Symmetry Analysis and Exact Solutions to the Space-Dependent Coefficient PDEs in Finance

The variable-coefficients partial differential equations (vc-PDEs) in finance are investigated by Lie symmetry analysis and the generalized power series method. All of the geometric vector fields of the equations are obtained; the symmetry reductions and exact solutions to the equations are presented, including the exponentiated solutions and the similarity solutions. Furthermore, the exact analytic solutions are provided by the transformation technique and generalized power series method, which has shown that the combination of Lie symmetry analysis and the generalized power series method is a feasible approach to dealing with exact solutions to the variable-coefficients PDEs.

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A Truncation Method for Solving the Time-Fractional Benjamin-Ono Equation

Journal of Applied Mathematics ◽

10.1155/2019/3456848 ◽

2019 ◽

Vol 2019 ◽

pp. 1-7 ◽

Cited By ~ 11

Author(s):

Mohamed R. Ali

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Power Series ◽

Fractional Order ◽

Nonlinear Ordinary Differential Equation ◽

Lie Symmetry ◽

Series Method ◽

Lie Symmetry Analysis ◽

Power Series Method ◽

Bo Equation

We deem the time-fractional Benjamin-Ono (BO) equation out of the Riemann–Liouville (RL) derivative by applying the Lie symmetry analysis (LSA). By first using prolongation theorem to investigate its similarity vectors and then using these generators to transform the time-fractional BO equation to a nonlinear ordinary differential equation (NLODE) of fractional order, we complete the solutions by utilizing the power series method (PSM).

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Lie symmetry analysis, exact solutions, conservation laws of variable-coefficients Calogero–Bogoyavlenskii–Schiff equation

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887822500220 ◽

2021 ◽

Author(s):

Feng Zhang ◽

Yuru Hu ◽

Xiangpeng Xin ◽

Hanze Liu

Keyword(s):

Conservation Laws ◽

Exact Solutions ◽

Expansion Method ◽

Variable Coefficients ◽

Lie Symmetry ◽

Symmetry Analysis ◽

Lie Symmetry Analysis ◽

Interaction Solution ◽

Group Invariant Solutions ◽

Dimensional Variable

In this paper, a [Formula: see text]-dimensional variable-coefficients Calogero–Bogoyavlenskii–Schiff (vcCBS) equation is studied. The infinitesimal generators and symmetry groups are obtained by using the Lie symmetry analysis on vcCBS. The optimal system of one-dimensional subalgebras of vcCBS is computed for determining the group-invariant solutions. On this basis, the vcCBS is reduced to two-dimensional partial differential equations (PDEs) by similarity reductions. Furthermore, the reduced PDEs are solved to obtain the two-soliton interaction solution, the soliton-kink interaction solution and some other exact solutions by the [Formula: see text]-expansion method. Moreover, it is shown that vcCBS is nonlinearly self-adjoint and then its conservation laws are calculated.

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Lie symmetry analysis, conservation laws and some exact solutions of the time-fractional Buckmaster equation

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887820500401 ◽

2020 ◽

Vol 17 (03) ◽

pp. 2050040

Author(s):

Mahdieh Yourdkhany ◽

Mehdi Nadjafikhah ◽

Megerdich Toomanian

Keyword(s):

Power Series ◽

Conservation Laws ◽

Fractional Derivatives ◽

Nonlinear Ordinary Differential Equation ◽

Lie Symmetry ◽

Symmetry Analysis ◽

Lie Symmetry Analysis ◽

Power Series Method ◽

Power Series Solutions ◽

Liouville Fractional Derivative

This paper systematically investigates the Lie symmetry analysis of the time-fractional Buckmaster equation in the sense of Riemann–Liouville fractional derivative. With the aid of infinitesimal symmetries, this equation is transformed into a nonlinear ordinary differential equation of fractional order (FODE), where the fractional derivatives are in Erdelyi–Kober sense. The reduced FODE is solved with the explicit power series method and some figures for the obtained power series solutions are also depicted. Finally, Ibragimov’s method and Noether’s theorem have been employed to conclude the conservation laws of this equation.

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Lie Symmetry Analysis of the Time Fractional Generalized KdV Equations with Variable Coefficients

Symmetry ◽

10.3390/sym11101281 ◽

2019 ◽

Vol 11 (10) ◽

pp. 1281 ◽

Cited By ~ 1

Author(s):

Cheng Chen ◽

Yao-Lin Jiang ◽

Xiao-Tian Wang

Keyword(s):

Variable Coefficients ◽

Lie Symmetry ◽

Symmetry Analysis ◽

Boundary Values ◽

Lie Symmetry Analysis ◽

Power Series Method ◽

Kdv Equations ◽

Fractional Differential Operator ◽

Generalized Kdv Equations

The group classification of a class of time fractional generalized KdV equations with variable coefficient is presented. The Lie symmetry analysis method is extended to the certain subclasses of time fractional generalized KdV equations with initial and boundary values. Under the corresponding similarity transformation with similarity invariants, KdV equations with initial and boundary values have been transformed into fractional ordinary differential equations with initial value. Then we use the power series method to obtain the exact solution of the reduced equation with the Erdélyi-Kober fractional differential operator.

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Lie Symmetry Reductions and Exact Solutions to the Rosenau Equation

Advances in Mathematical Physics ◽

10.1155/2014/714635 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6

Author(s):

Ben Gao ◽

Hongxia Tian

Keyword(s):

Exact Solutions ◽

Analytic Solutions ◽

Lie Symmetry ◽

Series Method ◽

Lie Symmetry Analysis ◽

Power Series Method ◽

Exact Analytic Solutions ◽

Rosenau Equation ◽

Physical Phenomena ◽

Similarity Reductions

The Lie symmetry analysis is performed on the Rosenau equation which arises in modeling many physical phenomena. The similarity reductions and exact solutions are presented. Then the exact analytic solutions are considered by the power series method.

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