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Article Contents

# Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator

• * Corresponding author: Sushila
• In this paper, an effective analytical scheme based on Sumudu transform known as homotopy perturbation Sumudu transform method (HPSTM) is employed to find numerical solutions of time fractional Schrödinger equations with harmonic oscillator.These nonlinear time fractional Schrödinger equations describe the various phenomena in physics such as motion of quantum oscillator, lattice vibration, propagation of electromagnetic waves, fluid flow, etc. The main objective of this study is to show the effectiveness of HPSTM, which do not require small parameters and avoid linearization and physically unrealistic assumptions. The results reveal that proposed scheme is a powerful tool for study large class of problems. This study shows that the results obtained by the HPSTM are accurate and effective for analysis the nonlinear behaviour of complex systems and efficient over other available analytical schemes.

Mathematics Subject Classification: Primary: 26A33, 35Q41, 35R11.

 Citation:

• Figure 1.  Surface shows the imaginary part of the 1-D wave function $\psi \left(\xi ,\eta \right)$ at (a) $\alpha = 1$, (b) $\alpha = 0.75$ and (c) $\alpha = 0.50$

Figure 2.  Comparison of profile of imaginary part of $\psi \left(\xi ,\eta \right)$ at different values of $\alpha$ (a) $-5\le \xi \le 5$ and $\eta = 0.01$(b) $0\le \eta \le 1$ and $\xi = 1.$

Figure 3.  Surface shows the real part of the 1-D wave function $\psi \left(\xi ,\eta \right)$ at (a) $\alpha = 1$, (b) $\alpha = 0.75$ and (c) $\alpha = 0.50$

Figure 4.  Comparison of profile of real part of $\psi \left(\xi ,\eta \right)$ at different values of $\alpha$ (a) $-5\le \xi \le 5$ and $\eta = 0.01$(b) $0\le \eta \le 1$ and $\xi = 1.$

Figure 5.  Surface shows the imaginary part of the 2-D wave function $\psi \left(\xi ,\zeta, \eta \right)$ at (a) $\alpha = 1$, (b) $\alpha = 0.75$ and (c) $\alpha = 0.50$

Figure 6.  Comparison of profile of imaginary part of $\psi \left(\xi ,\zeta,\eta \right)$ for $\zeta = 1$ at different values of $\alpha$ (a) $-5\le \xi \le 5$ and $\eta = 0.01$(b) $0\le \eta \le 1$ and $\xi = 1.$

Figure 7.  Surface shows the real part of the 2-D wave function $\psi \left(\xi ,\zeta, \eta \right)$ at (a) $\alpha = 1$, (b) $\alpha = 0.75$ and (c) $\alpha = 0.50$

Figure 8.  Comparison of profile of real part of $\psi \left(\xi ,\zeta,\eta \right)$ for $\zeta = 1$at different values of $\alpha$ (a) $-5\le \xi \le 5$ and $\eta = 0.01$(b) $0\le \eta \le 1$ and $\xi = 1.$

Figure 9.  Surface shows the imaginary part of the 3-D wave function $\psi \left(\xi ,\zeta,\chi, \eta \right)$ at (a) $\alpha = 1$, (b) $\alpha = 0.75$ and (c) $\alpha = 0.50$

Figure 10.  Comparison of profile of imaginary part of $\psi \left(\xi ,\zeta,\chi, \eta \right)$ for $\zeta = \chi = 1$ at different values of $\alpha$ (a) $-5\le \xi \le 5$ and $\eta = 0.01$ (b) $0\le \eta \le 1$ and $\xi = 1.$

Figure 11.  Surface shows the real part of the 3-D wave function $\psi \left(\xi ,\zeta,\chi, \eta \right)$ at (a) $\alpha = 1$, (b) $\alpha = 0.75$ and (c) $\alpha = 0.50$

Figure 12.  Comparison of profile of real part of $\psi \left(\xi ,\zeta,\chi, \eta \right)$ for $\zeta = \chi = 1$ at different values of $\alpha$ (a) $-5\le \xi \le 5$ and $\eta = 0.01$(b) $0\le \eta \le 1$ and $\xi = 1.$

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