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<p>In this work, we construct a unified linearized Galerkin FEM for a Kirchhoff type doubly fractional PDE having a memory term. In this method, we approximate the Caputo time-fractional derivative by a linear or quadratic polynomial interpolation on a non-uniform graded time mesh and the space direction by a conforming Galerkin FEM. The nonlocal nonlinear Kirchhoff term causes high computational cost, which is reduced by linearising the nonlinearity without affecting the convergence rate. A combination of the rectangle rule and the composite trapezoidal rule is used to approximate the memory term. The resulting numerical scheme has a unique solution, assured by a consequence of the Broüwer fixed point theorem. The presence of memory term restricts the applications of discrete fractional Gronwall’s inequality to deduce the stability and the convergence estimates of the proposed numerical scheme. We overcome this issue by presenting an alternative approach which uses the standard discrete Gronwall’s inequality. finally, we utilize the best approximation properties of the modified fractional Ritz-Volterra projection operator to derive the convergence estimates of the developed numerical scheme. These convergence estimates are illustrated by implementing some numerical experiments.</p>

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A unified linearized Galerkin FEM for Kirchhoff type space-time fractional PDE with memory

  • 期刊论文

  • 发表日期：

    2025年4月14日

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  • Lalit Kumar,
  • Sivaji Ganesh Sista,
  • Konijeti Sreenadh

摘要

In this work, we construct a unified linearized Galerkin FEM for a Kirchhoff type doubly fractional PDE having a memory term. In this method, we approximate the Caputo time-fractional derivative by a linear or quadratic polynomial interpolation on a non-uniform graded time mesh and the space direction by a conforming Galerkin FEM. The nonlocal nonlinear Kirchhoff term causes high computational cost, which is reduced by linearising the nonlinearity without affecting the convergence rate. A combination of the rectangle rule and the composite trapezoidal rule is used to approximate the memory term. The resulting numerical scheme has a unique solution, assured by a consequence of the Broüwer fixed point theorem. The presence of memory term restricts the applications of discrete fractional Gronwall’s inequality to deduce the stability and the convergence estimates of the proposed numerical scheme. We overcome this issue by presenting an alternative approach which uses the standard discrete Gronwall’s inequality. finally, we utilize the best approximation properties of the modified fractional Ritz-Volterra projection operator to derive the convergence estimates of the developed numerical scheme. These convergence estimates are illustrated by implementing some numerical experiments.