Test set added

src/derivative_operators/derivative_operator.jl

@@ -26,8 +26,6 @@ struct DerivativeOperator{T<:Real,N,Wind,T2,S1,S2<:SArray,T3,F} <: AbstractDeriv
     coeff_func              :: F
 end
 
-struct NonLinearDiffusion! end
-
 function NonLinearDiffusion!(du::AbstractVector{T}, second_differential_order::Int, first_differential_order::Int, approx_order::Int,
     p::AbstractVector{T}, q::AbstractVector{T}, dx::Union{T , AbstractVector{T} , Real},
     nknots::Int) where {T<:Real, N}

test/Fast_Diffusion.jl

@@ -0,0 +1,50 @@
+# # Fast-Diffusion Problem
+#
+# This example demonstrates the use of 'NonLinearDiffusion' operator to solve time-dependent non-linear diffusion PDEs with coefficient having dependence on the unknown function. 
+# Here we consider a fast diffusion problem with Dirichlet BCs on unit interval:
+# ∂ₜu = ∂ₓ(k*∂ₓu) 
+# k = 1/u²   
+# u(x=0,t) = exp(-t)
+# u(x=1,t) = 1/(1.0 + exp(2t))
+# u(x, t=0) = u₀(x)
+using Test
+using DiffEqOperators, OrdinaryDiffEq
+
+@testset "1D Nonlinear fast diffusion equation" begin
+    # The analytical solution for this is given by :
+
+    u_analytic(x, t) = 1 / sqrt(x^2 + exp(2*t))
+
+    #
+    # Reproducing it numerically 
+    #
+
+
+    nknots = 100
+    h = 1.0/(nknots+1)
+    knots = range(h, step=h, length=nknots)
+    n = 1                                   # Outer differential order
+    m = 1                                   # Inner differential order
+    approx_ord = 2                               
+
+    u0 = u_analytic.(knots,0.0)
+    du = similar(u0)
+    t0 = 0.0
+    t1 = 1.0
+
+    function f(du,u,p,t)
+        bc = DirichletBC(exp(-t),(1.0 + exp(2*t))^(-0.5))
+        l = bc*u
+        k = l.^(-2)                        # Diffusion Coefficient
+        NonLinearDiffusion!(du,n,m,approx_ord,k,l,h,nknots)
+    end
+
+    prob = ODEProblem(f, u0, (t0, t1))
+    alg = KenCarp4()
+    sol = solve(prob,alg)
+
+    for t in t0:0.1:t1
+        @test u_analytic.(knots, t) ≈ sol(t) rtol=1e-3
+    end
+
+end