Numerical Approach to a Cantilever Beam Equation With One End Simply
Supported and Other End Fixed
Abstract
The existence of an approximate solution to the fourth-order boundary
value problem (BVP) \begin{eqnarray} u^{(4)}(t) =
h(t)f(u(t))+g(t), \quad t \in [0, 1],
\nonumber\\ u(0) =
\alpha, u^{\prime}(1)=
\beta, u(1) = \gamma,
u^{\prime\prime}(0) =
\delta, \nonumber
\end{eqnarray} is investigated using a new inverse
operator, where $f,g,h \in C(
\mathbb{R}, \mathbb{R})$, and
$\alpha, \beta, \gamma$
and $\delta$ are real numbers. As an application, we
apply the inverse operator and ADM to study the existence of an
approximate solution of the Cantilever beam equation whose one end
simply-supported with other end fixed \begin{eqnarray}
u^{(4)}(t) = h(t)f(u(t))+g(t), \quad t
\in [0, 1],
\nonumber\\ u(0) = 0,
u^{\prime}(1)= 0, u(1) = 0,
u^{\prime\prime}(0) = 0,
\nonumber \end{eqnarray} where $f,g$
and $h \in C( \mathbb{R},
\mathbb{R})$. Our examples shows that the proposed
inverse operator and the application ADM gives very less errors in the
obtained approximate solution when compared to the exact solution of the
Cantilever beam problems.