# On Conformable Laplace’s Equation

Authors: Francisco Martínez; Inmaculada Martínez; Mohammed K. A. Kaabar; Silvestre Paredes
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The most important properties of the conformable derivative and integral have been recently introduced. In this paper, we propose and prove some new results on conformable Laplace’s equation. We discuss the solution of this mathematical problem with Dirichlet-type and Neumann-type conditions. All our obtained results will be applied to some interesting examples.


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## Body

## 1. Introduction

The idea of fractional derivative was first raised by L’Hospital in 1695. After introducing this idea, many new definitions have been formulated. The most well-known ones are Riemann–Liouville and Caputo fractional definitions. For more background information about these definitions, we refer the reader to [1, 2]. A new definition of derivative and integral has been recently formulated by Khalil et al. in [3]. This new definition is a type of local fractional derivative [4]. This definition was proposed to overcome some of difficulties associated with solving the equations formulated in the sense of classical nonlocal fractional definitions where the solutions can be difficult to obtain or even impossible to obtain. As a result, various research studies have been conducted on the mathematical analysis of functions of a real variable formulated in the sense of conformable definition such as Rolle’s theorem, mean value theorem, chain rule, power series expansion, and integration by parts formulas [3, 5, 6]. In [7], the conformable partial derivative of the order α∈(0, 1] of the real-valued functions of several variables and the conformable gradient vector has been proposed, and conformable Clairaut’s theorem for partial derivative has also been investigated. In [8], the Jacobian matrix has been defined in the context of conformable definition, and the chain rule for multivariable conformable derivative has been also proposed. In [9], conformable Euler’s theorem on homogeneous has been successfully introduced.Furthermore, many research studies have been conducted on the theoretical and practical elements of conformable differential equations shortly after the proposition of this new definition [4, 10–26]. Conformable derivative has also been applied in modeling and investigating phenomena in applied sciences and engineering such as the deterministic and stochastic forms of coupled nonlinear Schrödinger equations [27] and regularized long wave Burgers equation [28] and the analytical and numerical solutions for (1 + 3)-Zakharov–Kuznetsov equation with power-law nonlinearity [29].Laplace’s equation is used as indicator of the equilibrium in applications such as heat conduction and heat transfer [30]. Generally, to solve the Laplace equation, Legendre’s differential equation, particularly the Legendre function or as commonly known as Legendre polynomials, is used to find a solution to the Laplace equation that indicates spherical symmetry in the physical systems [31]. Laplace equation can be widely seen in the field of heat transfer where the temperature is at different locations when the body's heat transfer is at the equilibrium point [30]. According to our knowledge, there are not many research studies that have been done on investigating Laplace’s equation in the sense of conformable derivative; therefore, all our results are considered new and worthy.This paper is organized as follows. In the next section, the main concepts of conformable fractional calculus are presented. Next, we successively discuss the solution of conformable Laplace’s partial differential equation with Dirichlet and Neumann conditions. Finally, the above results will be applied in some interesting examples to validate their applicability.


## 2. Basic Definitions and Tools

Definition 1.
Given a functionf:[0,∞)⟶R. Then, the conformable derivative of order α [3] is defined by(1)Tαft=limε⟶0ft+εt1−α−ftε,for all t > 0, 0 < α ≤ 1. If f is α− differentiable in some (0, a), a > 0, and limt⟶0h+Taft exists, then it is defined as(2)Tαf0=limt⟶0+Tαft.Theorem 1 (see [3]).
If a functionf:[0, ∞)⟶R is α-differentiable at t0 > 0, 0α ≤ 1, then f is continuous at t0.Theorem 2 (see [3]).
Let 0α ≤ 1 and let f, g be α−differentiable at a point t > 0. Then, we have(i)
Tαaf+bg=aTαf+bTαg,∀a,b∈R.(ii)
Tαtp=ptp−α,∀p∈R.(iii)
Tαλ=0,for all constant functionsft=λ.(iv)
Tαfg=fTαg+gTαf.(v)
Tαf/g=gTαf−fTαg/g2.(vi)
If, in addition,f is differentiable, then Tαft=t1−αdf/dtt.
The conformable derivative of certain functions using the above definition is given as follows:(i)
Tα1=0.(ii)
Tαsinat=at1−αcosat.(iii)
Tαcosat=−at1−αsinat.(iv)
Tαeat=aeat,a∈R.Definition 2.
The (left) conformable derivative starting froma of a given function f:[a,∞)⟶R of order 0 < α ≤ 1 [5] is defined by(3)Tαaft=lim∈⟶0ft+εt−a1−α−ftε.
Whena = 0, it is expressed as (Tαf) (t). If f is α−differentiable in some a, b, then the following can be defined:(4)Tαafa=limt⟶a+Tαaft.Theorem 3 (chain rule) (see [5]).
Letf, g:(a,∞)⟶R be (left) α-differentiable functions, where 0 < α ≤ 1. By letting h (t) = f (g(t)), h (t) is α-differentiable for all t ≠ a and g (t) ≠ 0; therefore, we have the following:(5)Tαaht=Tαafgt⋅Tαagt⋅gtα−1.
Ift = a, then we obtain(6)Tαaha=limt⟶a+Tαafgt⋅Tαagt⋅gtα−1.Theorem 4 (see [5]).
Assumef is infinitely α-differentiable function, for some 0 < α ≤ 1 at the neighborhood of a point t0. Then, f has the following fractional power series expansion:(7)ft=∑k=0∞kTαt0t0akk!t−t0kα,t0<t<t0+1Rα,
Here,kTαt0t0 means the application of the conformable derivative k times.The following definition is the conformableα-integral of a function f starting from a ≥ 0.Definition 3.
Iαaft=∫k=0tfx/x1−α⋅dx, where the integral is the usual Riemann improper integral, and α ∈ 0, 1 [2].
According to the above definition, the following can be shown.Theorem 5.
TαaIαaft=ft,fort≥a, where f is any continuous function in the domain of Iα [3].Lemma 6.
Letf: (a, b)⟶R be differentiable, and α ∈ 0, 1. Then, for all a > 0, we have [5](8)IαaTαaft=ft−fa.
From [7, 8], the conformable partial derivative of a real-valued function with several variables is defined as follows.Definition 4.
Letf be a real-valued function with n variables and a=a1,…,an∈Rn be a point whose ith component is positive. Then, the limit can be expressed as follows:(9)lim∈⟶0fa1,…,ai+εai1−α,…an−fa1,…,anε.
If the above limit exists, then we have theith conformable partial derivative of f of the order α∈(0, 1] at a, denoted by ∂α/∂xiαfa.
Finally, some results on conformable Fourier series will be recalled [22] as follows.
Letα∈(0, 1, and φ:0,∞)⟶R be defined by(10)φt=tαα,and g:[0, ∞) ⟶ R be any function. Let f: [0,∞)⟶R be defined by ft=gφt.Definition 5.
A function ft is calledα-periodical with period p if we have(11)ft=gφt=gφt+pαα.Definition 6.
Two functionsf, h are called α-orthogonal on [0, b] if ∫0bftht/t1−αdt=0.Definition 7.
Letf: [0, ∞)⟶R be a given piecewise continuous α-periodical with a period p. Then, we define the following:(i)
The cosineα-Fourier coefficients of f are expressed as an=2α/pα∫0pftcosntα/αdt/t1−α,n=1,2,3,….(ii)
The sineα-Fourier coefficients of f are expressed as bn=2α/pα∫0pftsinntα/αdt/t1−α,n=1,2,3,….Remark 1.
The following can be proven easily:(i)
cosntα/αandcosmtα/α are orthogonal on 0,α2π1/α, for all n≠m.(ii)
sinntα/αandsinmtα/α are orthogonal on 0,α2π1/α, for all n≠m.(iii)
sinntα/αandcosmtα/α are orthogonal on 0,α2π1/α, for all n, m.Definition 8.
Letf: [0, ∞)⟶R be a given piecewise continuous function which is α-periodical with period p. Then, the conformable α−Fourier series of f associated with the interval [0, p] is expressed as(12)Sft=a02+∑n=1∞ancosntαα+bnsinntαα,where an and bn, are as stated in Definition 7.Theorem 6.
The conformable Fourier series of a piecewise continuousα-periodical function converges pointwise to the average limit of the function at each point of discontinuity and to the function at each point of continuity.


## 3. Conformable Laplace’s Partial Differential Equation

In this section, we solve the two-dimensional conformable Laplace’s partial differential equation which is expressed in the following form:(13)∂α∂xα∂αux,y∂xα+∂α∂yα∂αux,y∂yα=0.As in the classical case, we propose this equation only with boundary conditions at the limit of the enclosure where the equation is fulfilled, which must have a certain regularity. These boundary conditions can be of two types:(i)
Dirichlet conditions: these are conditions in the functionu (x, y).(ii)
Neumann conditions: these are conditions imposed on the conformable partial derivatives ofu (x, y) of the order ∂αux,y/∂xαor ∂αux,y/∂yα.The geometry of the regionR where equation (13) is satisfied is very important, and we can only calculate solutions if they have certain regularity conditions.


### 3.1. Dirichlet Conditions

Let us discuss the solution of the following conformable Laplace’s partial differential equation:(14)∂α∂xα∂αux,y∂xα+∂α∂yα∂αux,y∂yα=0,0≤x≤a,0≤y≤b,ux,0=ux,b=0,0≤x≤a,u0,y=0,0≤y≤b,ua,y=fy,0≤y≤b.We will use the separation of variables technique [22].So, letux,y=PxQy. By substituting it in equation (13), we obtain the following:(15)dαdxαdαpxdxαQy+PxdαdyαdαQydyα=0.By ignoring the trivial solutionu≡0 and assuming that Px≠0 and Qx≠0, we have(16)1PxdαdxαdαPxdxα=−1QydαdyαdαQydyα.Hence, for some constantλ,(17)1PxdαdxαdαPxdxα=−1QydαdyαdαQydyα=λ.Consequently, we have(18)dαdxαdαPxdxα−λPx=0,dαdyαdαQydyα+λQy=0.The boundary conditions can be written as follows:(19)ux,0=PxQ0=0,ux,b=PxQb=0.Sincex is arbitrary, it follows that(20)Q0=Qb=0.Thus, we have the following contour problem:(21)dαdyαdαQydyα+λQy=0,Q0=0,Qb=0,whose solution depends on the separation parameter, λ. Now, we have the following:(1)
λ = 0. Then, equation (18) becomes dα/dyαdαQy/dyα=0, whose general solution is obtained by integrating twice with respect to x.(22)Qy=Ayαα+B.By using the following boundary conditions, we have:(23)Q0=0⟹B=0,Qb=0⟹Ab+B=0.Sinceb≠0λ=−μ2, the solution of the previous system is A = B = 0, and we obtain Q (y) = 0. Hence, there is no nontrivial solution when λ = 0.(2)
λ < 0, say λ=−μ2. Then, equation (18) becomes dα/dyαdαQy/dyα−μ2Qy=0, which has a general solution as follows:(24)Qy=Aeμyα/α+Be−μyα/α.By using the following boundary conditions, we have:(25)Q0=0⟹A+B=0,Qb=0⟹Aeμbα/α+Be−μbα/α=0.The previous equations form a homogeneous linear system in the unknownsA and B. The determinant of the matrix of the coefficients is expressed as(26)11eμbα/αe−μbα/α=eμbα/α−e−μbα/α=2sinhμbαα,and since μ≠0, the only solution of the system is the trivial A = B = 0, and we obtain Q (y) = 0. Hence, there is no nontrivial solution when λ < 0.(3)
λ > 0, say λ=μ2 Then, equation (18) becomes dα/dyαdαQy/dyα+μ2Qy=0, which has a general solution as follows:(27)Qy=Acosμyαα+Bsinμyαα.By using the following boundary conditions, we have:(28)Q0=0⟹A=0,Qb=0⟹Acosμbαα+Bsinμbαα=0,where(29)Bsinμbαα=0.Since we do not want the trivial solution,B = 0 and(30)sinμbαα=0⟺μbαα=nπ,n∈N,and then we obtain(31)μ=nπαbα,and the value of λ=μ2 is written as(32)λ=n2π2α2b2α,n∈N.Sinceλ was an arbitrary constant, then for each n ∈ N, we would have a possible solution of the conformable ordinary differential equation as follows:(33)λn=n2π2α2b2α⟹Qny=Bnsinnπybα.Substituting these values forλn in the other conformal differential equation, we have(34)dαdxαdαPxdxα−n2π2α2b2αpx=0,whose solution for each n ∈ N is of the following form:(35)Pnx=Cnenπx/bα+Dne−nπx/bα,Cn,Dn∈R.By using the initial conditionu (0, y) = 0, we have(36)u0,y=P0Qy=0,which by arbitrary y leads to P (0) = 0, and therefore, we obtain(37)Pn0=Cn+Dn=0⟹Dn=−Cn,and the function Pn (x) is given by(38)Pnx=Cnenπx/bα−e−nπx/bα=2Cnsinhnπxbα.The solution of the partial derivative equation will be, for eachn, of the following form:(39)unx,y=PnxQny=2CnsinhnπxbαBnsinnπybα=cnsinhnπxbαsinnπybα,with cn = 2CnBn.Since the equation is linear, any linear combination of solutions is another solution; therefore, we can consider it as a formal general solution:(40)ux,y=∑n=1∞Cnsinhnπxbαsinnπybα,and using the last boundary condition ua,y=fy, we have(41)ua,y=∑n=1∞cnsinhnπabαsinnπybα=∑n=1∞dnsinnπybα=fy.Finally, we can calculate the value of the coefficients dn, if we observe the expression as the conformable α− Fourier series of the odd extension of f (y); therefore, we obtain(42)dn=cnsinhnπabα=2αbα∫0bfysinnπybαdyy1−α,where(43)dn=2αbαsinhnπa/bα∫0bfysinnπybαdyy1−α.


### 3.2. Neumann Conditions

Let us discuss the solution of the following problem with Neumann-type conditions:(44)∂α∂xα∂αux,y∂xα+∂α∂yα∂αux,y∂yα=0,0≤x≤a,0≤y≤b,(45)∂αux,0∂yα=0,0≤x≤a,∂αux,b∂yα=0,0≤x≤a,∂αu0,y∂xα=fy,0≤y≤b,∂αua,y∂xα=0,0≤y≤b.We can see in this case that the boundary conditions involve the conformable partial derivatives ofu.All conditions are boundary. As we did previously, we use the method of separation of variables [22]:(46)ux,y=PxQy,which will lead us to the following two conformable ordinary differential equations:(47)dαdxαdαPxdxα−λPx=0,(48)dαdyαdαQydyα+λQy=0.The differences with the Dirichlet-type conditions appear when establishing the boundary conditions of these problems. Observe that in this case,∂αux,0/∂yα=0 and ∂αux,b/∂yα=0; therefore, the boundary conditions for the conformable differential equations are obtained as follows:(49)∂αux,y∂yα=PxdαdyαdαQydyα⟹∂αux,0∂yα=PxdαdyαdαQ0dyα⟹dαdyαdαQ0dyα=0,∂αux,b∂yα=PxdαdyαdαQbdyα⟹dαdyαdαQbdyα=0.We verify the following:(50)∂αua,y∂xα=dαdxαdαPadxαQy=0⟹dαdxαdαPadxα=0.Using equation (48) and the conditions found for dα/dyαdαQy/dyα, we have the following boundary problem:(51)dαdyαdαQydyα+λQy=0,dαQ0dyα=0,dαQbdyα=0.We distinguish according to the value ofλ. Now, we obtain the following:(1)
λ = 0. Then, equation (48) becomes dα/dyαdαQy/dyα=0, whose general solution is obtained by integrating twice with respect to y:(52)Qy=Ayαα+B,withA, B ∈ R arbitrary constants. By using the following boundary conditions, we have:(53)dαQ0dyα=0,dαQbdyα=0,⇔A=0.Therefore,Q (y) = B, and then u (x, y) = P (x) B.Using equation (44), we obtain(54)∂αux,y∂xα=dαPxdxαB⟹∂α∂xα∂αux,y∂xα=dαdxαdαPxdxαB∂αux,y∂yα=0⟹∂α∂yα∂αux,y∂yα=0,∂α∂xα∂αux,y∂xα+∂α∂yα∂αux,y∂yα=dαdxαdαPxdxαB+0=0,where either B = 0, but then we should have the null solution, or dα/dxαdαPx/dxα=0, and therefore, we have(55)Px=Cx+D.We have as a possible solution:(56)ux,y=PxQy=ACx+AD=ρx+σ,with ρ = AC and σ = AD. If we now use the boundary condition ∂αua,y/∂xα=0, we have(57)∂αux,y∂xα=ρ⟹∂αua,y∂xα=0⟹ρ=0.In this case, equation (44) has the following solution:(58)ux,y=σ.(2)
λ < 0, say λ=−μ2. Then, the equations are written as follows:(59)dαdyαdαQydyα−μ2Qy=0,dαQ0dyα=0,dαQbdyα=0,which has the following general solution:(60)Qy=Aeμyα/α+Be−μyα/α,By using the following boundary conditions, we obtain:(61)dαQydyα=Aμeμyα/α−Bμe−μyα/α.So,(62)dαQ0dyα=Aμ−Bμ=0⟹μA−B=0⟹A−B=0dαQbdyα=Aμeμbα/α−Bμe−μbα/α=0⟹μAeμbα/α−Be−μbα/α=0⟹Aeμbα/α−Be−μbα/α=0.From the first equation above,A = B, and by substituting it in the second equation, we have(63)Aeμbα/α−Be−μbα/α=0⟹Ae−μbα/α−e−μbα/α=0.In this case, we will have two options:(64)A=0⟹trivial solution,eμbα/α−e−μbα/α=0⟹eμbα/α=e−μbα/α⟹b=0⟹trivial solution.(3)
λ > 0, say λ = µ2.Then, the equations are expressed as follows:(65)dαdyαdαQydyα−μ2Qy=0,dαQ0dyα=0,dαQbdyα=0,which has the following general solution:(66)Qy=Acosμyαα+Bsinμyαα.We need the following equation:(67)dαQydyα=−μAsinμyαα+Bμcosμyαα,to be able to use the boundary conditions as follows:(68)dαQydyα=μB=0dαQbdyα=−μAsinμbαα+Bμcosμbαα⟹B=0−μAsinμbαα=0⟹μbαα=nπ,and therefore, we have(69)μ=nπαbα,n∈N,and for each value ofn, we will have the following function:(70)Qny=Ancosnπybα.With these values and equation (47), we obtain(71)dαdxαdαPxdxα−μ2Px=0⟹dαdxαdαPxdxα−n2π2αb2αPx=0,which has as a general solution as follows:(72)Px=Ceμxα/α+De−μxα/α.As we have seen before:dαPa/dxα=0.(73)dαPxdxα=Cμeμxα/α−Dμe−μxα/α⟹dαPadxα=Cμeμaα/α−Dμe−μaα/α=0⟹Ceμaα/α=De−μaα/α.Then for eachn, we have(74)Pnx=Cnenπx/bα+e−nπx/bαe2nπa/bα.The formal solution is the linear combination of all solutions that we have obtained in the caseλ = 0.(75)ux,y=σ+∑n=1∞AncosnπybαCnenπx/bα+e−nπx/bα,=σ+∑n=1∞Cncosnπybαenπx/bα+e−nπx/bαe2nπa/bα,where cn = AnCn has been taken.The values of cn are obtained using the boundary condition: ∂αu0,y/∂xα=fy(76)∂αux,y∂xα=∑n=1∞cnnπαbαcosnπybαenπx/bα−e−nπx/bαe2nπa/bα,∂αu0,y∂xα=∑n=1∞Cnnπαbα1−e2nπa/bαcosnπybα=∑n=1∞dncosnπybα=fy,dn=Cnnπαbα1−e2nπa/bα,All values ofcn are obtained from the boundary condition: ..., which are known as the coefficients of the conformable α-Fourier series of the even extension of the even extension of f (y); therefore, we obtain(77)dn=Cnnπαbα1−e2nπa/bα=2αbα∫0afycosnπybαdyy1−α,where(78)cn=2nπ1−e2nπa/bα∫0afycosnπybαdyy1−α.Note that in this case, it must be satisfied that the independent term of the conformableα−Fourier series d0/2 must be 0; therefore, for the solution of the problem with the Neumann conditions to be u (x, y), the following compatibility condition must be verified as follows:(79)d0=2αbα∫0afydyy1−α=0⟹∫0afydyy1−α=0.


## 3.1. Dirichlet Conditions

Let us discuss the solution of the following conformable Laplace’s partial differential equation:(14)∂α∂xα∂αux,y∂xα+∂α∂yα∂αux,y∂yα=0,0≤x≤a,0≤y≤b,ux,0=ux,b=0,0≤x≤a,u0,y=0,0≤y≤b,ua,y=fy,0≤y≤b.We will use the separation of variables technique [22].So, letux,y=PxQy. By substituting it in equation (13), we obtain the following:(15)dαdxαdαpxdxαQy+PxdαdyαdαQydyα=0.By ignoring the trivial solutionu≡0 and assuming that Px≠0 and Qx≠0, we have(16)1PxdαdxαdαPxdxα=−1QydαdyαdαQydyα.Hence, for some constantλ,(17)1PxdαdxαdαPxdxα=−1QydαdyαdαQydyα=λ.Consequently, we have(18)dαdxαdαPxdxα−λPx=0,dαdyαdαQydyα+λQy=0.The boundary conditions can be written as follows:(19)ux,0=PxQ0=0,ux,b=PxQb=0.Sincex is arbitrary, it follows that(20)Q0=Qb=0.Thus, we have the following contour problem:(21)dαdyαdαQydyα+λQy=0,Q0=0,Qb=0,whose solution depends on the separation parameter, λ. Now, we have the following:(1)
λ = 0. Then, equation (18) becomes dα/dyαdαQy/dyα=0, whose general solution is obtained by integrating twice with respect to x.(22)Qy=Ayαα+B.By using the following boundary conditions, we have:(23)Q0=0⟹B=0,Qb=0⟹Ab+B=0.Sinceb≠0λ=−μ2, the solution of the previous system is A = B = 0, and we obtain Q (y) = 0. Hence, there is no nontrivial solution when λ = 0.(2)
λ < 0, say λ=−μ2. Then, equation (18) becomes dα/dyαdαQy/dyα−μ2Qy=0, which has a general solution as follows:(24)Qy=Aeμyα/α+Be−μyα/α.By using the following boundary conditions, we have:(25)Q0=0⟹A+B=0,Qb=0⟹Aeμbα/α+Be−μbα/α=0.The previous equations form a homogeneous linear system in the unknownsA and B. The determinant of the matrix of the coefficients is expressed as(26)11eμbα/αe−μbα/α=eμbα/α−e−μbα/α=2sinhμbαα,and since μ≠0, the only solution of the system is the trivial A = B = 0, and we obtain Q (y) = 0. Hence, there is no nontrivial solution when λ < 0.(3)
λ > 0, say λ=μ2 Then, equation (18) becomes dα/dyαdαQy/dyα+μ2Qy=0, which has a general solution as follows:(27)Qy=Acosμyαα+Bsinμyαα.By using the following boundary conditions, we have:(28)Q0=0⟹A=0,Qb=0⟹Acosμbαα+Bsinμbαα=0,where(29)Bsinμbαα=0.Since we do not want the trivial solution,B = 0 and(30)sinμbαα=0⟺μbαα=nπ,n∈N,and then we obtain(31)μ=nπαbα,and the value of λ=μ2 is written as(32)λ=n2π2α2b2α,n∈N.Sinceλ was an arbitrary constant, then for each n ∈ N, we would have a possible solution of the conformable ordinary differential equation as follows:(33)λn=n2π2α2b2α⟹Qny=Bnsinnπybα.Substituting these values forλn in the other conformal differential equation, we have(34)dαdxαdαPxdxα−n2π2α2b2αpx=0,whose solution for each n ∈ N is of the following form:(35)Pnx=Cnenπx/bα+Dne−nπx/bα,Cn,Dn∈R.By using the initial conditionu (0, y) = 0, we have(36)u0,y=P0Qy=0,which by arbitrary y leads to P (0) = 0, and therefore, we obtain(37)Pn0=Cn+Dn=0⟹Dn=−Cn,and the function Pn (x) is given by(38)Pnx=Cnenπx/bα−e−nπx/bα=2Cnsinhnπxbα.The solution of the partial derivative equation will be, for eachn, of the following form:(39)unx,y=PnxQny=2CnsinhnπxbαBnsinnπybα=cnsinhnπxbαsinnπybα,with cn = 2CnBn.Since the equation is linear, any linear combination of solutions is another solution; therefore, we can consider it as a formal general solution:(40)ux,y=∑n=1∞Cnsinhnπxbαsinnπybα,and using the last boundary condition ua,y=fy, we have(41)ua,y=∑n=1∞cnsinhnπabαsinnπybα=∑n=1∞dnsinnπybα=fy.Finally, we can calculate the value of the coefficients dn, if we observe the expression as the conformable α− Fourier series of the odd extension of f (y); therefore, we obtain(42)dn=cnsinhnπabα=2αbα∫0bfysinnπybαdyy1−α,where(43)dn=2αbαsinhnπa/bα∫0bfysinnπybαdyy1−α.


## 3.2. Neumann Conditions

Let us discuss the solution of the following problem with Neumann-type conditions:(44)∂α∂xα∂αux,y∂xα+∂α∂yα∂αux,y∂yα=0,0≤x≤a,0≤y≤b,(45)∂αux,0∂yα=0,0≤x≤a,∂αux,b∂yα=0,0≤x≤a,∂αu0,y∂xα=fy,0≤y≤b,∂αua,y∂xα=0,0≤y≤b.We can see in this case that the boundary conditions involve the conformable partial derivatives ofu.All conditions are boundary. As we did previously, we use the method of separation of variables [22]:(46)ux,y=PxQy,which will lead us to the following two conformable ordinary differential equations:(47)dαdxαdαPxdxα−λPx=0,(48)dαdyαdαQydyα+λQy=0.The differences with the Dirichlet-type conditions appear when establishing the boundary conditions of these problems. Observe that in this case,∂αux,0/∂yα=0 and ∂αux,b/∂yα=0; therefore, the boundary conditions for the conformable differential equations are obtained as follows:(49)∂αux,y∂yα=PxdαdyαdαQydyα⟹∂αux,0∂yα=PxdαdyαdαQ0dyα⟹dαdyαdαQ0dyα=0,∂αux,b∂yα=PxdαdyαdαQbdyα⟹dαdyαdαQbdyα=0.We verify the following:(50)∂αua,y∂xα=dαdxαdαPadxαQy=0⟹dαdxαdαPadxα=0.Using equation (48) and the conditions found for dα/dyαdαQy/dyα, we have the following boundary problem:(51)dαdyαdαQydyα+λQy=0,dαQ0dyα=0,dαQbdyα=0.We distinguish according to the value ofλ. Now, we obtain the following:(1)
λ = 0. Then, equation (48) becomes dα/dyαdαQy/dyα=0, whose general solution is obtained by integrating twice with respect to y:(52)Qy=Ayαα+B,withA, B ∈ R arbitrary constants. By using the following boundary conditions, we have:(53)dαQ0dyα=0,dαQbdyα=0,⇔A=0.Therefore,Q (y) = B, and then u (x, y) = P (x) B.Using equation (44), we obtain(54)∂αux,y∂xα=dαPxdxαB⟹∂α∂xα∂αux,y∂xα=dαdxαdαPxdxαB∂αux,y∂yα=0⟹∂α∂yα∂αux,y∂yα=0,∂α∂xα∂αux,y∂xα+∂α∂yα∂αux,y∂yα=dαdxαdαPxdxαB+0=0,where either B = 0, but then we should have the null solution, or dα/dxαdαPx/dxα=0, and therefore, we have(55)Px=Cx+D.We have as a possible solution:(56)ux,y=PxQy=ACx+AD=ρx+σ,with ρ = AC and σ = AD. If we now use the boundary condition ∂αua,y/∂xα=0, we have(57)∂αux,y∂xα=ρ⟹∂αua,y∂xα=0⟹ρ=0.In this case, equation (44) has the following solution:(58)ux,y=σ.(2)
λ < 0, say λ=−μ2. Then, the equations are written as follows:(59)dαdyαdαQydyα−μ2Qy=0,dαQ0dyα=0,dαQbdyα=0,which has the following general solution:(60)Qy=Aeμyα/α+Be−μyα/α,By using the following boundary conditions, we obtain:(61)dαQydyα=Aμeμyα/α−Bμe−μyα/α.So,(62)dαQ0dyα=Aμ−Bμ=0⟹μA−B=0⟹A−B=0dαQbdyα=Aμeμbα/α−Bμe−μbα/α=0⟹μAeμbα/α−Be−μbα/α=0⟹Aeμbα/α−Be−μbα/α=0.From the first equation above,A = B, and by substituting it in the second equation, we have(63)Aeμbα/α−Be−μbα/α=0⟹Ae−μbα/α−e−μbα/α=0.In this case, we will have two options:(64)A=0⟹trivial solution,eμbα/α−e−μbα/α=0⟹eμbα/α=e−μbα/α⟹b=0⟹trivial solution.(3)
λ > 0, say λ = µ2.Then, the equations are expressed as follows:(65)dαdyαdαQydyα−μ2Qy=0,dαQ0dyα=0,dαQbdyα=0,which has the following general solution:(66)Qy=Acosμyαα+Bsinμyαα.We need the following equation:(67)dαQydyα=−μAsinμyαα+Bμcosμyαα,to be able to use the boundary conditions as follows:(68)dαQydyα=μB=0dαQbdyα=−μAsinμbαα+Bμcosμbαα⟹B=0−μAsinμbαα=0⟹μbαα=nπ,and therefore, we have(69)μ=nπαbα,n∈N,and for each value ofn, we will have the following function:(70)Qny=Ancosnπybα.With these values and equation (47), we obtain(71)dαdxαdαPxdxα−μ2Px=0⟹dαdxαdαPxdxα−n2π2αb2αPx=0,which has as a general solution as follows:(72)Px=Ceμxα/α+De−μxα/α.As we have seen before:dαPa/dxα=0.(73)dαPxdxα=Cμeμxα/α−Dμe−μxα/α⟹dαPadxα=Cμeμaα/α−Dμe−μaα/α=0⟹Ceμaα/α=De−μaα/α.Then for eachn, we have(74)Pnx=Cnenπx/bα+e−nπx/bαe2nπa/bα.The formal solution is the linear combination of all solutions that we have obtained in the caseλ = 0.(75)ux,y=σ+∑n=1∞AncosnπybαCnenπx/bα+e−nπx/bα,=σ+∑n=1∞Cncosnπybαenπx/bα+e−nπx/bαe2nπa/bα,where cn = AnCn has been taken.The values of cn are obtained using the boundary condition: ∂αu0,y/∂xα=fy(76)∂αux,y∂xα=∑n=1∞cnnπαbαcosnπybαenπx/bα−e−nπx/bαe2nπa/bα,∂αu0,y∂xα=∑n=1∞Cnnπαbα1−e2nπa/bαcosnπybα=∑n=1∞dncosnπybα=fy,dn=Cnnπαbα1−e2nπa/bα,All values ofcn are obtained from the boundary condition: ..., which are known as the coefficients of the conformable α-Fourier series of the even extension of the even extension of f (y); therefore, we obtain(77)dn=Cnnπαbα1−e2nπa/bα=2αbα∫0afycosnπybαdyy1−α,where(78)cn=2nπ1−e2nπa/bα∫0afycosnπybαdyy1−α.Note that in this case, it must be satisfied that the independent term of the conformableα−Fourier series d0/2 must be 0; therefore, for the solution of the problem with the Neumann conditions to be u (x, y), the following compatibility condition must be verified as follows:(79)d0=2αbα∫0afydyy1−α=0⟹∫0afydyy1−α=0.


## 4. Examples

In this section, we will use the above results to solve some conformable Laplace partial differential equations.Example 1.
Let us solve the solution of the following problem with Dirichlet-type conditions:(80)∂α∂xα∂αux,y∂xα+∂α∂yα∂αux,y∂yα=0,0≤x≤1,0≤y≤1,ux,0=ux,1=0,0≤x≤1,u0,y=00≤y≤1,u1,y=100,0≤y≤1,where α ∈ (0, 1].Solution. Using equations (40) and (43), we get(81)ux,y=∑n=1∞cnsinhnπxbαsinnπybα,where (82)cn=200αsinhnπ∫01sinnπyαdyy1−α=2001−−1nnπsinhnπ,n∈N.Example 2.
Let us solve the solution of the following problem with Neumann-type conditions:(83)∂α∂xα∂αux,y∂xα+∂α∂yα∂αux,y∂yα=0,0≤x≤a,0≤y≤b,∂αux,0∂yα=0,0≤x≤a,∂αux,b∂yα=00≤x≤a,∂αu0,y∂xα=bα2−yα,0≤y≤b,∂αua,y∂xα=0,0≤y≤b,where α∈0,1.
Solution. First, note that the functionfy=bα/2−yα satisfies the compatibility condition given by equation (30).(84)∫0bbα2−yαdyy1−α=0.
The formal solution of this problem is given by(85)ux,y=σ+∑n=1∞cncosnπybαenπx/bα+e−nπx/bαe2nπa/bα,where(86)cn=2nπ1−e2nπa/bα∫0bbα2−yαcosnπybαdyy1−α=2b2α−1n−1αn3π31−e2nπa/bα.


## 5. Conclusion

In this research paper, we have proposed some results referring to the conformable Laplace’s partial differential equation. The definitions of conformable derivative and integral have been applied to construct some of the results and relationships in our study. The solution of conformable Laplace’s partial differential equation with Dirichlet-type and Neumann-type conditions has been successfully established. The findings of this research study indicate that the results obtained in the sense of conformable derivative coincide with the results obtained in classical integer-order case. Finally, some interesting examples are presented to show the validity and potentiality of our obtained results to be applied in future research works in various applications in the field of natural sciences or engineering.



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