Abstract
A computational framework for the construction of solutions to linear homogenous partial differential equations (PDEs) with variable coefficients is developed in this paper. The considered class of PDEs reads: ∂p∂t−∑j=0m∑r=0njajrtxr∂jp∂xj=0 F-operators are introduced and used to transform the original PDE into the image PDE. Factorization of the solution into rational and exponential parts enables us to construct analytic solutions without direct integrations. A number of computational examples are used to demonstrate the efficiency of the proposed scheme.
Highlights
The Fourier transform, as one of the most important concepts in signal analysis, is widely used for the construction of solutions to partial differential equations (PDEs).While there exist numerous classical techniques for the construction of solutions to PDEs based on the Fourier transform, recent research demonstrates that new approaches in this field remain an important area of investigation.The immersed boundary smooth extension method for solving PDEs in general domains is presented in [1]
A computational framework for the construction of analytic solutions to linear homogenous PDEs with variable coefficients is developed in this paper
A common approach for the construction of solutions to such PDEs consists of applying the wave variable transformation ξ “ tαx; α P R to transform the PDE into an ODE
Summary
The Fourier transform, as one of the most important concepts in signal analysis, is widely used for the construction of solutions to partial differential equations (PDEs). The immersed boundary smooth extension method for solving PDEs in general domains is presented in [1] This high-order accuracy numerical scheme smoothly extends the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of Fourier spectral methods. An efficient computational scheme based on a Laplace transform-based exponential time integrator combined with a flexible Krylov subspace approach is proposed in [3] to solve linear, time-dependent, parabolic PDEs. Fourier wavelets are used to construct solutions to partial differential equations in [4]. Application of the F-operator scheme enables the construction of analytical solutions without direct integration of the original PDE.
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