# The Approximate Solution of Nonlinear Flexure of a Cantilever Beam with the Galerkin Method
## 1 Introduction  
Differential equations are widely used for the description of scientific and engineering phenomena with frequent appearances of nonlinear types [1; 2; 3; 4]. Consequently, differential equations and solution methods have a major role in modeling transmission dynamics and rendering control strategies for communicable diseases such as tuberculosis and COVID-19 [5; 6]. There have been tremendous efforts to solve differential equations with generalized and ad hoc methods, but there are always problems that cannot be effectively solved for many reasons [7; 8; 9]. Some known solutions of some specialized differential equations, such as those related to Bessel functions and hypergeometric functions, can be effectively used, but there are still differential equations that cannot be solved with existing methods and known functions [10; 11]. With classical methods of solutions failing in these contexts, in recent decades, some special methods with gradual sophistication have been developed, such as the multiscale method [10], homotopy analysis method [12; 13; 14; 15], homotopy perturbation method [16], and others dealing with emerging differential equations in many fields [17; 18]. Generally, most of these methods are asymptotic in nature, and can find solutions in series form with an improved converging rate and numerical stability over a relatively large interval. In combination with the algorithm development and computing tools for symbolic computation, these methods can powerfully and efficiently solve many practical and classical nonlinear equations. As a result, many techniques have been applied to the latest problems, and the improved solution techniques have been widely promoted and utilized. Studies on the large deflection of beams mostly focused on different boundary conditions, cross-section characteristics, and loading conditions [19], among others. Formulated with the angle enclosed by the axes of the deformed and undeformed configurations, thenonlinear equation of an elastic beam subjected to concentrated forces or couples resembles the equation of motion of a rigid pendulum according to [PERSON]'s kinetic analogy [20]. Large deflections of an end-supported beam subjected to a point load are solved using both the elliptic integral method and the shooting-optimization technique [21]. The homotopy analysis method was used to solve this problem with enough accuracy [13]. In recent attempts to find an optimal method for nonlinear vibrations of structures, it was discovered that the Galerkin method and its alternative form, the Rayleigh-Ritz method, can be effective for approximate analysis of nonlinear vibrations in certain circumstances [22; 23; 24; 25]. The Legendre-Galerkin matrix (LGM) method is used to solve the vibration equations of single (SDOF) and multiple degree-of-freedom (MDOF) structural systems [26]. The nonresonance and primary resonance of microbeams with consideration of small-scale effects and nonlinear terms were obtained by the Galerkin and multiple-scale methods [27]. With the objective of finding an effective method and taking the advantage of the widespread availability of powerful symbolic computing tools and resources, it is thought that the Galerkin method can be used for obtaining relatively accurate solutions to nonlinear differential equations with a simple procedure [28; 29]. The procedure can be developed with the appropriate trial solutions satisfying essential boundary and initial conditions, making the computing process more efficient for accelerated convergence and improving accuracy [30; 31]. Furthermore, if the orthogonality of trial solutions is guaranteed, the calculation will be further optimized. These concepts are already practiced in linear problems with the [PERSON] method and its modifications, and further improvements can be made with nonlinear problems as an innovative implementation.