The Solvability of First Type Boundary Value Problem for a Schrödinger Equation

Nigar Yildirim Aksoy 1
  • 1 Department of Mathematics, Faculty of Sciences and Arts, 36100, Kars, Turkey
Nigar Yildirim Aksoy
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  • Department of Mathematics, Faculty of Sciences and Arts, Kafkas University, 36100, Kars, Turkey
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Abstract

The paper presents an first type boundary value problem for a Schrödinger equation. The aim of paper is to give the existence and uniqueness theorems of the boundary value problem using Galerkin’s method. Also, a priori estimate for its solution is given.

1 Introduction

The fundamental equation of quantum mechanics, Schrödinger eqution, is the basic non-relativistic wave equation which describes the behaviour of a single particle (on systems of particles) in a field of force. Its solution is called a wave function which give us information about the particle’s behavior in time and space and the square of the wave function states the probability of finding the location of an particle in a given area. Schrödinger equation is roughly similar to Newton’s equation. Schrödinger equation does for a quantum mechanical particle what Newton’s Second Law does for a classical particle. When we solve Newton’s equation we can find the position of a particle as depend on time. But, when we solve Schrödinger’s equation we get a wave function stated the probability of finding the particle in some region in space varies as a function of time.

As known, Schrödinger equation is a partial differential equation, that is, its solution is a function, not a number. This means that some partial differential equations can be solved, some can’t, and some can only be estimated. The exact and numerical solutions of the various partial differential equations have been investigated by using the different methods in works [1, 2, 3, 4, 5], [7, 8, 9, 10, 11], [13, 14], [16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. In this study, we analyze a Schrödinger equation whose solution is an estimate of particle. Many researchers have been studied the approximate and exact solutions of Schrödinger equation by using different methodologies such as Adomian Decomposition Method (ADM) [10, 16], Homotopy Perturbation Method (HPM) [3, 5, 14], Homotopy Analysis Method (HAM) [2, 4], Variational Iteration Method (VIM) [7, 18], Galerkin’s Method [1, 8, 9, 11, 13, 17, 19, 20, 21]. Besides, there is various solution techniques for Schrödinger equation.

In this paper, we regard a first type boundary value problem for linear Schrödinger equation in the form:

iψt+a02ψx2+ia1(x,t)ψxa2(x)ψ+v(t)ψ=f(x,t)
ψ(x,0)=ϕ(x),x(0,l)
ψ(0,t)=ψ(l,t)=0,t(0,T),
where ψ = ψ(x, t) is a wave function, l and T are positive numbers. Here, we will use the notations: xI = (0, l), t ∈ [0, T ], Ωt = (0, l) × (0, t), Ω= ΩT, i=1. a0 > 0 is a given real number, the functions a1(x, t), a2(x), v(t) ∈ L2(0, T) are the measurable real-valued which satisfy the conditions, respectively,
|a1(x,t)|μ1,|a1(x,t)x|μ2,|2a1(x,t)x2|μ3foralmostall(a.a)(x,t)Ω,μ1,μ2,μ3=const.>0,
0<a2(x)μ4fora.axI,μ4=const.>0,
v(t)L2(0,T)b0,b0=const.>0,
ϕ(x) and f (x, t) are given complex-valued functions such that
ϕW22(I),fW22,0(Ω).
Here, the spaces W22(I), W22,0(Ω) are Sobolev space and are defined in [12] widely.

We investigate the solutions of the boundary value problem (1.1) (1.3) (BVP) under conditions (1.4)(1.7). For this purpose, we shall apply a well known Galerkin’s method to BVP. Also, we obtain an estimate for solutions of equation (1.1).

According to Galerkin’s method, some fundamental system of linearly independent functions in the studied spaces is choosen and the approximate solutions by means of linearly independent functions are constituted. The solution of BVP is obtained as limits of approximate solutions calculated by this method [11].

2 The Existence and Uniqueness of Solutions

In this section, we define the solution of BVP and prove a theorem which states the existence and uniqueness of the solution of BVP.

Definition 2.1

A functionψ(x,t)W22,1(Ω)is said to be a solution of BVP, if it holds the conditions (1.1)(1.3) for a.a (x, t) ∈ Ω, whereW22,1(Ω)is a Sobolev space of all elements in L2(Ω) having generalized derivatives up to order 2 and 1 with respect to variables x and t, respectiely, inclusive in L2(Ω). Also, W22,1(Ω)is a subspace of W22,1(Ω) andW22,1(Ω)=W22,1(Ω)W21,0(Ω).

Theorem 2.1

Let the conditions (1.4)(1.7) be satisfied. Then, BVP has a unique solutionψ(x,t)W22,1(Ω)which holds the estimate

ψW22,1(Ω)2c0(ϕW22(I)2+fW22,0(Ω)2),
where c0is a positive constant independent from ϕ and f.

Proof

We will prove the theorem (2.1) by the Galerkin’s method. By this method, the approximate solutions are searched in the form:

ψN(x,t)=k=1NCkN(t)uk(x),
where the functions uk = uk(x) for k = 1,2,.. generate a fundamental system in the space W22(I) and are eigenfunctions corresponding to the eigenvalues λ k of the problem:
Luk(x)=a0d2uk(x)dx2+a2(x)uk(x)=λkuk(x),xIuk(0)=uk(l)=0fork=1,2,
This is a Sturm-Liouville problem. So, its eigenvalues are real and nonnegative and the eigenfunctions uk = uk(x) corresponding to the eigenvalues λk are real and ortogonal in the spaces L2(I), W21(I), W22(I). Assume
ukW22(I)dkfork=1,2,..,
where dk are positive constants and let uk = uk(x) for k = 1,2,.. be an orthonormal basis in the space L2(I). Also, in (2.2), the coefficients CkN(t)=(ψN(.,t),uk)L2(I)=(ψN,uk) hold the system
(iψNt,uk)=(LψN,uk)i(a1(x,t)ψNx,uk)(v(t)ψN,uk)+(f,uk),
and initial conditions
CkN(0)=(ψN(.,0),uk)L2(I)=(ϕ,uk)=ϕk
for k = 1,2,..,N, where ϕkϕ strongly in W22(I) . The system (2.4) is a system of first order linear nonhomogeneous ordinary differential equations with constant coefficients with respect to the unknowns CkN(t) and system (2.4) with (2.5) is a Cauchy problem. From [15], it is written that the problem (2.4)(2.5) has locally at least one solution on [0, T ].

We assert that the problem (2.4)(2.5) has global solution on [0, T ]. To prove it, we give the next lemma:

Lemma 2.1

The coefficientsCkN(t)provide the estimate

0Tk=1N|CkN(t)|2+0Tk=1N|dCkN(t)dt|2ψNW22,1(Ω)2c1(ϕW22(I)2+fW22,0(Ω)2)
for N = 1,2,.., where the positive constant c1does not depend on N.

The proof of lemma (2.1) is carried out as in [20].

We now turn to proof of the theorem (2.1). It follows from the Lemma (2.1) that all possible solutions of the Cauchy problem (2.4)(2.5) are uniformly bounded on [0, T ], which implies that problem (2.4)(2.5) has one global solution on [0, T ].

Let’s define a family of functions lN,k(t) for k,N = 1,2,... such that lN,k(t) = ψN(.,t),uk)L2(I). From the estimate (2.6), it is seen that the functions lN,k(t) for k,N = 1,2,... are uniformly bounded on [0, T ]. So, the inequality

max0tT|lN,k(t)|c2,
is written, where the positive constant c2 is independent from N,k.

Now, let’s show that the functions lN,k(t) are equicontinuous for fixed k and Nk, N, k = 1,2,... on the interval [0, T ]. For this purpose, we integrate the kth equation in (2.4) on (t, t + Δt) and so, we get

i(lN,k(t+Δt)lN,k(t))=a0tt+Δt0lψNxdukdxdxdτitt+Δt0la1(x,t)ψNxukdxdτ+tt+Δt0la2(x)ψNukdxdτtt+Δt0lv(τ)ψNukdxdτ+tt+Δt0lfukdxdτ.
After taking the absolute value of (2.7), applying the Cauchy-Schwarz inequality, we obtain
|lN,k(t+Δt)lN,k(t)|a0tt+ΔtψN(.,τ)xL2(I)dukdxL2(I)dτ+μ1tt+ΔtψN(.,τ)xL2(I)ukL2(I)dτ+μ4tt+ΔtψN(.,τ)L2(I)ukL2(I)dτ+tt+Δt|v(τ)|ψN(.,τ)L2(I)ukL2(I)dτ+tt+Δtf(.,τ)L2(I)ukL2(I)dτ
by means of the conditions (1.4), (1.5). Using (2.3) in (2.8), if we take account the inequality
tt+Δt|v(τ)|ψN(.,τ)L2(I)ukL2(I)dτdktt+Δt|v(τ)|ψN(.,τ)L2(I)dτdk(max0τTψN(.,τ)L2(I))tt+Δt|v(τ)|dτdkΔt(max0τTψN(.,τ)L2(I))(0T|v(τ)|2dτ)12b0dkΔt(max0τTψN(.,τ)L2(I))
and apply Cauchy-Schwarz inequality with respect to t to all term at right-hand side of (2.8), we obtain
|lN,k(t+Δt)lN,k(t)|dkΔta0(0TψNxL2(I)2dτ)12+dkΔtμ1(0TψNxL2(I)2dτ)12+dkΔtμ4(0TψNL2(I)2dτ)12+b0dkΔt(max0τTψN(.,τ)L2(I))+dkΔt(0TfL2(I)2dτ)12.
In above inequality, using the estimate (2.6), we get
|lN,k(t+Δt)lN,k(t)|c3dk(Δt)12
for k,N = 1,2,..., where the positive constant c3 does not depend on N,kt. It follows from (2.9), that the functions lN,k(t) are equicontinuous on [0, T ].

Thus, from Ascoli-Arzela’s theorem [6], we can extract the subsequence {lNm,k(t)} from sequences {lN,k(t)} for fixed k and m = 1,2,.. such that

lNm,k(t)uniformlylk(t)on[0,T]
Let
ψ(x,t)=k=1lk(t)uk(x)
and we claim that the subsequence {ψNm} weakly converges to ψ(x, t) in L2(I), which this convergence are uniformly with respect to the variable t. That is, there is a positive number ɛ such that |(ψNm (x, t) − ψ(x, t), g)L2(I)|< ɛ for all t ∈ [0, T ], ∀gL2(I). Since the space L2(I) is a separable Hilbert space, we can write any element g of L2(I) in the form g=k=1(g,uk)L2(I)uk. Thus, it is written that
(ψNm(x,t)ψ(x,t),g)L2(I)=(ψNmψ,k=1(g,uk)L2(I)uk)L2(I)=(ψNmψ,k=1s(g,uk)L2(I)uk)L2(I)+(ψNmψ,k=s+1(g,uk)L2(I)uk)L2(I)=k=1s(g,uk)L2(I)(ψNm(x,t)ψ(x,t),uk)L2(I)+(ψNmψ,k=s+1(g,uk)L2(I)uk)L2(I)(k=1s|(g,uk)L2(I)|2)12(k=1s|lNm,k(t)lk(t)|2)12+ψNmψL2(I)k=s+1(g,uk)L2(I)ukL2(I).
Since gL2(I), (k=1s|(g,uk)L2(I)|2)12=gL2(I)<+ for big enough values of s. Also, since lNm,k(t) → lk(t) as Nm → ∞ it is written that for any ɛ > 0
(k=1s|(g,uk)L2(I)|2)12(k=1s|lNm,k(t)lk(t)|2)12c3ɛ2
for big enough values of s, where c3 > 0 is independent from Nm. Similarly, it is clear that
ψNmψL2(I)k=s+1(g,uk)L2(I)ukL2(I)c4(k=s+1|(g,uk)L2(I)|2)12,
where c4 > 0 is independent from Nm. Since the series k=s+1|(g,uk)L2(I)|2 is the rest of Fourier series of the function gL2(I), if we regard the converging of the series k=1|(g,uk)L2(I)|2 we can write k=s+1|(g,uk)L2(I)|2ɛ24 for any ɛ > 0. Thus, consideringly this inequality if we use (2.13) and (2.14) in (2.12), we achieve
|(ψNm(x,t)ψ(x,t),g)L2(I)|<ɛ
as Nm → ∞ for ∀gL2(I), ∀t ∈ [0, T ] and ∀ɛ > 0, which follows that the sequence {ψNm(x, t)} weakly converges to ψ(x, t) in L2(I) as uniformly with respect to t.

For N = Nm, since the subsequence {ψNm } is uniformly bounded from (2.6), we can extract a subsequence which weakly converges in W22,1(Ω) to ψ(x, t) defined by formula (2.11). For simplicity, let’s denote this subsequence as {ψNm(x, t)}. That is, limit relations

{ψNm}weaklyψ(x,t)inL2(Ω)
{ψNmx}weaklyψ(x,t)xinL2(Ω)
{2ψNmx2}weakly2ψ(x,t)x2inL2(Ω)
{ψNmt}weaklyψ(x,t)tinL2(Ω).
are written. Thus, by using the limit relations (2.15)(2.18) and the weakly lower semicontinuity of the norm on L2(Ω), if we take the lower limit of estimate (2.6) for N = Nm and as m → ∞ we have the inequalities
ψL2(Ω)2limm¯(ψNmL2(Ω)2)limm¯(c1(ϕW22(I)2+fW22,0(Ω)2))ψxL2(Ω)2limm¯(ψNmxL2(Ω)2)limm¯(c1(ϕW22(I)2+fW22,0(Ω)2))2ψx2L2(Ω)2limm¯(2ψNmx2L2(Ω)2)limm¯(c1(ϕW22(I)2+fW22,0(Ω)2))ψtL2(Ω)2limm¯(ψNmtL2(Ω)2)limm¯(c1(ϕW22(I)2+fW22,0(Ω)2))
which is equivalent to
ψW22,1(Ω)24c1(ϕW22(I)2+fW22,0(Ω)2),
it follows that the limit function ψ(x, t) provides the estimate (2.1) and ψW22,1(Ω).

Now, let’s show that the function ψ(x, t) provides the equation (1.1) for a.a (x, t) ∈ Ω. After multiplying the k-th equation in (2.4) for N = Nm with any continuous function η¯k(t) and let’s sum the obtained equalities on k from 1 to N′Nm and finally integrate over [0, T ]. Ultimately, we achieve the identity

Ω[iψNmt+a02ψNmx2+ia1(x,t)ψNmxa(x)ψNm+v(t)ψNmf]η¯N(x,t)dx=0,
where η¯N(x,t)=k=1Nη¯k(t)uk(x), N′Nm. Thus, taking the limit of (2.19) for N = Nm as m → ∞ and then using the limit relations (2.15)(2.18), we obtain the identity
Ω[iψt+a02ψx2+ia1(x,t)ψxa(x)ψ+v(t)ψf]η¯N(x,t)dx=0,
where η¯N(x,t)=k=1Nη¯k(t)uk(x),N′Nm. Since the set of functions η¯N(x,t) are dense in L2(Ω), if we take the limit of above integral identity for N′→ ∞ we get the following identity for any η(x, t) ∈ L2(Ω)
Ω[iψt+a02ψx2+ia1(x,t)ψxa(x)ψ+v(t)ψf]η¯(x,t)dx=0.
From (2.20), we can easily say that the limit function ψ(x, t) holds (1.1) for a.a (x, t) ∈ Ω.

Similarly to the paper [20], we prove that the conditions (1.2) and (1.3) is fulfilled by limit function ψ(x, t). Thus, we arrive ψW22,1(Ω).

Finally, let’s prove the uniquenes of solution of BVP in W22,1(Ω). To that end, we consider two different solutions ψ and ζ in W22,1(Ω). Let’s denote ρ(x, t) = ψ(x, t) − ζ (x, t). Then, the function ρ(x, t) satisfies the following boundary value problem:

iρt+a02ρx2+ia1(x,t)ρxa(x)ρ+v(t)ρ=0
ρ(x,0)=0,xI
ρ(0,t)=ρ(l,t)=0,t(0,T).

To obtain the uniquenes of the solution of BVP in W22,1(Ω), if we multiply the equation (2.21) by ρ¯(x,t) and later integrate over Ωt, we have

Ωt[iρtρ¯+a02ρx2ρ¯+ia1(x,t)ρxρ¯a(x)|ρ|2+v(τ)|ρ|2]dxdτ=0.
In above equality, if we apply the formula of partial integration we obtain
Ωt[iρtρ¯a0|ρx|2+ia1(x,t)ρxρ¯a(x)|ρ|2+v(τ)|ρ|2]dxdτ=0.
Then, subtracting the complex conjugate of (2.24) from itself, we get
Ωt[(|ρ|2)t+a1(x,t)(ρxρ¯+ρ¯xρ)]dxdτ=0,
which is equivalent to
ρ(.,t)L2(I)2+Ωtx(a1(x,t)|ρ|2)dxdτ=Ωta1(x,t)x|ρ|2dxdτ.
If we use (2.22), (2.23) in (2.25), we get
ρ(.,t)L2(I)2Ωt|a1(x,t)x|ρ|2dxdτ|μ30tρ(.,τ)L2(I)2dτ.
Thus, if we apply Gronwall’s lemma to the above inequality, we have
0ρ(.,t)L2(I)20.
The inequality (2.26) implies that ρ(.,t)L2(I)2=0 for any t ∈ [0, T ]. So, ψ(x, t) = ζ (x, t) for any t ∈ [0, T ] and a.a xI. i.e., BVP has a unique solution in W22,1(Ω). We arrive at the conslusion of the theorem (2.1).

3 Conclusion

In this paper, Galerkin’s method have been succesfully applied to the linear Schrödinger equation with special gradient term. It was shown that the solution of BVP exists and it is unique. Also, an estimate satisfied by the solution function is obtained. Studied problem consists of a special gradient term and the coefficients of equation are more general than the former works. Especially, the coefficient a1 depends on both the variables x and t. Because of the distinctness of considered equation with conditions, our problem differs from previous works in the literature.

References

  • [1]

    G. D. Akbaba, (2011), The optimal control problem with lions functional for the Schrödinger equation containing gradient with imaginary coefficient, Kafkas University, Inst. of Sci.&Tech., Master Thesis.

  • [2]

    A. K. Alomari, M. S. M. Noorani and R. Nazar, (2009), Explicit series solutions of some linear and nonlinear Schrödinger equations via the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 14 (4), 1196–1207.

  • [3]

    J. Biazar and H. Ghazvini, (2007), Exact solutions for non-linear Schrödinger equations by He’s homotopy perturbation method, Physics Letters A., 366 (1–2), 79–84.

  • [4]

    B. Ghanbari, (2014), An analytical study for (2+1)- Dimensional Schrödinger Equation, The Scientific World Journal, http://dx.doi.org/10.1155/2014/438345.

  • [5]

    J. H. He, (2005), Application of homotopy perturbation method to nonlinear wave equation, Chaos, Solitons and Fractals, 26, 695–700.

  • [6]

    P. F. Hsieh and Y. Sibuya, (1999), Basic Theory of Ordinary Differential Equations, Springer Verlag, 468p., New York.

  • [7]

    Kh. Hosseinzadeh, (2017), An analytic approximation to the solution of Schrödinger equation by VIM, Applied Mathematical Sciences, 11(16), 813–818.

  • [8]

    A. D. Iskenderov and G.Y. Yagubov, (2007), Optimal control Problem with unbounded potential for multidimensional, nonlinear and nonstationary Schrödinger equation// Proceedings of the Lankaran State University, Natural Sciences series, pp. 3–56.

  • [9]

    A. D. Iskenderov, G. Y. Yagubov, M. A. Musayeva, (2012), Identification of Quantum Potentials, Baku, Casıoglu, 552 p.

  • [10]

    S. A. Khuri, (1998), A new approach to the cubic Schrödinger equation: an application of the decomposition technique, Appl. Math. and Comput., 97, 251–254.

  • [11]

    O. A. Ladyzhenskaya, (1985), The Boundary Value Problems of Mathematical Physics, Springer Verlag.

  • [12]

    O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’ceva, (1968), Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. (English Trans.), Providence, RI.

  • [13]

    N. M. Mahmudov, (2007), Solvability of Boundary Value Problems for a Schrödinger Equation with Pure Imaginary Coefficient in the Nonlinear Part of This Equation, Proceedings of IMM of NAS of Azerbaijan, Vol.27, issue 35, pp.25–36.

  • [14]

    M. M. Mousa, S. F. Ragab and Z. Nturforsch, (2008), Application of the homotopy perturbation method to linear and nonlinear Schrödinger equation, Zeitschrift Fur Naturforschung A, 63 (3–4), 140–144.

  • [15]

    L.S. Pontryagin, (1962), Ordinary Differential Equtions, Addison-Wesley Publishing Comp.. (translated from the Russian)

  • [16]

    A. Sadighi and D.D. Ganji, (2008), Analytic treatment of linear and nonlinear Schrödinger equation: A study with homotopy perturbation and Adomian’s decomposition method, Pyhs. Lett. A, 372, 465–469.

  • [17]

    F. Toyoglu, (2012), Optimal control problems for the two-dimensional Schrödinger equation and their numerical solutions, Atatürk University, Inst. of Sci.&Tech., Ph. D. Thesis.

  • [18]

    A. M. Wazwaz, (2008), A study on linear and nonlinear Schrödinger Equation by variational iteration method, Chaos, Solitons and Fractals, 37, 1136–1142.

  • [19]

    G. Yagub, E. Aksoy, (2017), The Solvability of Initial Boundary Value Problem for Three Dimensional Nonlinear Schrödinger Equation With a Special Gradient Term, AIP Conference Proceedings, 1833, 020042.

  • [20]

    N. Yildirim Aksoy, Y. Kocak,Y. Ozeroglu, (2016), The Solvability of Initial Boundary Value Problems for Nonlinear Time-Dependent Schrödinger Equations, Quaestiones Mathematicae, 39 (6), 751–771.

  • [21]

    N. Yildirim Aksoy, (2017), Solution of a nonlinear Schrödinger equation with Galerkin’s Method, Igdır Univ. J. Inst. Sci. &Tech., 7(2), 225–239.

  • [22]

    A. Yokus, (2018), Numerical solution for space and time fractional order Burger type equation, Alexandria Engineering Journal, 57(3), 2085–2091.

  • [23]

    A. Yokus, (2018), Comparison of Caputo and Conformable derivatives for time-fractional Kortewegde Vries equation via the finite difference method, International Journal of Modern Physics B, 32(29) 1850365.

  • [24]

    A. Yokus and H. Bulut, (2019), On the numerical investigations to the Cahn-Allen equation by using finite difference method, An International Journal of Optimization and Control: Theories & Applications, 9(1) 18–23.

  • [25]

    A. Yokus, H. M. Baskonus, T. A. Sulaiman, H. Bulut, (2018), Numerical simulation and solutions of the two-component second order KdV evolutionary system, Numerical Methods for Partial Differential Equations, 34(1), 211–227.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    G. D. Akbaba, (2011), The optimal control problem with lions functional for the Schrödinger equation containing gradient with imaginary coefficient, Kafkas University, Inst. of Sci.&Tech., Master Thesis.

  • [2]

    A. K. Alomari, M. S. M. Noorani and R. Nazar, (2009), Explicit series solutions of some linear and nonlinear Schrödinger equations via the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 14 (4), 1196–1207.

  • [3]

    J. Biazar and H. Ghazvini, (2007), Exact solutions for non-linear Schrödinger equations by He’s homotopy perturbation method, Physics Letters A., 366 (1–2), 79–84.

  • [4]

    B. Ghanbari, (2014), An analytical study for (2+1)- Dimensional Schrödinger Equation, The Scientific World Journal, http://dx.doi.org/10.1155/2014/438345.

  • [5]

    J. H. He, (2005), Application of homotopy perturbation method to nonlinear wave equation, Chaos, Solitons and Fractals, 26, 695–700.

  • [6]

    P. F. Hsieh and Y. Sibuya, (1999), Basic Theory of Ordinary Differential Equations, Springer Verlag, 468p., New York.

  • [7]

    Kh. Hosseinzadeh, (2017), An analytic approximation to the solution of Schrödinger equation by VIM, Applied Mathematical Sciences, 11(16), 813–818.

  • [8]

    A. D. Iskenderov and G.Y. Yagubov, (2007), Optimal control Problem with unbounded potential for multidimensional, nonlinear and nonstationary Schrödinger equation// Proceedings of the Lankaran State University, Natural Sciences series, pp. 3–56.

  • [9]

    A. D. Iskenderov, G. Y. Yagubov, M. A. Musayeva, (2012), Identification of Quantum Potentials, Baku, Casıoglu, 552 p.

  • [10]

    S. A. Khuri, (1998), A new approach to the cubic Schrödinger equation: an application of the decomposition technique, Appl. Math. and Comput., 97, 251–254.

  • [11]

    O. A. Ladyzhenskaya, (1985), The Boundary Value Problems of Mathematical Physics, Springer Verlag.

  • [12]

    O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’ceva, (1968), Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc. (English Trans.), Providence, RI.

  • [13]

    N. M. Mahmudov, (2007), Solvability of Boundary Value Problems for a Schrödinger Equation with Pure Imaginary Coefficient in the Nonlinear Part of This Equation, Proceedings of IMM of NAS of Azerbaijan, Vol.27, issue 35, pp.25–36.

  • [14]

    M. M. Mousa, S. F. Ragab and Z. Nturforsch, (2008), Application of the homotopy perturbation method to linear and nonlinear Schrödinger equation, Zeitschrift Fur Naturforschung A, 63 (3–4), 140–144.

  • [15]

    L.S. Pontryagin, (1962), Ordinary Differential Equtions, Addison-Wesley Publishing Comp.. (translated from the Russian)

  • [16]

    A. Sadighi and D.D. Ganji, (2008), Analytic treatment of linear and nonlinear Schrödinger equation: A study with homotopy perturbation and Adomian’s decomposition method, Pyhs. Lett. A, 372, 465–469.

  • [17]

    F. Toyoglu, (2012), Optimal control problems for the two-dimensional Schrödinger equation and their numerical solutions, Atatürk University, Inst. of Sci.&Tech., Ph. D. Thesis.

  • [18]

    A. M. Wazwaz, (2008), A study on linear and nonlinear Schrödinger Equation by variational iteration method, Chaos, Solitons and Fractals, 37, 1136–1142.

  • [19]

    G. Yagub, E. Aksoy, (2017), The Solvability of Initial Boundary Value Problem for Three Dimensional Nonlinear Schrödinger Equation With a Special Gradient Term, AIP Conference Proceedings, 1833, 020042.

  • [20]

    N. Yildirim Aksoy, Y. Kocak,Y. Ozeroglu, (2016), The Solvability of Initial Boundary Value Problems for Nonlinear Time-Dependent Schrödinger Equations, Quaestiones Mathematicae, 39 (6), 751–771.

  • [21]

    N. Yildirim Aksoy, (2017), Solution of a nonlinear Schrödinger equation with Galerkin’s Method, Igdır Univ. J. Inst. Sci. &Tech., 7(2), 225–239.

  • [22]

    A. Yokus, (2018), Numerical solution for space and time fractional order Burger type equation, Alexandria Engineering Journal, 57(3), 2085–2091.

  • [23]

    A. Yokus, (2018), Comparison of Caputo and Conformable derivatives for time-fractional Kortewegde Vries equation via the finite difference method, International Journal of Modern Physics B, 32(29) 1850365.

  • [24]

    A. Yokus and H. Bulut, (2019), On the numerical investigations to the Cahn-Allen equation by using finite difference method, An International Journal of Optimization and Control: Theories & Applications, 9(1) 18–23.

  • [25]

    A. Yokus, H. M. Baskonus, T. A. Sulaiman, H. Bulut, (2018), Numerical simulation and solutions of the two-component second order KdV evolutionary system, Numerical Methods for Partial Differential Equations, 34(1), 211–227.

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