A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

Abstract

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.

Formation of solitary waves solutions and dynamic visualization of the nonlinear schrödinger equation with efficient techniques

“…After substituting equation (25) along with (26) in equation (24), we get an algebraic system by equating the coefficients of distinct powers of ( )…”

“…Various techniques and methods are available for obtaining accurate soliton solutions in non-linear PDEs. These methodologies encompass a variety of approaches, including the unified auxiliary equation method used by Mathanaranjan et al [22], the sine-Gordon expansion method [23], the simple extended [24], the bilinear neural network method [25], variational iteration techniques [26], the extended exponential function technique [27], the power series methodology [28], the Hirota bilinear technique [29]. Numerous others [30][31][32][33][34][35][36].…”

Non-singular complexiton, singular complexiton and complex N-soliton solutions of the new extended (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

“…Solitons have important applications in physical and mathematical sciences, such as fluid mechanics, optics, elasticity and plasma science [1][2][3][4][5][6][7][8][9][10][11][12][13]. Recently, many new methods have emerged for exploring the soliton solutions of the PDEs, for instance, the Jacobi elliptic-function technique [14,15], general integral approach [16,17], trial equation approach [18,19], Bäcklund transformation approach [20], exp-function approach [19,21,22], Sardar-subequation method [23][24][25], extended F-expansion approach [26][27][28], Kudryashov's method [29][30][31] and so on [32][33][34][35][36][37][38][39][40][41][42][43][44][45]. They are also used in scattering light waves, neural networks, and quantum mechanics.…”

Exact travelling wave solutions of time-fractional Clannish Random Walker's Parabolic equation and sensitive analysis

“…used toobtain exact solitonsolutions for non-linear partial differential equations, [30] variational iteration method [31], extended exponential function method [32], Hirota bilinear technique [33], power series method [34], F-expansion technique [35] as well as several others [36][37][38][39][40][41][42].…”