# Reduction an Inverse Problem to a System of Second Kind Fredholm Integral Equations with No Singularities

Authors: M. Sajjadmanesh; H. Aydi; J. Younis
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this article, we deal with an inverse problem concerning the two-dimensional Laplace equation with local boundary conditions on a bounded region. In this problem, the goal is to reduce it into a system of Fredholm integral equations of the second kind involving kernels with weakly/or no singularities (Fredholm property) by considering some additional conditions on the parameters of the problem. Finally, the method is carried out on an example, to show the simplicity and efficiency of the method.


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## Body

## 1. Introduction

The inverse problems arise in modeling of many physical and geophysical phenomena, such as elastography and medical imaging, seismology, potential theory, ion transport problems or chromatography, and finances (see, for example, [1–4]). In many cases of these problems which have been studied for the first time by Lavrentiev [5], the goal is to determine the coefficients or the right hand side of the differential equations for some known data about its solutions. There are many numerical methods to solve these problems. Among them, we cite the method of fundamental solutions (MFS) by Marin and Lesnic [6] and Wang et al. [7] and by Chen et al. [8] and Sun and He [9] for the two-dimensional and three-dimensional inverse problems, respectively, the boundary function method (BFM) by Wang et al. [10], the boundary particle method (BPM) by Chen and Fu [11], the variational iteration method (VIM) by Canon and Tatari [12], the globally convergent numerical method by Baysal [13], the weighted homotopy analysis method (WHAM) by Shidfar et al. [14], and the conjugate gradient method (CGM) by Lu et al. [15]. In some special cases, these problems are solved by the analytical methods by Liua and Tatar [16] or Liu [17].In 1997, Aliev and Jahanshahi [18] proposed a new approach for reducing the direct problem concerning the mixed PDE with nonlocal boundary conditions into a system of Fredholm integral equations with weak singularities, i.e., the boundary values of the unknown function and their derivatives satisfy some of the Fredholm integral equations with weak singularities. The system obtained can be solved by some of the numerical methods, such as semiorthogonal B-spline wavelet collocation method by Sahau and Saha Ray [19], the method based on Bernstein polynomial by Basit and Khan [20], triangular functions method by Almasieh and Roodaki [21], homotopy perturbation method by Javidi and Golbabai [22], and Taylor-series expansion method by Maleknejad et al. [23].In this article, based on the method presented in [18], we study an inverse boundary value problem for a two-dimensional Laplace equation which is a logical continuation of the paper [24] concerning the inverse problem of Cauchy-Riemann equation with nonlocal boundary conditions.In the bounded domainΩ⊂ℝ2 (see Figure 1) where its boundary Γ≔∂Ω is given as
(1)Γ≔Γ1∪Γ2,Γj:x2=γjx1,j=1,2,where γjj=1,2 are known functions satisfying
(2)γ1x1<γ2x1forallx1∈a,b,we consider an inverse boundary value problem of the two-dimensional Laplace equation
(3)Δux=0inΩ,subject to the boundary conditions
(4)∑i=12αijx1∂u∂xix1,γjx1=ϕjx1,(5)ux1,γjx1=ψjx1,x1∈a,b,j=1,2,where αij,ϕji,j=1,2 and ψ1 are known C1 functions on a,b. The interest is then to recover the unknown function ψ2 based on a Fredholm integral equation of the second kind with respect to boundary values of the unknown function u and its first-order partial derivatives.Figure 1
The domainΩ with boundaries Γ1 and Γ2.


## 2. Main Results

Our first main result is stated as follows:Theorem 1.
The solutionu of Equation (3) at internal points of the domain can be expressed by Green’s formula:
(6)uξ=∫Γux∂G∂nx;ξdΓx−∫Γ∂u∂nxGx;ξdΓx,ξ∈Ω,where n is the exterior normal to the boundary Γ and Gx;ξ is the fundamental solution of 2D Laplace operator satisfying
(7)ΔGx;ξ=δx−ξ,which is given by
(8)Gx;ξ=12πlnx−ξ,where δx−ξ is the Dirac measure at point ξ=ξ1,ξ2 [25]. Moreover, the boundary values uΓ and ∂u/∂ξjΓj=1,2 should satisfy the following boundary integral equations:
(9)12uξ=∫Γux∂G∂nx;ξdΓx−∫Γ∂u∂nxGx;ξdΓx,ξ∈Γ,(10)12∂uξ∂ξ1=∫Γ∂G∂x2x;ξ∂ux∂x1cosn,x2−∂ux∂x2cosn,x1dΓx+∫Γ∂u∂nx∂G∂x1x;ξdΓx,ξ∈Γ,(11)12∂uξ∂ξ2=∫Γ∂G∂x1x;ξ∂ux∂x2cosn,x1−∂ux∂x1cosn,x2dΓx+∫Γ∂u∂nx∂G∂x2x;ξdΓx,ξ∈Γ,where n,xj is the angle between the exterior normal vector n to the boundary Γ and coordinate axes xj,j=1,2.Proof.
The first-order partial derivatives of Equation (8) are as follows:
(12)∂G∂xjx;ξ=12π×xj−ξjx−ξ2,j=1,2.
Multiplying both sides of Equation (3) by (8) and (12), respectively, integrating over Ω, using the divergence theorem [25] (similar to [24]), the proof is completed.Letni,xj and τi,xj be the angles between exterior normal and unit tangent vectors on the boundaries Γii=1,2 with coordinate axes xjj=1,2, respectively. Then
(13)cosni,x1=−1i+1sinτi,x1,cosni,x2=−1icosτi,x1,(14)tanτi,x1=γi′x1,cosτi,x1=dx1dΓx,i=1,2.Taking into account arcs of the boundaryΓ as Γ1 and Γ2, plugging (12)–(14) in (9)–(11), we obtain the following boundary integral equations:
(15)uΓj=uξ1,γjξ1=∫abk1j1x1,ξ1uΓ1dx1−∫abk2j1x1,ξ1uΓ2dx1−2∫abγ1′x1Gx;ξΓ1Γj∂u∂x1Γ1dx1+2∫abGx;ξΓ1Γj∂u∂x2Γ1dx1+2∫abγ2′x1Gx;ξΓ2Γj∂u∂x1Γ2dx1−2∫abGx;ξΓ2Γj∂u∂x2Γ2dx1,j=1,2,(16)∂u∂ξ1Γj=∂u∂ξ1ξ1,γjξ1=∫abk1j1x1,ξ1∂u∂x1Γ1dx1−∫abk2j1x1,ξ1∂u∂x1Γ2dx1−∫abk1j2x1,ξ1∂u∂x2Γ1dx1+∫abk2j2x1,ξ1∂u∂x2Γ2dx1,j=1,2,(17)∂u∂ξ2Γj=∂u∂ξ2ξ1,γjξ1=∫abk1j2x1,ξ1∂u∂x1Γ1dx1−∫abk2j2x1,ξ1∂u∂x1Γ2dx1+∫abk1j1x1,ξ1∂u∂x2Γ1dx1−∫abk2j1x1,ξ1∂u∂x2Γ2dx1,j=1,2,where
(18)Gx;ξΓiΓj=14πlnx1−ξ12+γix1−γjξ12,i,j=1,2,(19)kij1x1,ξ1≔1πγ′ix1x1−ξ1x−ξ2ΓiΓj−x2−ξ2x−ξ2ΓiΓj,(20)kij2x1,ξ1≔1πγi′x1x2−ξ2x−ξ2ΓiΓj+x1−ξ1x−ξ2ΓiΓj,i,j=1,2.As we see, the kernelskiiji,j=1,2 in (20) contain singularities. To remove them, we state the following theorem.Theorem 2.
LetΩ⊂ℝ2 be a bounded domain with the boundary ∂Ω as Lyapunov curve in Cartesian coordinates x=x1,x2 which is convex in direction x2 and assume that (for i,j=1,2), ψ1,ϕj,αij∈C1a,b and α2jx1≠0 (for all x1∈a,b). Then the singularities in kernels kii1i=1,2 are regularized. Moreover, the unknown function ψ2 in (5) can be expressed as the following boundary integral equation with no singularities
(21)ψ2ξ1=∫abK010x1,ξ1ψ1x1dx1+∫abK020x1,ξ1ψ2x1dx1+∫abK110x1,ξ1∂u∂x1Γ1dx1+∫abK120x1,ξ1∂u∂x1Γ2dx1+∫abK10x1,ξ1ϕ1x1dx1+∫abK20x1,ξ1ϕ2x1dx1,where
(22)K010x1,ξ1≔1πγ1′x1x1−ξ1x−ξ2Γ1Γ2−x2−ξ2x−ξ2Γ1Γ2,(23)K020x1,ξ1≔−γ2′′x1π1+γ2′2x1,(24)K1j0x1,ξ1≔−1jπα2jx1α2jx1γj′x1+α1jx1lnx−ξΓjΓ2,(25)Kj0x1,ξ1≔−1j+1πα2jx1lnx−ξΓjΓ2,j=1,2.Proof.
Applying the mean value theorem twice onγii=1,2, we get the kernels kii1i=1,2 with no singularities,
(26)kii1x1,ξ1=1πx1−η1γ′′iζ1x1−ξ11+γi′2η1≃γ′′ix1π1+γi′2x1,i=1,2,where η1 and ζ1 are between x1,ξ1 and x1,η1, respectively.From the boundary condition (4), we get
(27)∂u∂x2Γj=1α2jx1ϕjx1−α1jx1∂u∂x1Γj,j=1,2.Taking into account thatα2jx1≠0j=1,2, for all x1∈a,b, plugging the boundary condition (5) and relations (27) in Equation (15) (for j=2) and considering (20) and (26) with k121 and k221, respectively, the integral equation (21) can be resulted.Similar to Equation (26), we obtain
(28)kii2x1,ξ1=1πγi′x1γi′η1+1x1−ξ11+γi′2η1≃1πx1−ξ1,i=1,2,where η1 is between x1 and ξ1.For regularization and removing singularities inkii2i=1,2, we construct the following two linear combinations with unknown coefficients from the boundary integral equations (16)–(17):
(29)A11ξ1∂u∂ξ1Γ1+A21ξ1∂u∂ξ2Γ1=−A11ξ1∫abk112x1,ξ1∂u∂x2Γ1dx1+A21ξ1∫abk112x1,ξ1∂u∂x1Γ1dx1+⋯=∫abk112x1,ξ1A11∗x1,ξ1∂u∂x2Γ1dx1−∫abk112x1,ξ1A11x1∂u∂x2Γ1dx1−∫abk112x1,ξ1A21∗x1,ξ1∂u∂x1Γ1dx1+∫abk112x1,ξ1A21x1∂u∂x1Γ1dx1+⋯,(30)A12ξ1∂u∂ξ1Γ2+A22ξ1∂u∂ξ2Γ2=−∫abk222x1,ξ1A12∗x1,ξ1∂u∂x2Γ2dx1+∫abk222x1,ξ1A12x1dx1+∫abk222x1,ξ1A22∗x1,ξ1∂u∂x1Γ2dx1−∫abk222x1,ξ1A22x1dx1+⋯,where Aij∗x1,ξ1≔Aijx1−Aijξ1,i,j=1,2.The notation"⋯" in the right-hand side of (29)–(30) stands for all integrals without singularities.PluggingA11=−α21,A21=α11 and A12=α22,A22=−α12 in (29)–(30), respectively, and considering boundary conditions (4), we get
(31)−α21ξ1∂u∂ξ1Γ1+α11ξ1∂u∂ξ2Γ1=−∫abα21ξ1k111x1,ξ1+α11∗x1,ξ1k112x1,ξ1∂u∂x1Γ1dx1+∫abα21ξ1k211x1,ξ1−α11ξ1k212x1,ξ1∂u∂x1Γ2dx1+∫abα11ξ1k111x1,ξ1−α21∗x,ξ1k112x1,ξ1∂u∂x2Γ1dx1−∫abα21ξ1k212x1,ξ1+α11ξ1k211x1,ξ1∂u∂x2Γ2dx1+∫abk112x1,ξ1ϕ1x1dx1,(32)α22ξ1∂u∂ξ1Γ2−α12ξ1∂u∂ξ2Γ2=∫abk121x1,ξ1α22ξ1−k122x1,ξ1α12ξ1∂u∂x1Γ1dx1−∫abk122x1,ξ1α22ξ1+k121x1,ξ1α12ξ1∂u∂x2Γ1dx1−∫abk221x1,ξ1α22ξ1+α12∗x1,ξ1k222x1,ξ1∂u∂x1Γ2dx1+∫abk221x1,ξ1α12ξ1−α22∗x1,ξ1k222x1,ξ1∂u∂x2Γ2dx1.+∫abk222x1,ξ1ϕ2x1dx1.Plugging (27) in (31)–(32), respectively, we obtain
(33)α112ξ1+α212ξ1∂u∂ξ1Γ1=∫abK111x1,ξ1∂u∂x1Γ1dx1+∫abK121x1,ξ1∂u∂x1Γ2dx1+∫abK11x1,ξ1ϕ1x1dx1+∫abK21x1,ξ1ϕ2x1dx1+α11ξ1ϕ1ξ1,(34)α122ξ1+α222ξ1∂u∂ξ1Γ2=∫abK112x1,ξ1∂u∂x1Γ1dx1+∫abK122x1,ξ1∂u∂x1Γ2dx1+∫abK12x1,ξ1ϕ1x1dx1+∫abK22x1,ξ1ϕ2x1dx1,+α12ξ1ϕ2ξ1,where
(35)K121x1,ξ1≔α21ξ1α22x1−α21ξ1α22x1+α11ξ1α12x1k211x1,ξ1+α11ξ1α22x1−α21ξ1α12x1k212x1,ξ1,(36)K11x1,ξ1≔α21ξ1α21x1α21∗x1,ξ1−α21x1k112x1,ξ1−α11ξ1k111x1,ξ1,(37)K21x1,ξ1≔α21ξ1α22x1α11ξ1k211x1,ξ1+α21ξ1k212x1,ξ1,(38)K112x1,ξ1≔α22ξ1α21x1α22ξ1α21x1+α11x1α12ξ1k121x1,ξ1+α22ξ1α11x1−α12ξ1α21x1k122x1,ξ1,(39)K122x1,ξ1≔−α22ξ1α22x1α22ξ1α22x1+α12x1α12ξ1k221x1,ξ1+α12∗x1,ξ1α22x1−α12x1α22∗x1,ξ1k222x1,ξ1,(40)K12x1,ξ1≔−α22ξ1α21x1α12ξ1k121x1,ξ1+α22ξ1k122x1,ξ1,(41)K22x1,ξ1≔α22ξ1α22x1α12ξ1k221x1,ξ1+α22x1−α22∗x1,ξ1k222x1,ξ1.Finally, plugging the kernels (20) (for i,j=1,2,i≠j), (26), and (28) in kernels (37)–(41), we get
(42)K111x1,ξ1=α21ξ1πα21x1α21ξ1α21x1+α11x1α11ξ1γ1′′x11+γ1′2x1+α11′x1α21x1−α21′x1α11x1,(43)K121x1,ξ1=−α21ξ1πα22x1x2−ξ2x−ξ2Γ2Γ1α12x1α21ξ1−α11ξ1α22x1γ2′x1−α22x1α21ξ1+α11ξ1α12x1+x1−ξ1x−ξ2Γ2Γ1α22x1α21ξ1+α11ξ1α12x1γ2′x1+α12x1α21ξ1−α11ξ1α22x1,(44)K11x1,ξ1=−α21ξ1πα21x1α11ξ1γ1′′x11+γ1′2x1−α21′x1−α21ξ1πx1−ξ1,(45)K21x1,ξ1=α21ξ1πα22x1x2−ξ2x−ξ2Γ2Γ1α21ξ1γ2′x1−α11ξ1+x1−ξ1x−ξ2Γ2Γ1α11ξ1γ2′x1+α21ξ1,(46)K112x1,ξ1=α22ξ1πα21x1x2−ξ2x−ξ2Γ1Γ2α11x1α22ξ1−α21x1α12ξ1γ1′x1−α21x1α22ξ1+α11x1α12ξ1+x1−ξ1x−ξ2Γ1Γ2α21x1α22ξ1+α11x1α12ξ1γ1′x1+α11x1α22ξ1−α21x1α12ξ1,(47)K122x1,ξ1=−α22ξ1πα22x1α22x1α22ξ1+α12ξ1α12x1γ2′′x11+γ2′2x1+α22x1α12′x1−α12x1α22′x1,(48)K12x1,ξ1=−α22ξ1πα21x1x1−ξ1x−ξ2Γ1Γ2α22ξ1+α12ξ1γ1′x1+x2−ξ2x−ξ2Γ1Γ2α22ξ1γ1′x1−α12ξ1,(49)K22x1,ξ1=α22ξ1πα22x1α12ξ1γ2′′x11+γ2′2x1−α22′x1+α22ξ1πx1−ξ1.According to what explained above, the following theorems can be resulted:Theorem 3.
Under the assumptions of Theorem2, and also
(50)ϕia=ϕib=0,i=1,2,the unknown function ψ2 in (5) can be recovered as a system of boundary integral equations (21) and (33)–(34) subject to kernels (25) and (45)–(49), respectively, with no singularities.Theorem 4.
Under the assumptions of Theorem3, the main problem (3)–(5) is reduced to a system of boundary integral equations, i.e., the unknown function u can be written as the integral equation Equation (6) where the existing boundary values can be expressed as a system of boundary integral Equations (21), (33)–(34) and relations (5) and (27) with no singularities.The application of the method described will be illustrated in the following example.Example 1.
Consider the Laplace equation(51)Δux=0inΩ,subject to the following boundary conditions
(52)∂u∂nΓj=ϕjx1,(53)uΓj=ψjx1,j=1,2,where the boundary Γ≔∂Ω of the elliptic domain Ω⊂ℝ2 has the following polar equation (see Example 1 in [26])
(54)γθ=191361+39sin2θ,0≤θ≤2π,or in the Cartesian coordinates,
(55)x2+y2b2=1,b=1920,and the boundary Γ has the following form
(56)Γ=Γ1∪Γ2,Γj:x2=γjx1≔−1jb1−x12,j=1,2.Moreover, the functionsϕjj=1,2 and ψ1 in (53) are known.The exact solution of (51) and also the unknown boundary value ψ2 in (53) are given by
(57)ux1,x2=x11−2x12−2x1x23+2x12+2x1x223+2x2,ψ2x1=x1−2x13−2bx13+2x121−x12+2b2x11−x123+2b1−x12.Comparing the boundary conditions (53) with (4)–(5) yields that
(58)α1jx1=−b2x1,α2jx1=−1j+1b1−x12,j=1,2,ϕjx1=b2x163+bx12−62+b−1+−1jbx11−x124x12b4+6b2+1−2b22b2+3+1,j=1,2,ψ1x1=x1−2x13+2bx12x12+31−x12+2b2x11−x123−2b1−x12.As we see, the domainΩ as well as αij,ϕji,j=1,2 and ψ1 satisfies the hypotheses of Theorem 4. Therefore, from (25) and (45)–(49), we obtain
(59)K010x1,ξ1=1π×11−x12×bx1−ξ12+b21−x12+1−ξ122×1−x1ξ1+1−x121−ξ12,(60)K020x1,ξ1=1π×b1+b2−1x12×11−x12,(61)K1j0x1,ξ1=0,j=1,2,(62)Kj0x1,ξ1=1π×12b1−x12lnx1−ξ12+b21−x12+1−ξ122,j=1,2,(63)K1iix1,ξ1=−1π×1−ξ121−x12b31−1−x121−ξ12+b2x1ξ1b2−1x12+1,i=1,2,(64)K1jix1,ξ1=−1π×1−ξ121−x12×b3b2−1x12+1x1−ξ12+b21−x12+1−ξ122×x1ξ1−1−x121−ξ12−1i,j=1,2,i≠j,(65)Kiix1,ξ1=−1π×1−ξ121−x12bx1−b2ξ1b2−1x12+1,i=1,2,(66)Kjix1,ξ1=−1π×1−ξ121−x12×bx1−ξ1x1−ξ12+b21−x12+1−ξ122×1−b21−x121−ξ12−b2,i,j=1,2,i≠j.Plugging the kernels (66) in (33)–(34), a system of second kind Fredholm integral equations with respect to ∂u/∂x1Γii=1,2 is obtained which can be solved by some of numerical methods.In addition, after substituting the kernels (62) as well as ∂u/∂x1Γii=1,2 into the integral equation (21), the unknown function ψ2 can be resulted.Finally, plugging the boundary values∂u/∂x1Γii=1,2 in (27) as well as ψ2 in (6), the unknown function u in (51) is obtained.


## 3. Conclusion

In this paper, we studied an inverse boundary value problem including Laplace equation with nonlocal boundary conditions in a two-dimensional convex bounded domain with the boundary as Lyapunov curve. By considering some of the extra conditions, the problem is reduced to a system of second kind Fredholm integral equations with no singularities. This method can be used to any inverse boundary value problem where its fundamental solution is known.



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