SYMMETRY CONSIDERATIONS FOR DIFFERENTIAL EQUATION FORMULATIONS FROM CLASSICAL AND FRACTIONAL LAGRANGIANS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The utility of Noether’s classical theorem on differential equations extended to a generalized nonclassical theorem is the focus of this paper. After addressing a couple of standard related Partial Differential Equation (P.D.E.) formulations from classical Lagrangians, it culminates into a non-classical formulation of the diffusion equation in one spatial dimension from a fractional Lagrangian. Comparisons and contrasts between techniques for the classical and fractional formulations, as done here, facilitate the basic computational methods required for building analytical results. A noteworthy interface between Distribution theory, Trace theory and Lie symmetry theory is a key point of interest in this study.


INTRODUCTION
Optimization of regular functionals on Banach spaces leads to formulation settings for differential equations, from which symmetry considerations can often reveal much about possible solutions. This observation is particularly useful in physical systems governed by such differential equations, as the qualitative and quantitative properties of equilibrium states of these systems can be inferred from admitted symmetry groups and the associated group invariant solutions. Much classical work has been dedicated to this profound concept, with Noether's theorem at the hub of its efficacy and development. Indeed, the concern about which differential equations can be formulated via optimization in the calculus of variations, remains very cogent in several modern computational scientific endeavors. Fractional calculus, being required for extrapolation to generalized non-classical versions of Noether's theorem, has paved the way for inclusion of 'conservation laws' for certain dissipative systems. Needless to mention, this has immensely expanded the utility of Noether's theorem in extensive analysis of differential equations, particularly in the 21st century. In this study, we shall visit the formulation of Laplace's equation and Poisson's equation, the latter which is often regarded as the static version of the diffusion equation. The transition across these formulations is a vital concept highlighted in [4] in appreciable detail. In conjunction with modern methods from the fractional calculus of variations and some key classical methods summarized in the reference by Olver, substantive symmetry considerations of the diffusion equation are examined from the vantage point of its fractional Lagrangian. Reconciliation between partial differential equations that can be formulated from the calculus of variations and those equations which possibly admit infinitesimal symmetries is pointed out as a vital link being sought.

FORMULATION OF LAPLACE'S EQUATION AND POISSON'S EQUATION
Classical related equations and their formulation techniques shall be elucidated in this section. We may begin with the task of optimizing a given functional defined on an improper subset of some Banach functional space. In this event, we have the optimization theorem below as a guarantor of existence of an extremal, given fulfillment of the included criteria.
Optimization Theorem (1) [ [3], pp.198 ] -Let E be a real reflexive Banach space, and the functional f : E → R ∪ {+∞} be convex, lower semi-continuous and proper. Then, i.) for any non-empty K ⊂ E that is weakly compact (closed, convex and norm-bounded) If in addition f is strictly convex, then the minimizer v would exist uniquely in either case mentioned above.
There are certain settings whereby this elegant optimization theorem is not quite relevant, such as when the Banach space of reference cannot be made reflexive without losing other crucial properties (such as continuous differentiability) of the test functions and candidates for the optimal solution v. Moreover, although this theorem can be tweaked by negating f to speak about local maxima, it does not reckon with criteria for seeking saddle points, which are evident legitimate critical points. By differentiation in the Banach spaces of reference and appropriate incorporation of the fundamental lemma of the calculus of variations, we are able to formulate differential equations which any solution v must satisfy [8]. There is providence for a reverse check via the Lax-Milgram theorem [ [2], pp.140 ] that a solution to the formulated differential equation is indeed an extremal of the function f in the above given optimization theorem (1). (2): Assume that a(u, v) is a continuous and coercive bilinear form on a Hilbert Space H. Then given any ϕ ∈ H * , there exists a unique u ∈ H such that

Lax-Milgram Theorem
Moreover, if a is symmetric, then u is characterized by In this event, we may begin with the boundary-constrained differential equation and then attempt to identify an associated optimization problem which its weak solution(s) must solve.
Hence, the Lax-Milgram theorem is a means to identify certain PDE's which can be formulated from optimization via the calculus of variations, with aid of standard multivariate integration formulas. As for equations of classically dissipative systems that are constructed from fractional Lagrangians, this particular scheme is not sufficient.
Intricacies of determination of admitted infinitesimal symmetries either starting from infinitesimal criteria of the PDE or the Lagrangian of its associated optimization problem shall be compared for the equations of choice here in due course. It is worthy of note that although sufficiently regular solutions of class C k : k ∈ N are often desired, the peculiarities of structures available in reflexive functional spaces compel us to first formulate equations weakly in larger Sobolev spaces (W k,p (Ω) : k ∈ N, p ≥ 1). Eventually, after establishing (unique) existence of weak solutions, we may check whether weak solutions are sufficiently regular as desirable.
These are the rigors of classically identified procedures for confirming existence and uniqueness of solutions to an appreciable class of partial differential equations. Hence, before even reckoning with explicit solution techniques, the links to variational problems already become evident in core P.D.E. analysis.

Formulation of Laplace's Equation:
Now, Laplace's equation is famously formulated via optimization of Dirichlet's energy functional. The Dirichlet energy functional on Ω is given by for Ω ⊂ R n an open, bounded set with C 1 topological boundary, where and h is a particular differentiable function defined on the compact set ∂ Ω.
The domain K of f in the weak setting is the pre-image of the singleton h ∈ L 2 (∂ Ω) under the continuous trace operator; so K is (norm-) closed in W 1,2 (Ω) := H 1 (Ω). Moreover, the set K is convex because and every u, v ∈ K . The functional f is continuous, coercive and strictly convex on K . For any function v 0 in K , it is easy to check that the set to the coercivity of f , giving us existence of a minimizer for f on B and thus also on K . The critical function v will exist uniquely due to strict convexity of f [8].
Given the minimizer v ∈ H 1 (Ω), the weak formulation for this problem is the following boundary value PDE: .
be the coordinates on Ω and dµ the volume element on Ω. Summarily, the technique of formulation of the associated differential equation via optimization of a sufficiently If Ω is an open subset of R n , then we may replace the divergence (div) operator above to express the formulation in (5) simply as These are the multivariate Euler-Lagrange equations, and we have that any critical point v of (4) must be a solution of (5).
The symmetries admitted by the Laplace equation (3) are identified as the conformal Lie groups and they include the infinitesimal rotations, translations and scalings; which may be relevantly utilized to simplify or explicitly solve the equation provided they leave the boundary More generic, engaging observations linking variational procedures and admitted symmetries in associated systems of differential equations shall be elucidated in each of the succeeding sections.

Formulation of Poisson's Equation:
Let u be a particular element of L 2 (Ω) for Ω in R n an open, bounded set with C 1 topological boundary. In the weak setting, Poisson's equation: is formulated by optimizing the functional Arguments using the given optimization theorem (1) above yield existence of a unique minimizer. Moreover, engagement of the multivariate Euler-Lagrange equations in the Lagrangian reveals that any critical point v satisfies the Poisson equation: on the reflexive Hilbert space H 1 0 (Ω).

INFINITESIMAL SYMMETRIES DETERMINED FROM LAGRANGIANS
The above variations also induce a variation of the gradient ∇v, which we denotẽ We require those variations which leave I invariant at its critical point(s) with respect to the parameter ε at ε = 0. This is to say, we require The higher order terms o(ε 2 ) in the variation [ṽx = ∇v + εη x + o(ε 2 ) ] are dropped because their derivatives evaluated at ε = 0 vanish, being a countable sum of zeros. Hence, In the above expression, [J] is the Jacobian for the coordinate transformation from the system which leads to the result: For the above integral to give the desired zero result, we require a pointwise null result for the integrand: At this juncture, we reckon that the vector field ξ i admitted by critical points of the functional I. Infinitesimal symmetries computed in this manner are identified as variational symmetries of I, with the prolongation vector coefficient(s) obtained as All the above are consistent with the definitions in formally established literature, and the opportunity to reconcile infinitesimal symmetries with variational techniques on the Lagrangian from the first principles presents us with a viable platform for symmetry considerations for differential equations formulated from Lagrangians. It is also well-known that infinitesimal symmetries from Lagrangians are admitted by the associated Euler-Lagrange equations. The result in (7) is relevant for computing symmetries of both equations (3) and (6) In quantitative terms, for a system of differential equations ∆(x, u (n) ) = 0, a conservation law is simply defined as a divergence expression Div P = 0 which vanishes for all solutions u = f (x) of the given system. We have P = (P 1 (x, u (n) ), · · · , P m (x, u (n) )) as an m-tuple of smooth functions of x, u and the derivatives of u, while Div P represents the total divergence. [ [7], pp.261]

FORMULATION OF THE DIFFUSION EQUATION
The diffusion equation The left Riemann-Liouville fractional integral of order α (0 < α < 1) of a function u(t) with respect to t is given as: Note that Γ denotes the hypergeometric Gamma function, which extends the usual factorial from the natural numbers to R − (Z − ∪ {0}), such that Hence, the left Riemann-Liouville fractional derivative of order α (0 < α < 1) of u(t) with respect to t is given as: The operators a I 1−α t and d dt do not commute, and switching their order from the Riemann-Liouville definition to a I 1−α t • d dt gives us the left Caputo fractional differential operator, denoted C a D α t . Importantly, we have that .
As for the right fractional differential operators, we have the right Riemann-Liouville derivative of order α : and the right Caputo derivative of order α: .
We shall feature only left fractional differential operators in the Lagrangians of this study, since we only consider evolutions toward the future and not the past. In [4], there is an effort by the authors to motivate development of a doubled phase space (u + , u − ) as an aspect of dynamical modeling of the future and past of an evolution process.
We shall impose the boundary value constraint [u(0, x) = 0] a-priori in any case, so that the left Caputo and Riemann-Liouville derivatives coincide. We shall hence employ the more con- respectively. With these observations, we have that the diffusion equation in one spatial dimension is formulated via optimization of the following fractional Lagrangian functional [4]: We may attempt to assess optimality conditions of the functional in (8) . For every v ∈ H 1 (Ω), we have that: In fractional integration by parts formula [1], [5] to make the term [ Ω u t .v] in the given expression for a(u, v) into a symmetric form with fractional derivatives: Speaking now of symmetries from the fractional Lagrangian, we have a similar setting to the classical case, outlined as follows. Consider a functional I = x, u, D α t u, u x ) dtdx, perturbed by one-parameter infinitesimal variations (t,x,ũ) of the independent variables (t, x) and the dependent variable u: (t,x,ũ) = (t, x, u) + ε(τ, ξ , η) .
The above variations also induce variations of the fractional derivative D α − u := u α t and the integer derivative u x , which we respectively denote We require those variations which leave I invariant at its critical point(s) with respect to the parameter ε at ε = 0 [5]. This is to say, we require In the above expression, [J] is the Jacobian for the coordinate transformation from the system (t, x, u) to (t,x,ũ), and the unperturbed Lagrangian is F = F(t, x, u, D α t u, u x ) . As such, we have the result: For the above integral to give the desired zero result, we require a pointwise null result for the integrand: Here, the vector field τ.
admitted by critical points of the functional I. The fractional prolongation coefficient in the above vector field [η α,t ] is obtained via the generalized chain and Leibniz rules of fractional calculus [9]. Its explicit expression relevant in our formulation for [α = 0.5] is hereby given as follows: where µ in the above expression is given as We refer the reader to [10] for details of how the above coefficient is determined.

COMPUTATIONAL RESULTS
Importantly, we reckon that each variational symmetry must also be a symmetry of the Euler- is the integrand of the Lagrangian associated to the Poisson equation, and the arbitrary constant k is ∂ η ∂ v for this case. At first glance, notice that the term k.F 1 above appears to be anomalous to the infinitesimal criterion (7) derived previously, while beginning symmetry computations from the Lagrangian.
For the diffusion equation, we find by way of computation that if [u t = u xx ] admits an infinites- is the integrand of the fractional Lagrangian (8) associated to the diffusion equation, and the coefficient in the apparently anomalous term [Q(x).F 2 ] for this case is given as Terminating our findings at this juncture would tend to suggest dearth of variational symmetries for each case. However, further considerations from the proposition below suggest a possible total or partial correspondence between existing variational symmetries and symmetries of the Euler-Lagrange equations, given appropriate imposition of the trace boundary constraints.
As the proposition, a compelling cue from the above computational results is that the boundary value constraints of well-posed P.D.E's must be suitably incorporated in the associated Lagrangian functionals in order to fully realize the correspondence between variational symmetries and symmetries of the Euler-Lagrange equations. When attempting to obtain the associated optimization problems from the boundary value P.D.E's using the mechanism available in the stated Lax-Milgram theorem (and its alternative versions) in each case, there is always a trace boundary term, which none of the herein identified Lagrangians actually incorporate. It is feasible and agreeable to concisely incorporate these boundary terms without leaving a second integral in the Lagrangian by way of Stoke's theorem for manifolds of finite volume: for an n-dimensional open set Ω and an (n − 1) differential form ρ. We have to also assume relevance of the given Stoke's theorem for weak exterior derivatives of ρ instead of just the classically analytic functional cases, because existence and uniqueness theorems for P.D.E's cannot be extrapolated to spaces C ∞ (Ω). The formulation of Poisson's equation is worth being given a final thought at this point. Because its functional space of formulation is H 1 0 (Ω), there should be zero trace contribution to the Lagrangian, when starting formulation from the Euler-Lagrange equation (that is, the P.D.E without imposed boundary value constraint). However, we must recall that we have only equivalence classes of functions in the setting of Sobolev spaces, meaning that v ≡ 0 almost everywhere on ∂ Ω in (6). Taking the weak exterior differential of the trace value under this consideration would then cause a non-zero contribution to the Lagrangian.
When imposed boundary values interact with the hypersurface ∂ Ω appropriately, then we have a greater chance of realizing more variational symmetries in P.D.E's formulated from the calculus of variations. As such, the observations made in this paper present a prospective frontier for a meaningful and interesting interface between distribution theory, trace theory and infinitesimal symmetry theory.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.