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Nonlinear dirac and diffusion equations in 1+1 dimensions from stochastic considerations
1Physics Department, Utkal University, Bhubaneswar 751004, India.
Summary
Researchers derived nonlinear partial differential equations from stochastic processes, generalizing diffusion and Schrodinger equations. Group classification was also performed on these novel nonlinear equations.
Area of Science:
- Mathematical Physics
- Stochastic Processes
- Nonlinear Differential Equations
Background:
- Fundamental linear partial differential equations (PDEs) like the diffusion, Schrodinger, Dirac, and telegrapher's equations are crucial in various scientific domains.
- Existing methods for deriving these PDEs often rely on deterministic approaches.
Purpose of the Study:
- To generalize the stochastic method for deriving fundamental linear PDEs.
- To obtain nonlinear forms of these important equations.
- To perform a group classification on the derived nonlinear PDEs.
Main Methods:
- Generalization of a stochastic consideration method.
- Application of one-parameter group transformations for classification.
Main Results:
- Successful derivation of nonlinear versions of fundamental PDEs from stochastic principles.
- Identification of group properties for two of the derived nonlinear equations.
Conclusions:
- Stochastic considerations can lead to nonlinear PDEs, expanding their applicability.
- Group classification provides insights into the symmetries and structures of these nonlinear equations.