  # It's likely that more efficient constructions exist.
+  mat = zeros(size(L))
+  for i = 1:L.M
+    mat[i,i] = 1.0
+    mat[i,i+1] = -2.0
+    mat[i,i+2] = 1.0
+  end
+  return mat / L.dx^2
+end
+
+struct GenericUpwindOperator{LT,QT,CT} <: AbstractDiffEqLinearOperator{Float64}
+  L::LT
+  QB::QT
+  coeff::CT
+end
+function GenericUpwindOperator(L, QB)
+  coeff = DiffEqScalar(1.0)
+  GenericUpwindOperator(L, QB, coeff)
+end
+size(LB::GenericDerivativeOperator) = (size(LB.L, 1), size(LB.QB, 2))
+function *(LB::GenericUpwindOperator, x::AbstractVector{Float64})
+  xbar = LB.QB * x
+  # This only covers the case when L.coeff is a scalar. Need to add support for
+  # vector coefficients later.
+  c = convert(Number, LB.coeff)
+  # This part should be changed to use the stencil coefficients in L
+  if c > 0 # backwards difference
+    return [xbar[i+1] - xbar[i] for i = 1:LB.L.M] * (c / LB.L.dx)
+  else # forward difference
+    return [xbar[i+2] - xbar[i+1] for i = 1:LB.L.M] * (c / LB.L.dx)
+  end
+end
+function convert(::Type{AbstractMatrix}, LB::GenericUpwindOperator)
+  Lmat = zeros(size(LB.L))
+  c = convert(Number, LB.coeff)
+  if c > 0 # backwards difference
+    for i = 1:L.LB.M
+      Lmat[i,i] = -1.0
+      Lmat[i,i+1] = 1.0
+    end
+  else # forward difference
+    for i = 1:L.LB.M
+      Lmat[i,i+1] = -1.0
+      Lmat[i,i+2] = 1.0
+    end
+  end
+  Lmat .*= (c / L.LB.dx)
+  Qmat = convert(AbstractMatrix, LB.QB)
+  return Lmat * Qmat
+end
+function *(a::DiffEqScalar, LB::GenericUpwindOperator)
+  # Need to define *(::DiffEqScalar, ::DiffEqScalar) for this to work
+  return GenericUpwindOperator(L.M, L.dx, a * L.coeff)
+end
+
+struct UniformDriftStencil <: AbstractDiffEqLinearOperator{Float64}
+  M::Int
+  dx::Float64
+end
+size(L::UniformDriftStencil) = (L.M, L.M+2)
+
+# TODO: stencil operators for irregular grids
+
+struct QRobin <: AbstractDiffEqLinearOperator{Float64}
+  M::Int
+  dx::Float64
+  al::Float64
+  bl::Float64
+  ar::Float64
+  br::Float64
+end
+size(Q::QRobin) = (Q.M+2, Q.M)
+function *(Q::QRobin, x::AbstractVector{Float64})
+  xl = x[2] + 2*Q.al*Q.dx/Q.bl * x[1]
+  xr = x[end-1] - 2*Q.ar*Q.dx/Q.br * x[end]
+  return [xl; x; xr] # could optimize using LazyArrays.jl
+end
+
+######################################
+# Mock user interface
+# DerivativeOperator serves as a factory method to generate derivative operators.
+# It generates either stencil operators (for which xgrid is the extended domain),
+# or a square operator if BC is provided (for which xgrid is the interior).
+# For the mock interface, BC will always be Robin and is specified as the named tuple
+# (al, bl, ar, bl).
+function DerivativeOperator(xgrid::AbstractRange{Float64}, dorder, aorder)
+  M = length(xgrid) - 2
+  dx = step(xgrid)
+  # This section will be replaced by DiffEqOperator's Fornberg functions
+  if dorder == 2 && aorder == 2
+    return UniformDiffusionStencil(M, dx)
+  elseif dorder == 1 && aorder == 1
+    return UniformDriftStencil(M, dx)
+  end
+end
+function DerivativeOperator(xgrid::AbstractRange{Float64}, dorder, aorder, BC)
+  M = length(xgrid) - 2
+  dx = step(xgrid)
+  if dorder == 2 && aorder == 2
+    L = UniformDiffusionStencil(M, dx)
+    QB = QRobin(M, dx, BC.al, BC.bl, BC.ar, BC.br)
+    return GenericDerivativeOperator(L, QB)
+  elseif dorder == 1 && aorder == 1
+    L = UniformDriftStencil(M, dx)
+    QB = QRobin(M, dx, BC.al, BC.bl, BC.ar, BC.br)
+    return GenericUpwindOperator(L, QB)
+  end
+end
+
+function DerivativeOperator(xgrid::AbstractVector{Float64}, dorder, aorder)
+  error("Irregular grid not yet supported") # TODO
+end
+function DerivativeOperator(xgrid::AbstractVector{Float64}, dorder, aorder, BC)
+  error("Irregular grid not yet supported") # TODO
+end