Petrov-Galerkin method for small deflections in fourth-order beam equations in civil engineering

This study explores the Petrov–Galerkin method’s application in solving a linear fourth-order ordinary beam equation of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo accent="true">″</m:mo> <m:mo accent="false">″</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>q</m:mi> <m:mi>u</m:mi> <m:mo>=</m:mo> <m:mi>f</m:mi> </m:math> u^{} ^{} +qu=f . The equation entails two distinct boundary conditions: pinned–pinned conditions on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> </m:math> u and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> <m:mo accent="false">′</m:mo> </m:math> u^{} , and clamped–clamped conditions on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>u</m:mi> </m:math> u and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo accent="true">″</m:mo> </m:mrow> </m:msup> </m:math> {u}^{^{} } . To satisfy these boundary conditions, we have built two sets of basis functions. The explicit forms of all spectral matrices were reported. The nonhomogeneous boundary conditions were easily handled using perfect transformations, ensuring the numerical solution’s accuracy. Detailed analysis of the method’s convergence was studied. Some numerical examples were presented, accompanied by comparisons with other existing methods in the literature.