Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to sol...Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.展开更多
An analytical method is derived for the thermal consolidation of a saturated, porous, hollow cylinder with infinite length. The solutions in Laplace transform space are first obtained and then numerically inverted by ...An analytical method is derived for the thermal consolidation of a saturated, porous, hollow cylinder with infinite length. The solutions in Laplace transform space are first obtained and then numerically inverted by Stehfest method. Two cases of boundary conditions are considered. First, variable thermal loadings are applied on the inner and outer pervious lateral surfaces of the hollow cylinder, and a variable mechanical loading with time is applied on the outer surface;while the displacement of the inner surface remains fixed. Secondly, variable thermal and mechanical loading are applied on the outer pervious surface, and the inner surface remains fixed, impervious and insulated. As two special problems, a solid cylinder with infinite length and a cylindrical cavity in a half-space body are also discussed. Finally, the evolutions of temperature, pore pressure and displacement with time along radial direction are analyzed by a numerical example.展开更多
文摘Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.
文摘An analytical method is derived for the thermal consolidation of a saturated, porous, hollow cylinder with infinite length. The solutions in Laplace transform space are first obtained and then numerically inverted by Stehfest method. Two cases of boundary conditions are considered. First, variable thermal loadings are applied on the inner and outer pervious lateral surfaces of the hollow cylinder, and a variable mechanical loading with time is applied on the outer surface;while the displacement of the inner surface remains fixed. Secondly, variable thermal and mechanical loading are applied on the outer pervious surface, and the inner surface remains fixed, impervious and insulated. As two special problems, a solid cylinder with infinite length and a cylindrical cavity in a half-space body are also discussed. Finally, the evolutions of temperature, pore pressure and displacement with time along radial direction are analyzed by a numerical example.
