Analogues to Lie Method and Noether’s Theorem in Fractal Calculus

: In this manuscript, we study symmetries of fractal differential equations. We show that using symmetry properties, one of the solutions can map to another solution. We obtain canonical coordinate systems for differential equations on fractal sets, which makes them simpler to solve. An analogue for Noether’s Theorem on fractal sets is given, and a corresponding conservative quantity is suggested. Several examples are solved to illustrate the results.


Introduction
In past centuries, discussions in physics included regular objects such as straight lines, squares, spheres, cones, etc. Functions were smooth or involved a few singularities.Fractal sets or curves are shapes that are irregular or whose Hausdorff dimension exceeds their topological dimension.Fractals are certain shapes that share a common feature such as irregularities on a large range of scales and whose properties, like density, length, area, and volume, are not meaningful.The surface of human lungs and snowflakes, the boundaries of clouds, and the folds of mammalian brains are in the category of fractals.The theory of heat and wave transfer in disordered systems was modeled by fractals and random walks on them, such as polymers, fractured and porous rocks, amorphous semiconductors, etc. [1][2][3][4][5][6][7][8][9][10].Clusters in nature are the points at which the density of points does not have meaning as a quantifier.In processes with fractal structures, we note that there are no perfect Cantor sets, von Koch curves, or Sierpinski gaskets in nature.But, they can be reasonable approximations to natural shapes and are simpler to analyze since they are systematic mathematical constructions [2,3,11,12].
Cantor-like sets are in the class of dusts and totally disconnected sets.Fractal von Koch curves can be used to model natural irregular curves that do not have tangents and for which using smaller yardsticks leads to an increase in the measured length of the curve.The fractal Sierpinski gasket is a good model for objects such as the backbones of percolating clusters [1,13,14].
To solve this problem, fractional local derivatives were defined and turn out to be applicable to the differentiability of graphs that are fractals [42,43].
Probabilistic approach: In this approach, the Laplacian is defined as an infinitesimal generator of Brownian motion on fractal sets.Using self-similarity properties, Laplacians on connected Sierpinski sets were suggested, and the solutions of equations were obtained by utilizing the self-similarities of fractals [16,44].Generalized constructions using Brownian motion are given for more general nested fractals [45].
Measure theory approach: Measure theory is used to define suitable derivatives for fractal subsets of .If m is a measure and its support is a fractal subset of the real line F ∈ [a, b] ∈ , then a function g : F → is called a m-derivative of a given function g : F → if the following condition is satisfied: Note that the function g (y) does not always exist and is not unique for a given g(y).If g ∈ L 2 (F, m), then it solves both of these problems [6,19,[46][47][48][49].
F α -Calculus approach: F α -Calculus (F α -C) is a simple, constructive, and algorithmic approach to the analysis of fractals [20,21].In [22], F α -C was formulated as a calculus framework on fractal sets and fractal curves in higher dimensions.As an application of F α -C in celestial mechanics, the motion of simple harmonic oscillators and Kepler's Third Law on fractal-time spaces were explained and obtained [50].The existence and uniqueness of the solutions have important roles in applications that were studied on fractal sets [51,52].
Recently, transport through pre-fractal and porous media has been modeled using Lévy flights, Lévy walks, scaled random walks, and corresponding diffusion equations [53,54].Fractal scaled Brownian motion and ultra-slow fractal scaled Brownian motion were studied and the corresponding fractal mean square displacements are suggested [55,56].Sub-, normal-, and super-diffusion on middle-ξ Cantor sets was characterized in view of F α -C [23].
Lie groups have an important role since they give symmetries of physical laws and help in finding conserved quantities in physics by using Noether's theorem.For example, Lie symmetries of the Lagrangian of a system give conserved quantities [57][58][59][60][61][62][63][64][65][66].In this work, we generalize Lie methods for differential equations on fractal Cantor-like sets and Noether's theorem.
The outline of the paper is as follows: Relevant background and various definitions are given in Section 2. In Section 3, we discuss symmetry and the Lie method for solving differential equations on fractal sets.In Section 4, we give a generalized Noether's theorem for fractal Cantor-like sets.Section 5 offers a conclusion.

Basic Tools
In this section, we review some of the basic definitions of fractal calculus, which was adapted for the middle-ξ Cantor sets.For details see [20,21,23].
and r, t ∈ is a middle-ξ Cantor set.Then the flag function of C ξ is defined by where 0 < α ≤ 1.Let δ > 0, then the coarse-grained mass function γ α δ (C ξ , r, t) is given by Here we take infimum over all subdivisions E satisfying |E| := max 1≤i≤n (x i − x i−1 ) ≤ δ.
The mass function γ α (C ξ , r, t) is defined by [20,21,23]: The integral staircase function S α C ξ (x) of order α for a fractal set C ξ is defined in [20,21] by where r 0 is an arbitrary real number and fixed.
In Figure 1, we present middle-ξ Cantor sets and their staircase functions.
Remark 1.For a given ξ, then we have dim β (C ξ ∩ [r, t]) = α.For example, for ξ = 1/5 we get α ≈ 0.77.The plot in Figure 2 indicates the approximation of γ α δ 2 /γ α δ 1 where δ 2 < δ 1 .This value leads to the β-dimension since it shows convergence to the finite valve while δ → 0, which can be seen by choosing different various pairs of (δ 1 , δ 2 ).The C α -Limits: Suppose h : C ξ → R and x ∈ C ξ .Then l is called to be the limit of h through the points of The C α -Differentiation: The C α -derivative of a function u defined on C ξ at a point x is [20,21,23]: In view of infinitesimal calculus and non-standard analysis, Equation ( 2) is written [67] h where A more general form of the Taylor expansion formula is The C α -Integration: For a bounded function h on C ξ , one can define [20,21,23]: and similarly, The upper C α -sum and lower C α -sum for a function h over the subdivision E are given respectively by [20,21,23] and We say that h is C α -integrable on C ξ if [20,21,23] the following equations are equal.
In that case, the C α -integral of h on C ξ is denoted by t r h(x)d α C ξ x and is given by the common value of ( 4) and (5).Fundamental Theorems of C α -Calculus.Suppose that h(x) :

The Lie Method on C α -Calculus
In this section, we study Sophus Lie's method for solving linear and non-linear fractal differential equations [57][58][59].An infinitesimal generator is defined on the middle-ξ Cantor sets.A fractal Lie group is a set of maps with parameter η such that with the following properties [57][58][59]: (1) L η onto and one-to-one; (2) Symmetry condition of fractal differential equations: Consider a fractal differential equation of the form In order to find the fractal symmetry conditions, we write where By Equation (8), we have where might be called the fractal total derivative operator.From Equation (10), we obtain Substituting Equation ( 9) into Equation (11), we obtain which might be called the fractal symmetry condition.
Example 1.The following fractal differential equation has a fractal symmetry To show this, we substitute Equation (14) into Equation (12).Hence we get Then Equation (15) looks like Consequently, Equation (12) holds.

Orbit of a point in the fractal differential equations:
If H is a point on the solution of the fractal differential equation, then given a fractal symmetry map by choosing different values of η, we get an orbit of the point H.We demonstrate this by giving the following example.
Figure 3 shows the orbit of point H under the symmetry of Equation (17).Analogues to tangent vectors on fractal orbit: Analogues to the tangent vectors/fractal tangent vectors for any given orbit at the point (S α C ξ (t ), x ) are defined as follows: For the initial point (S α C ξ (t), x), we set η = 0, namely In what follows, we want to obtain an invariant solution of Equation ( 7) by using fractal tangent vectors.Therefore, we can write which is called a fractal symmetric equation.In view of Equation ( 20), we define Q as follows: which might be called a fractal characteristic function.Utilizing Equation ( 20), we get which might be called a fractal reduced characteristic function.Under the given symmetry, we conclude Q = 0.
Example 3. Consider the fractal Riccati differential equation as follows: Hence, Equation ( 22) has the following symmetry By Equation (19), the fractal tangent vectors are Therefore, if , which is the fractal invariant solution of Equation ( 22) under symmetry equation (23).
Example 4. Suppose one parameter fractal Lie group as follows: Then, the associated fractal tangent vectors is given by Equation (18), that is, and Evaluating Equations ( 24) and ( 25) at η = 0, we get To solve Equation (26), we use the conjugacy of C α -calculus with standard calculus; that is, Then, it is straightforward to get x(t) = c t k .
By inverse conjugacy, we have which might be called fractal canonical coordinates.
Example 5. Consider the fractal equation Riccati equation as follows: with the symmetry (S α C ξ (t ), x ) = (e η S α C ξ (t), e −2η x).In the same manner, using Equations ( 24) and (25), we can see that Using the conjugacy of C α -calculus with ordinary calculus, we can write Then, inverting the conjugacy leads to which is the solution of Equation (28).
Remark 2. Note that by setting k = 0 in Equation (29), we can obtain the invariant solution.
Linearized symmetry condition for the fractal differential equations: Solving the fractal symmetry condition in Equation ( 12) is often very difficult or impossible.Therefore, we linearize Equation ( 12) by using Taylor series expansion, namely, Here, O(η 2 ) = e(η) describes the error function of Taylor series expansions, such that lim η→0 e(η) Substituting Equation (30) into Equation ( 12) and disregarding terms of η 2 or higher orders, we have Example 6.Consider a fractal differential equation Substituting Equation (32) into Equation (31), we get Then, to solve Equation (33), let φ = 0 so that we have Conjugacy of the fractal calculus with standard calculus gives If we set c = 1, then we get the fractal canonical coordinates ).
Moreover, since we have we can write Substituting Equation (39) into Equation (38), one can obtain following: where k is constant.Replacing S α C ξ (t) and x back into Equation (40), we obtain

Infinitesimal fractal generator:
In view of the fractal symmetry group of Equation ( 6), the infinitesimal fractal generator is defined by where X C ξ is called an infinitesimal fractal generator.
Example 7. Consider the fractal Lie group of the fractal differential equation as follows: The associated the fractal tangent vectors are Then, the fractal infinitesimal generator is Example 8. Consider the fractal infinitesimal generator We can calculate the fractal tangent vectors using Equation (18) as follows: In addition, we have Hence, the fractal Lie symmetry of Equation (42) will be

Noether's Theorem for Lagrangians with Fractal Set Support
Noether's theorem presents the connection between conservation laws and Lie symmetries.For every Lie symmetry, there is a conserved quantity in the system.The Lagrangian on middle-ξ Cantor sets is not a differentiable manifold in the sense of standard calculus.Here, we consider fractal calculus to generalize Noether's Theorem to include the wider class of the Lagrangian.Consider a fractal Lagrangian as follows: where which might be called a fractal Lie group of Lagrangian.The existence of Equation ( 44) leads to conserved quantities of the system.Equation ( 44) is a functional equality, so that one can write If we expand L C ξ using Taylor series, we have where φ, D α C ξ φ, and ρ are defined by Local fractal symmetries are given by Equation (47).Fractal Noether's Theorem is given by where which is called Hamiltonian on fractal sets.
The fractal Hamiltonian is obtained by Equation (49): which is conserved.

Conclusions
In this paper, the Lie method for solving differential equations was extended to C α -calculus.Analogues for the orbit of a point in view of fractal differential equations were defined.Using linearized symmetry conditions, canonical coordinates on fractal differential equations were derived.Analogues to tangent vectors utilizing C α -C were suggested.Infinitesimal generators applying symmetry properties were presented.Noether's Theorem was expanded to non-differentiable manifolds such as Lagrangians with middle-ξ Cantor sets.Some examples were worked out to show the details.