# Two Numerical Methods for Solving the Schrödinger Parabolic and Pseudoparabolic Partial Differential Equations

Authors: Mahmut Modanli; Bushra Bajjah; Sevgi Kuşulay
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this work, the initial-boundary value problems for one-dimensional linear time-dependent Schrödinger parabolic and pseudoparabolic partial differential equations are studied. The modified double Laplace decomposition method is applied to get the semianalytic solutions and the explicit finite difference method to get the approximate solutions of the problems. The von Neumann stability analysis of the presented problems is also investigated.


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## Body

## 1. Introduction

The parabolic partial differential equations (PPDEs) take place in several areas of applied mathematics, for instance in the heat diffusion equation and in fluid mechanics [1]. Pseudoparabolic partial differential equations (PPPDEs) arise in modelling various phenomena, such as diffusion, heat conduction, thermoelectricity, chemical engineering, subsurface water flow, population dynamics, and plasma physics [2]. Several analytical approximation methods have been structured to treat such problems. For instance, the modified double Laplace decomposition method (MDLDM) has been used to solve the singular one-dimensional PPPDE with initial conditions by Gadain [3]. Ilhan et al. [4] constructed a family of travelling wave solutions to the nonlinear PPPDEs utilizing the modified exponential function method. In [5], the authors employed the variational iteration method with He’s polynomials to solve a coupled PPPDE. A three-layer finite difference scheme (FDS) to a linear PPPDE with a delay has been dealt by Amirali and Gabil [6]. In [7], Yang has studied a finite volume element approximation of PPPDEs in three spatial dimensions. Singh et al. [8] solved approximately the two-dimensional diffusion equation with Dirichlet boundary conditions using an Euler matrix method. The authors in [9] studied a nonlinear PPPDE with initial and Dirichlet boundary conditions using spectral discretization in space. Huntul et al. [10] solved the inverse initial-Neumann boundary value problem for a third-order PPPDE using the CB-spline collocation method. In [11], the researcher used a finite time blow-up criterion to solve a semilinear PPPDE. The Jacobi pseudospectral method has been used to solve numerically the two-dimensional linear heat diffusion equations subject to the Neumann and Robin boundary conditions by Yang et al. [12].The famous time-dependent one-dimensional linear Schrödinger partial differential equation (SPDE) in quantum physics is known as,(1)iζutt,x=−ζ22muxxt,x+ft,xut,x,where i=−1 is the imaginary number, ζ is Planck’s constant, and ut,x and ft,x represent the wave function and the potential function, respectively. Equation (1) has been reformulated by the use of the path integral approach considering the Gaussian probability distribution by Feynman and Hibbs [13].In this work, an initial-boundary value problem for one-dimensional time-dependent linear Schrödinger parabolic and pseudoparabolic partial differential equation in the domain0,T×0,L is considered as follows:(2)iutt,x=αutxxt,x+uxxt,x+ft,x,α≥0,0<x<L,0<t<T,u0,x=ψx,0≤x≤L,ut,0=ut,L=0,0≤t≤T,where f and ψ are known sufficiently smooth functions, which may be complex-valued. Equation (2) is referred to as a Schrödinger parabolic partial differential equation (SPPDE) when α=0, and as a Schrödinger pseudoparabolic partial differential equation (SPPPDE) when α>0.The MDLDM will be employed to find the semianalytic solution of problem (2). Many researchers consider it to be one of the best integral transforms. Jaradat et al. [14] used a Laplace–Adomian decomposition method (LADM) to obtain the semianalytical solution to the nonlinear SPDE with a harmonic oscillator in one and two dimensions. A nonlinear SPDE has been solved utilizing a double Laplace decomposition method by Gündoğdu and Ömer F [15]. The combination of the double Laplace transform and the Adomian decomposition method is called MDLDM.On the other hand, the approximate solution of problem (2) will be obtained using the explicit FDS. Ghafouri et al. [16] have applied FDS to find the approximate solution of SPDE in a nanoscale side-contacted field effect diode. For solving multidimensional coupled damped Schrödinger systems in Bose–Einstein condensates, Oruç [17] used the radial basis function-FDS approach. The accuracy of FDS for the smooth solutions of SPDEs has been proven by Li et al. [18]. In [19], linear SPDE has been solved by the Sinc–Galerkin and Sinc collocation methods. The approximate solution of the cubic nonlinear SPDE in one and two dimensions has been achieved using the Haar wavelet collocation method in combination with the Crank–Nicolson scheme by Pervaiz and Imran [20]. Subaşi has proposed three distinct FDSs for the numerical solution of two-dimensional SPDE [21]. Fairweather and Khebchareon [22] have presented numerical methods for solving linear and nonlinear SPDE in one and various space variables. Lehtovaara et al. [23] have used the imaginary time propagation method to obtain the eigenvalues and eigenvectors of large matrices originating from the discretization of linear and nonlinear SPDEs. In [24], the authors have solved the Schrödinger eigenvalue equation by the imaginary time propagation technique using splitting methods with complex coefficients.The paper is organized as follows. In Section2, some information about the double Laplace transform is given. In Section 3, the MDLDM is explained with two examples to get the semianalytic solution for problems of type (2). In Section 4, the explicit FDS for problem (2) is described and the stability is proved. In Section 5, numerical examples are presented. In Section 6, the paper ends with a brief conclusion.


## 2. Preliminaries on Double Laplace Transform

Some important definitions and theorems of the double Laplace transform are given which are used further in this work.Definition 1.
Let us assumeut,x, a function of two variables t and x. The double Laplace transform of ut,x is defined by the following double improper integral:(3)LtLxut,x≔Us,p≔∫0∞∫0∞e−st+pxdtdx≔limn,m⟶∞∫0n∫0me−st+pxdxdt,where s and p are complex numbers and Res>0 and Rep>0.Definition 2.
The inverse double Laplace transform ofUs,p is defined as follows:(4)ut,x≔Lt−1Lx−1Us,p≔12iπ∫σ−i∞σ+i∞estdt12iπ∫δ−i∞δ+i∞epxUs,pdp,where σ and δ are large enough that Us,p is defined for the real parts of s≤σ and p≤δ.Definition 3.
A functionut,x is said to be exponential order α>0 and β>0 on 0≤t,x<∞, if there exists a nonnegative constant M such that ut,x≤Meαt+βx.Theorem 1.
Ifut,x is of an exponential order and a piecewise continuous function on 0,T and 0,X, then its double Laplace transform exists for Res>α and Rep>β.Theorem 2.
Letu∈C†ℝ+×ℝ+ and †=maxk,n, there exist M,α,β>0 such that(5)∂j+iut,x∂tj∂xi≤Meαt+βx,j=0,1,…,k,i=0,1,…,n.Then, the following formulas yield:(6)LtLx∂nut,x∂xn=pnUs,p−∑i=0n−1pn−1−iLt∂iu0,t∂xi,LtLx∂kut,x∂tk=skUs,p−∑j=0k−1sk−1−jLx∂jux,0∂tj,LtLx∂k+nut,x∂tk∂xn=skpnUs,p−∑j=0k−1s−1−jLx∂jux,0∂tj−∑i=0n−1p−1−iLt∂iu0,t∂xi+∑j=0k−1∑i=0n−1s−1−jp−1−i∂j+iu0,0∂tj∂xi.


## 3. Modified Double Laplace Decomposition Method

In this section, the MDLDM will be constructed to solve the following initial-boundary value problem for one-dimensional time-dependent linear SPPPDE in the domain0,T×0,L,(7)iutt,x=αuxxtt,x+uxxt,x+ft,x,t,x∈0,T×0,L,α>0,with the following initial and boundary conditions:(8)u0,x=ψx,x∈0,L,ut,0=ut,L=0,t∈0,T.Firstly, the double Laplace transform with respect tot and x is applied to both sides of (7), and the following is obtained:(9)LtLxiutt,x=LtLxαuxxtt,x+uxxt,x+ft,x.In the next step, using the linearity and the differentiation properties of the double Laplace transform to equation (9), we obtain,(10)LtLxut,x=Gps+1isFs,p+1isLtLxαuxxtt,x+uxxt,x,where Gp and Fs,p symbolize the double Laplace transforms of ψx and ft,x, respectively.The LADM is used to define the solution of one-dimensional time-dependent linear SPPPDEut,x by the infinite series as follows:(11)ut,x=∑n=0∞unt,x.After this step, by applying the inverse double Laplace transform to both sides of equation (10) and using equation (11), as well as the linearity of the inverse transform, it holds,(12)∑n=0∞unt,x=ψx−iLt−1Lx−1Fs,ps−iLt−1Lx−11sLtLxα∂3∂t∂x2∑n=0∞unt,x+∂2∂x2∑n=0∞unt,x.From equation (12), the following recursive relationship can be acquired:(13)u0t,x=ψx−iLt−1Lx−1Fs,ps,(14)un+1t,x=−iLt−1Lx−11sLtLxα∂3∂t∂x2unt,x+∂2∂x2unt,x,n≥0.Finally, the resulting componentsu0,u1,u2,… are substituted into the decomposition series for ut,x, and then the required solution ut,x is obtained. Equations (13) and (14) are called the MDLDM of ut,x.


### 3.1. Applications of the Method

We will now analyze some examples to demonstrate the applicability of the MDLDM for solving the one-dimensional time-dependent linear SPPPDEs and SPPDEs.Example 1.
We consider the following initial-boundary value problem for a one-dimensional linear time-dependent SPPPDE:(15)iutt,x=αutxxt,x+uxxt,x+ft,x,α>0,ft,x=i+αcost+sintsinx,0<t,x<π,u0,x=0,0≤x≤π,ut,0=ut,π=0,0≤t≤π.
On applying the double Laplace transform to system (15) with respect to t and x, the following is obtained:(16)LtLxut,x=−ii+αs2+1+1ss2+11p2+1−isLtLxαutxxt,x+uxxt,x.
When both sides of equation (16) are transformed with the inverse double Laplace transform, the result is as follows:(17)ut,x=−αi+1sint−i1−costsinx−iLt−1Lx−11sLtLxαutxxt,x+uxxt,x.
Putting the expansion (11) into equation (17), it leads to the following:(18)∑n=0∞unt,x=−αi+1sint−i1−costsinx−iLt−1Lx−11sLtLxα∂3∂t∂x2∑n=0∞unt,x+∂2∂x2∑n=0∞unt,x.
If both sides of equation (18) are compared, we get the following recursive relation:(19)u0t,x=−αi+1sint−i1−costsinx,un+1t,x=−iLt−1Lx−11sLtLxα∂3∂t∂x2unt,x+∂2∂x2unt,x,n≥0.
Using the above mentioned recursive relation, the other components are given as follows:(20)u0t,x=−αi+1sint−i1−costsinx,u1t,x=αi−1+αsint+2α+i1−cost+tsinx,u2t,x=−α2−3αi+α2i+1sint+2α2i+αi−2α−i1−cost+3α−1t+it22!sinx,u3t,x=−α3i−α3+α2+α+1sint+−5α2−3αi−α+1t−t33!+−2α3−3α2i−2α2+4α+i1−cost−4α+it22!sinx.
The noise terms±αisintsinx and ±i1−costsinx between the components u0t,x and u1t,x can be taken away, and system (15) is still satisfied by the remaining term of u0t,x. As a result, the required solution is given by,(21)ut,x=∑n=0∞unt,x=sintsinx.
For the SPPDE of system (15), that is, α=0, the same solution has been obtained.Example 2.
Consider the following initial-boundary value problem for a one-dimensional linear time-dependent SPPPDE:(22)iutt,x=αutxxt,x+uxxt,x+ft,x,α>0,ft,x=2itx2−x−4αt−2t2−1,0<x,t<1,u0,x=x−x2,0≤x≤1,ut,0=ut,1=0,0≤t≤1.
Using the steps mentioned in the previous example, the following components are obtained:(23)u0t,x=x2−xt2−1+23itt2+3αt−3,u1t,x=−23itt2+3αt−3,u2t,x=u3t,x=⋯=0.
Thus, the desired solution is as follows:(24)ut,x=∑n=01unt,x=x2−xt2−1.
For the SPPDE of system (22), we obtain the same solution.


## 3.1. Applications of the Method

We will now analyze some examples to demonstrate the applicability of the MDLDM for solving the one-dimensional time-dependent linear SPPPDEs and SPPDEs.Example 1.
We consider the following initial-boundary value problem for a one-dimensional linear time-dependent SPPPDE:(15)iutt,x=αutxxt,x+uxxt,x+ft,x,α>0,ft,x=i+αcost+sintsinx,0<t,x<π,u0,x=0,0≤x≤π,ut,0=ut,π=0,0≤t≤π.
On applying the double Laplace transform to system (15) with respect to t and x, the following is obtained:(16)LtLxut,x=−ii+αs2+1+1ss2+11p2+1−isLtLxαutxxt,x+uxxt,x.
When both sides of equation (16) are transformed with the inverse double Laplace transform, the result is as follows:(17)ut,x=−αi+1sint−i1−costsinx−iLt−1Lx−11sLtLxαutxxt,x+uxxt,x.
Putting the expansion (11) into equation (17), it leads to the following:(18)∑n=0∞unt,x=−αi+1sint−i1−costsinx−iLt−1Lx−11sLtLxα∂3∂t∂x2∑n=0∞unt,x+∂2∂x2∑n=0∞unt,x.
If both sides of equation (18) are compared, we get the following recursive relation:(19)u0t,x=−αi+1sint−i1−costsinx,un+1t,x=−iLt−1Lx−11sLtLxα∂3∂t∂x2unt,x+∂2∂x2unt,x,n≥0.
Using the above mentioned recursive relation, the other components are given as follows:(20)u0t,x=−αi+1sint−i1−costsinx,u1t,x=αi−1+αsint+2α+i1−cost+tsinx,u2t,x=−α2−3αi+α2i+1sint+2α2i+αi−2α−i1−cost+3α−1t+it22!sinx,u3t,x=−α3i−α3+α2+α+1sint+−5α2−3αi−α+1t−t33!+−2α3−3α2i−2α2+4α+i1−cost−4α+it22!sinx.
The noise terms±αisintsinx and ±i1−costsinx between the components u0t,x and u1t,x can be taken away, and system (15) is still satisfied by the remaining term of u0t,x. As a result, the required solution is given by,(21)ut,x=∑n=0∞unt,x=sintsinx.
For the SPPDE of system (15), that is, α=0, the same solution has been obtained.Example 2.
Consider the following initial-boundary value problem for a one-dimensional linear time-dependent SPPPDE:(22)iutt,x=αutxxt,x+uxxt,x+ft,x,α>0,ft,x=2itx2−x−4αt−2t2−1,0<x,t<1,u0,x=x−x2,0≤x≤1,ut,0=ut,1=0,0≤t≤1.
Using the steps mentioned in the previous example, the following components are obtained:(23)u0t,x=x2−xt2−1+23itt2+3αt−3,u1t,x=−23itt2+3αt−3,u2t,x=u3t,x=⋯=0.
Thus, the desired solution is as follows:(24)ut,x=∑n=01unt,x=x2−xt2−1.
For the SPPDE of system (22), we obtain the same solution.


## 4. Finite Difference Scheme and Its Stability

### 4.1. Construction of the FDS

LetΩ=0,T×0,L. For some nonnegative integers M and N, the uniformly grid steps are introduced as,(25)Ωτ≔tk:tk=kτ,k=1,…,N−1,τ=TN,Ωh≔xn:xn=nh,n=1,…,M−1,h=LM.The values of the functionut,x at the grid points are represented as utk,xn≡usnk.In order to construct an explicit FDS,utnk is approximated by the forward Euler scheme and the two derivatives uxxnkand uxxtnk by the backward Euler scheme. Because the solution ut,x is assumed to be smooth enough, the following three-layer explicit FDS at any point tk,xn∈Ω can be written as follows:(26)αh2τun+1k−1−1h2+αh2τun+1k+2h2−2αh2τ−iτunk+2αh2τ+iτunk+1+αh2τun−1k−1−1h2+αh2τun−1k≈fnk,fnk=ftk,xn,α>0,tk,xn∈Ω,un0=ψxn,xn∈Ω¯h,u0k=uMk=0,tk∈Ω¯τ.Ignoring discretization errors of orderOτ2,h2, the explicit FDS for the SPPPDE of problem (2) can be written as,(27)lαh2τun+1k−1−1h2+αh2τun+1k+2h2−2αh2τ−iτunkl+2αh2τ+iτunk+1+αh2τun−1k−1−1h2+αh2τun−1k≈fnk,lfnk=ftk,xn,α>0,tk,xn∈Ω,lun0=ψxn,xn∈Ω¯h,lu0k=uMk=0,tk∈Ω¯τ.However, the explicit FDS for the SPPDE of problem (2) can be obtained as,(28)−1h2un+1k+2h2−iτunk+iτunk+1−1h2un−1k≈fnk,fnk=ftk,xn,tk,xn∈Ω,un0=ψxn,xn∈Ω¯h,u0k=uMk=0,tk∈Ω¯τ,where Ω¯h≔x0,x1,…,xM and Ω¯τ≔t0,t1,…,tN.


### 4.2. Stability and Convergence of the FDM

The von Neumann analysis will be used to investigate the stability of the two explicit FDSs (27) and (28). We start with the following usual assumption:(29)unk=rkeinθ,−π<θ<π.Inserting (29) into scheme (27) with supposing k=1and n=0, the following is obtained:(30)ir2−rτ=ατreiθ−2r2+re−iθh2−eiθ−2r+e−iθh2+reiθ−2r+re−iθh2.After several transformations, equation (30) becomes just a quadratic equation for the amplification factor r, namely,(31)ih2+2αr2−ih2+4αcos2θ2−4τsin2θ2r+2αcosθ=0.And we obtain,(32)r1r2=2αcosθih2+2α,<2αh4+4α2<1,(33)r1+r2=ih2+4αcos2θ/2−4τsin2θ/2ih2+2α,<h4+16α2τ2h4+4α2<1,where r1 and r2 are the solutions of equation (31). It can be concluded from inequalities (32) and (33) that r1<1 and r2<1, and follows that explicit FDS (27) is unconditionally stable.In the same way, the stability of scheme (28) is verified, and we obtain the following:(34)r1=0,r2=1+16τ2h4sin4θ2≥1.So, explicit FDS (28) is stable at r1=0, unconditionally unstable at r2>1, and marginally stable at r2=1.


## 4.1. Construction of the FDS

LetΩ=0,T×0,L. For some nonnegative integers M and N, the uniformly grid steps are introduced as,(25)Ωτ≔tk:tk=kτ,k=1,…,N−1,τ=TN,Ωh≔xn:xn=nh,n=1,…,M−1,h=LM.The values of the functionut,x at the grid points are represented as utk,xn≡usnk.In order to construct an explicit FDS,utnk is approximated by the forward Euler scheme and the two derivatives uxxnkand uxxtnk by the backward Euler scheme. Because the solution ut,x is assumed to be smooth enough, the following three-layer explicit FDS at any point tk,xn∈Ω can be written as follows:(26)αh2τun+1k−1−1h2+αh2τun+1k+2h2−2αh2τ−iτunk+2αh2τ+iτunk+1+αh2τun−1k−1−1h2+αh2τun−1k≈fnk,fnk=ftk,xn,α>0,tk,xn∈Ω,un0=ψxn,xn∈Ω¯h,u0k=uMk=0,tk∈Ω¯τ.Ignoring discretization errors of orderOτ2,h2, the explicit FDS for the SPPPDE of problem (2) can be written as,(27)lαh2τun+1k−1−1h2+αh2τun+1k+2h2−2αh2τ−iτunkl+2αh2τ+iτunk+1+αh2τun−1k−1−1h2+αh2τun−1k≈fnk,lfnk=ftk,xn,α>0,tk,xn∈Ω,lun0=ψxn,xn∈Ω¯h,lu0k=uMk=0,tk∈Ω¯τ.However, the explicit FDS for the SPPDE of problem (2) can be obtained as,(28)−1h2un+1k+2h2−iτunk+iτunk+1−1h2un−1k≈fnk,fnk=ftk,xn,tk,xn∈Ω,un0=ψxn,xn∈Ω¯h,u0k=uMk=0,tk∈Ω¯τ,where Ω¯h≔x0,x1,…,xM and Ω¯τ≔t0,t1,…,tN.


## 4.2. Stability and Convergence of the FDM

The von Neumann analysis will be used to investigate the stability of the two explicit FDSs (27) and (28). We start with the following usual assumption:(29)unk=rkeinθ,−π<θ<π.Inserting (29) into scheme (27) with supposing k=1and n=0, the following is obtained:(30)ir2−rτ=ατreiθ−2r2+re−iθh2−eiθ−2r+e−iθh2+reiθ−2r+re−iθh2.After several transformations, equation (30) becomes just a quadratic equation for the amplification factor r, namely,(31)ih2+2αr2−ih2+4αcos2θ2−4τsin2θ2r+2αcosθ=0.And we obtain,(32)r1r2=2αcosθih2+2α,<2αh4+4α2<1,(33)r1+r2=ih2+4αcos2θ/2−4τsin2θ/2ih2+2α,<h4+16α2τ2h4+4α2<1,where r1 and r2 are the solutions of equation (31). It can be concluded from inequalities (32) and (33) that r1<1 and r2<1, and follows that explicit FDS (27) is unconditionally stable.In the same way, the stability of scheme (28) is verified, and we obtain the following:(34)r1=0,r2=1+16τ2h4sin4θ2≥1.So, explicit FDS (28) is stable at r1=0, unconditionally unstable at r2>1, and marginally stable at r2=1.


## 5. Numerical Experiments

In this part, the explicit FDM is applied to solve the algebraic systems (15) and (22). MATLAB code is presented in the Appendix. To analyze the error between the obtained analytical ut,x and approximate unk solutions, we use,(35)ε=maxk=0,1,…,Nn=0,1,…,Mut,x−unk.We have shown that the analytic solution of system (15) is as follows:(36)ut,x=sintsinx,and of system (22) is as follows:(37)ut,x=x2−xt2−1.We ran multiple simulations with different input parameters to compare the approximate and analytical solutions to quantitatively demonstrate the accuracy and efficiency of our schemes. The analytical solution was approximated using the MDLDM, while the approximate solution was approximated using the explicit FDS. The error analysisε between the two solutions for systems (15) and (22) is shown in Tables 1 and 2, respectively.Table 1
Error analysis for system (15) with τ=π/N and h=π/M.
N,Mαε40,5000.13650.000010.13620.0010.114765,6500.08570.000010.08530.0010.067870,9000.07950.000010.07890.0010.0900150,10000.03860.000010.03820.0010.0655Table 2
Error analysis for system (22) with τ=1/N and h=1/M.
N,Mαε50,5000.02310.000020.02310.000050.023070,7000.01730.000020.01710.000050.0169120,10800.01060.000020.01030.000050.0098250,23000.00530.000020.00520.000050.0063Figures1 and 2 show a comparison of the profile evolution of the analytical and approximate solutions for systems (15) and (22), respectively.Figure 1
Profile evolution solution for system (15) at N = 70 and M = 90. (a) Approximate simulation for α=0. (b) Approximate simulation for α=0.001. (c) Analytic solution.
(a)(b)(c)Figure 2
Profile evolution solution for system (22) at N = 120 and M = 108. (a) Approximate simulation for α=0. (b) Approximate simulation for α=0.00005. (c) Analytic solution.
(a)(b)(c)Since the exact solutions obtained for the mentioned systems did not contain the parameterα, then the graph of the analytic solution does not change as the value of α changes.


## 6. Conclusion

In this study, the initial-boundary value problems for one-dimensional linear time-dependent SPPPDE and SPPDE are discussed. The semianalytic and approximate solutions for the SPPPDE and SPPDE of problem (2) were obtained using MDLDM and explicit FDS, respectively. The von Neumann stability analysis of the submitted problems was investigated. The abovementioned tables and figures indicate that the values of α have an impact on the solutions.



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