Solutions of Fractional Partial Differential Equations of Quantum Mechanics

2013 ◽  
Vol 5 (05) ◽  
pp. 639-651 ◽  
Author(s):  
S. D. Purohit
Keyword(s):  
Quantum Mechanics ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourier Transforms ◽  
Fractional Partial Differential Equations ◽  
Laplace And Fourier Transforms ◽  
Partial Differential ◽  
One Dimensional ◽  
Fractional Equations ◽  
Special Cases

AbstractThe aim of this article is to investigate the solutions of generalized fractional partial differential equations involving Hilfer time fractional derivative and the space fractional generalized Laplace operators, occurring in quantum mechanics. The solutions of these equations are obtained by employing the joint Laplace and Fourier transforms, in terms of the Fox’sH-function. Several special cases as solutions of one dimensional non-homogeneous fractional equations occurring in the quantum mechanics are presented. The results given earlier by Saxena et al. [Fract. Calc. Appl. Anal., 13(2) (2010), pp. 177–190] and Purohit and Kalla [J. Phys. A Math. Theor., 44 (4) (2011), 045202] follow as special cases of our findings.

scholarly journals Explicit solutions of some fractional partial differential equations via stable subordinators

2006 ◽  
Vol 2006 ◽  
pp. 1-18 ◽  
Author(s):  
Latifa Debbi
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fundamental Solutions ◽  
Explicit Solutions ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Stable Law ◽  
One Dimensional ◽  
Fractional Equations ◽  
Canonical Processes

The aim of this work is to represent the solutions of one-dimensional fractional partial differential equations (FPDEs) of order (α∈ℝ+\ℕ) in both quasi-probabilistic and probabilistic ways. The canonical processes used are generalizations of stable Lévy processes. The fundamental solutions of the fractional equations are given as functionals of stable subordinators. The functions used generalize the functions given by the Airy integral of Sirovich (1971). As a consequence of this representation, an explicit form is given to the density of the 3/2-stable law and to the density of escaping island vicinity in vortex medium. Other connected FPDEs are also considered.

Considerations on Computational Accuracy of the Solution for a Partial Differential Equation

2020 ◽  
Vol 896 ◽  
pp. 59-66
Author(s):  
Ionica Valeriu ◽  
Cosmin Mihai Miriţoiu ◽  
Alexandru Bolcu ◽  
Dan Gheorghe Bagnaru ◽  
Dumitru Bolcu ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourier Transforms ◽  
Algebraic Approach ◽  
Integral Transforms ◽  
Computational Accuracy ◽  
Laplace And Fourier Transforms ◽  
Partial Differential ◽  
Mechanical Models ◽  
Implicit Finite Difference

In this paper, we will compare the methods of solving with explicit or implicit finite difference of the partial differential equations that define the mechanical models of hydrodynamics movements, thermodynamics or those that define the vibration movements with the ones that use integral transforms. By applying the Laplace and Fourier transforms, finite in sine or cosine, depending on the boundary conditions of the real physical problem, it leads to the algebraic approach of the problem, which reduces the difficulty of solving partial differential equations. The errors obtained for the solution of partial differential equations using different methods are within the standard norms. However, in terms of calculus precision, the use of integral transforms is more advantageous.

scholarly journals Homotopy-Sumudu transforms for solving system of fractional partial differential equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
A. K. Alomari
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
General Framework ◽  
Analysis Method ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Fractional Equations ◽  
Convergence Results ◽  
Homotopy Analysis ◽  
Sumudu Transforms

Abstract In this paper, we investigate the Sumudu transforms and homotopy analysis method (S-HAM) for solving a system of fractional partial differential equations. A general framework for solving such a kind of problems is presented. The method can also be utilized to solve systems of fractional equations of unequal orders. The algorithm is reliable and robust. Existence and convergence results concerning the proposed solution are given. Numerical examples are introduced to demonstrate the efficiency and accuracy of the algorithm.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

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