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<p>This manuscript deals with the Euler-Bernoulli beam equation with a delayed forcing term, subject to both boundary and history conditions. Using analytic semigroup theory, we reformulate the problem as a system of coupled nonlinear integral equations. By introducing projection operators, we derive a system of approximate coupled nonlinear integral equations. We then prove the existence and uniqueness of their solutions using the contraction mapping theorem. Furthermore, we establish the convergence of both the Faedo-Galerkin approximations and the approximate integral equations to the mild solution and the original integral equations, respectively. Finally, we examine the regularity of the solution.</p>

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Approximation of Solutions to Delayed Euler-Bernoulli Beam Like Equation

  • 期刊论文

  • 发表日期：

    2025年8月22日

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  • Nishi Gupta,
  • Md. Maqbul

摘要

This manuscript deals with the Euler-Bernoulli beam equation with a delayed forcing term, subject to both boundary and history conditions. Using analytic semigroup theory, we reformulate the problem as a system of coupled nonlinear integral equations. By introducing projection operators, we derive a system of approximate coupled nonlinear integral equations. We then prove the existence and uniqueness of their solutions using the contraction mapping theorem. Furthermore, we establish the convergence of both the Faedo-Galerkin approximations and the approximate integral equations to the mild solution and the original integral equations, respectively. Finally, we examine the regularity of the solution.