Draft PR: Will be working on differential equations function transformations.

src/systems/diffeqs/basic_transformations.jl

@@ -46,6 +46,7 @@ Optimal Transport Approach
 Abhishek Halder, Kooktae Lee, and Raktim Bhattacharya
 https://abhishekhalder.bitbucket.io/F16ACC2013Final.pdf
 """
+
 function liouville_transform(sys::AbstractODESystem)
     t = get_iv(sys)
     @variables trJ
@@ -55,3 +56,107 @@ function liouville_transform(sys::AbstractODESystem)
     vars = [unknowns(sys); trJ]
     ODESystem(neweqs, t, vars, parameters(sys), checks = false)
 end
+
+"""
+$(TYPEDSIGNATURES)
+
+Transforms an ODE system by applying a set of variable transformations and their inverses (back-transformations). 
+Returns the new set of differential equations and the updated list of variables after substitution and simplification. This is useful for redefining an ODE system in terms of new dependent variables while preserving the dynamics.
+
+For example, transforming `x` to `z = exp(x)` redefines the system in terms of `z`, adjusting the equations accordingly.
+
+Example:
+
+```julia
+using ModelingToolkit
+
+@parameters t α
+@variables x(t) z(t)
+D = Differential(t)
+eqs = [D(x) ~ α*x]
+sys = ODESystem(eqs, t, [x], [α])
+transformations = [z => exp(x)]
+back_transformations = [x => log(z)]
+new_eqs, new_vars = transform_eqs(sys, transformations, back_transformations)
+# new_eqs: [D(z) ~ α*z], new_vars: [z]
+"""
+
+
+function transform_eqs(sys::ODESystem, transformations, back_transformations)
+    all_dept_vars = unknowns(sys)
+    diff_eqs = equations(sys)
+    diff_eqs_map = Dict(eq.lhs => eq.rhs for eq in diff_eqs)
+    eq_transformations = [k ~ v for (k, v) in transformations]
+
+    new_vars = [] # transformed variables
+    mutable_vars = [] # mutable variables
+    for tr in transformations
+        push!(new_vars, tr.first)
+        change_var = get_variables(tr.second)
+
+        # change_var can also be a parameter like α. hence the check. 
+        for every_change_var in change_var
+            for all_var in all_dept_vars
+                if isequal(every_change_var,all_var)
+                    push!(mutable_vars, all_var)
+                end
+            end
+        end
+    end
+
+    current_vars = setdiff(vcat(all_dept_vars, new_vars), mutable_vars)
+
+    # Differentiating on both sides of the transformation
+    dzdt = Symbolics.derivative(last.(transformations), t)
+    dz = Symbolics.derivative(first.(transformations), t)
+    new_eqs = Equation[]
+    for (dzi, dzdti) in zip(dz, dzdt)
+        updated_dzdti = substitute(dzdti, diff_eqs_map)
+        exp = dzi ~ updated_dzdti
+        updated_exp = substitute(exp, Dict(back_transformations))
+        push!(new_eqs, Symbolics.simplify(updated_exp))
+    end
+
+    return new_eqs, current_vars
+end
+
+### `substitute_defaults`
+
+
+"""
+$(TYPEDSIGNATURES)
+
+Substitutes default values into transformation expressions for an ODE system, returning a dictionary of new default values for transformed variables. Optionally evaluates at a specific time `t_val`. Useful for initializing transformed systems with consistent initial conditions.
+
+For example, if `x => 1.0` is a default and `z = exp(x)` is a transformation, it computes `z => exp(1.0)`.
+
+Example:
+
+```julia
+using ModelingToolkit
+
+@parameters t α
+@variables x(t) z(t)
+eqs = [Differential(t)(x) ~ α*x]
+sys = ODESystem(eqs, t, [x], [α]; defaults=Dict(x => 1.0))
+transformations = [z => exp(x)]
+new_defaults = substitute_defaults(sys, transformations)
+# new_defaults: Dict(z => 2.718281828459045)
+"""
+
+function substitute_defaults(sys::ODESystem, transformations, t_val=missing)
+    existing_defaults = ModelingToolkit.get_defaults(sys)
+    new_defs = Dict()
+    for tr in transformations
+        ex = substitute(last(tr), existing_defaults)
+        if !ismissing(t_val)
+            ex = substitute(ex, t => t_val)
+        end
+        vars = get_variables(ex)
+        if length(vars) == 0
+            new_defs[first(tr)] = ex
+        end
+    end
+
+    return new_defs
+end