General Basset–Boussinesq–Oseen equation: existence, uniqueness, approximation and regularity of solutions

The Basset–Boussinesq–Oseen equation (BBO equation) is the mathematical formulation of the Lagrangian acceleration of a spherical particle, moving with Lagragian velocity, in an unsteady flow as the sum of the viscous, gravitational, buoyancy, virtual mass, and Basset forces acting respectively on the particle. Despite the widespread use of the equation in applications, the basic properties of its solutions have remained unexplored. Here we fill this gap by proving global existence and uniqueness of solutions in an appropriate partially ordered Banach space. The method also gives an approximate solution to the problem with accurate result which is easy to implement. Meanwhile, regularity properties of the solutions are proved under some conditions which gives a representation of the solutions. Illustrative examples exhibit the efficiency of our method.

[1]  Juan J. Nieto,et al.  Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations , 2005, Order.

[2]  M. Li,et al.  An efficient difference scheme for the coupled nonlinear fractional Ginzburg–Landau equations with the fractional Laplacian , 2018, Numerical Methods for Partial Differential Equations.

[3]  Mouffak Benchohra,et al.  Advanced fractional differential and integral equations , 2015 .

[4]  Margarita Rivero,et al.  Fractional differential equations as alternative models to nonlinear differential equations , 2007, Appl. Math. Comput..

[5]  Meng Li,et al.  A fast energy conserving finite element method for the nonlinear fractional Schrödinger equation with wave operator , 2018, Appl. Math. Comput..

[6]  Yong Zhou.  Basic Theory of Fractional Differential Equations , 2014 .

[7]  Chengming Huang,et al.  ADI Galerkin FEMs for the 2D nonlinear time-space fractional diffusion-wave equation , 2017, Int. J. Model. Simul. Sci. Comput..

[8]  Chengming Huang,et al.  A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations , 2019, Numerical Algorithms.

[9]  Hossein Fazli,et al.  Nonlinear sequential fractional differential equations in partially ordered spaces , 2018 .

[10]  Juan J. Nieto,et al.  Fractional Langevin equation with anti-periodic boundary conditions , 2018, Chaos, Solitons & Fractals.

[11]  A. Haselbacher,et al.  Generalized Basset-Boussinesq-Oseen equation for unsteady forces on a sphere in a compressible flow. , 2011, Physical review letters.

[12]  Vasily E. Tarasov,et al.  Generalized Memory: Fractional Calculus Approach , 2018, Fractal and Fractional.

[13]  Fariba Bahrami,et al.  A new approach on fractional variational problems and Euler-Lagrange equations , 2015, Commun. Nonlinear Sci. Numer. Simul..

[14]  Hao Guo,et al.  A superlinear convergence scheme for nonlinear fractional differential equations and its fast implement , 2019, Int. J. Model. Simul. Sci. Comput..

[15]  Jincheng Ren,et al.  L1 scheme on graded mesh for the linearized time fractional KdV equation with initial singularity , 2019, Int. J. Model. Simul. Sci. Comput..

[16]  Ernst Hairer,et al.  Numerical methods for singular nonlinear integro-differential equations , 1987 .

[17]  Juan J. Nieto,et al.  Fixed point theorems in ordered abstract spaces , 2007 .

[18]  Fariba Bahrami,et al.  On the Steady Solutions of Fractional Reaction-Diffusion Equations , 2017 .

[19]  Mouffak Benchohra,et al.  Topics in Fractional Differential Equations , 2012 .

[20]  G. H. Hardy,et al.  Some properties of fractional integrals. I. , 1928 .

[21]  Juan J. Nieto,et al.  An investigation of fractional Bagley-Torvik equation , 2019, Open Mathematics.

[22]  T. Kaczorek,et al.  Fractional Differential Equations , 2015 .

[23]  V. Lakshmikantham,et al.  Fixed point theorems in partially ordered metric spaces and applications , 2006 .

[24]  J. Riley,et al.  Equation of motion for a small rigid sphere in a nonuniform flow , 1983 .

[25]  Zerner.  Neuere Methoden und Ergebnisse in der Hydrodynamik , 1928 .

[26]  Xinwei Su,et al.  Boundary value problem for a coupled system of nonlinear fractional differential equations , 2009, Appl. Math. Lett..

[27]  Behrouz Parsa Moghaddam,et al.  An integro quadratic spline approach for a class of variable-order fractional initial value problems , 2017 .

[28]  Hermann Brunner,et al.  Polynomial spline collocation methods for the nonlinear basset equation , 1989 .

[29]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[30]  Meng Li,et al.  Galerkin finite element method for nonlinear fractional Schrödinger equations , 2017, Numerical Algorithms.

[31]  S. McKee,et al.  Product Integration Methods for the Nonlinear Basset Equation , 1983 .