# Laplace Transform Method for Parabolic Problems with Time-Dependent Coefficients

@article{Lee2013LaplaceTM, title={Laplace Transform Method for Parabolic Problems with Time-Dependent Coefficients}, author={Hyoseop Lee and Jinwoo Brian Lee and Dongwoo Sheen}, journal={SIAM J. Numer. Anal.}, year={2013}, volume={51}, pages={112-125} }

The Laplace transform method has proven to be very efficient for dealing with parabolic problems whose coefficients are time independent, and it is easily parallelizable. However, the method has not been proven to be applicable to linear problems whose coefficients are time dependent. The reason is that the Laplace transform of two time-dependent functions leads to a convolution of the Laplace transformed functions in the dual variable. In this paper, we propose a Laplace transform method to…

## 17 Citations

### Laplace transform method for parabolic problems with time-dependent and nonlinear coefficients: Magnus integrator and linearization

- Mathematics
- 2016

We apply the Laplace transform method, which is efficient for dealing with high-dimensional parabolic problems, to the case of time-dependent and nonlinear coefficients. With rep sect to…

### Fast Laplace Transform Methods for Free-Boundary Problems of Fractional Diffusion Equations

- MathematicsJ. Sci. Comput.
- 2018

A fast Laplace transform method for solving a class of free-boundary fractional diffusion equations arising in the American option pricing, which outperforms the full finite difference methods in regard to the accuracy and complexity.

### Fast Laplace Transform Methods for Free-Boundary Problems of Fractional Diffusion Equations

- MathematicsJournal of Scientific Computing
- 2017

In this paper we develop a fast Laplace transform method for solving a class of free-boundary fractional diffusion equations arising in the American option pricing. Instead of using the time-stepping…

### Fast Numerical Contour Integral Method for Fractional Diffusion Equations

- MathematicsJournal of Scientific Computing
- 2015

The numerical contour integral method with hyperbolic contour is exploited to solve space-fractional diffusion equations. By making use of the Toeplitz-like structure of spatial discretized matrices…

### Fast Numerical Contour Integral Method for Fractional Diffusion Equations

- MathematicsJ. Sci. Comput.
- 2016

The numerical contour integral method with hyperbolic contour is exploited to solve space-fractional diffusion equations and the regions that the spectra of resulting matrices lie in are derived by making use of the Toeplitz-like structure of spatial discretized matrices.

### Second-Order and Nonuniform Time-Stepping Schemes for Time Fractional Evolution Equations with Time–Space Dependent Coefficients

- MathematicsJournal of Scientific Computing
- 2021

By means of the novel technique, the nonuniform Alikhanov type schemes are constructed and analyzed for the sub-diffusion and diffusion-wave problems and the second-order convergence is obtained with respect to discreteH-norm.

### Subdiffusion with a time-dependent coefficient: Analysis and numerical solution

- MathematicsMath. Comput.
- 2019

In this work, a complete error analysis is presented for fully discrete solutions of the subdiffusion equation with a time-dependent diffusion coefficient, obtained by the Galerkin finite element…

### Convergence Analysis of Iterative Laplace Transform Methods for the Coupled PDEs from Regime-Switching Option Pricing

- MathematicsJ. Sci. Comput.
- 2018

This paper provides the rigorous error analysis for the iterative Laplace transform methods by proving that the method has a second-order convergence rate in space and exponential- order convergence rate with respect to the number of the quadrature nodes for the Laplace inversion.

### Convergence Analysis of Iterative Laplace Transform Methods for the Coupled PDEs from Regime-Switching Option Pricing

- MathematicsJournal of Scientific Computing
- 2017

This paper aims to analyze the convergence rates of the iterative Laplace transform methods for solving the coupled PDEs arising in the regime-switching option pricing. The so-called iterative…

## References

SHOWING 1-10 OF 32 REFERENCES

### Parallel Methods for Solving Time-Dependent Problems Using the Fourier-Laplace Transformation

- Mathematics
- 2000

In this paper we summarize recent progresses on the parallel method for solving time-dependent problems using the Fourier-Laplace transformation. These problems arise in the study of elastic wave…

### A Parallel Method for Backward Parabolic Problems Based on the Laplace Transformation

- MathematicsSIAM J. Numer. Anal.
- 2006

A parallel method for time discretization of backward parabolic problems is proposed, obtaining a regularized solution with high frequency terms cut off by the inverse Laplace transforms without requiring the knowledge of the eigenfunctions of the differential operator.

### A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature

- Mathematics
- 2003

We consider the discretization in time of an inhomogeneous parabolic equation in a Banach space setting, using a representation of the solution as an integral along a smooth curve in the complex left…

### Time discretization via Laplace transformation of an integro-differential equation of parabolic type

- MathematicsNumerische Mathematik
- 2006

Abstract.We consider the discretization in time of an inhomogeneous parabolic integro-differential equation, with a memory term of convolution type, in a Banach space setting. The method is based on…

### Numerical Methods for Laplace Transform Inversion

- Mathematics
- 2007

Operational methods have been used for over a century to solve problems such as ordinary and partial differential equations. When solving such problems, in many cases it is fairly easy to obtain the…

### A New Numerical Method for Backward Parabolic Problems in the Maximum-Norm Setting

- MathematicsSIAM J. Numer. Anal.
- 2002

A new method for solving numerically backward parabolic problems is proposed, which is valid in the maximum-norm setting, and rigorous estimates are derived.

### A high order parallel method for time discretization of parabolic type equations based on Laplace transformation and quadrature

- Mathematics, Computer Science
- 2005

The discretization in time of a parabolic equation is considered, using a representation of the solution as an integral along a smooth curve in the complex left half plane, which reduces the problem to a finite set of elliptic equations, which may be solved in parallel.

### Exponentially Convergent Parallel Discretization Methods for the First Order Evolution Equations

- Mathematics
- 2001

Abstract We propose a new discretization of an initial value problem for differen- tial equations of the first order in a Banach space with a strongly P-positive operator coefficient. Using the…

### Numerical solution of a heat diffusion problem by boundary element methods using the Laplace transform

- MathematicsNumerische Mathematik
- 2005

A heat diffusion problem in a half-space which is motivated by the detection of material defects using thermal measurements is solved by inverting the Laplace transform with respect to time on a contour in the complex plane using an exponentially convergent quadrature rule.

### Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces

- MathematicsMath. Comput.
- 2005

We consider the Cauchy problem for the first and the second order differential equations in Banach and Hilbert spaces with an operator coefficient A(t) depending on the parameter t. We develop…