Existence and non-existence of global solutions to a generalized modification of the improved Boussinesq equation

A Fourier pseudospectral method for a generalized improved Boussinesq equation

“…, we present the variation of I ( t ) in time, where the integral I ( t ) is calculated using trapezoidal rule. As it has been shown analytically in , the numerical results indicate that the approximation of the integral I ( t ) goes to as time increases. Good agreement between the numerical and analytical results shows that the Fourier pseudospectral method does not miss the blow‐up solutions.…”

A meshless finite point method for the improved Boussinesq equation using stabilized moving least squares approximation and Richardson extrapolation

“…To study the solution blow‐up phenomenon numerically, this example considers the IBq Equation (12) with the following initial conditions $$$$ {\displaystyle \begin{array}{ll}u\left(x,0\right)& =-3\sin \left(\pi x\right),\kern1em 0<x<1,\\ {}{u}_t\left(x,0\right)& =-\sin \left(\pi x\right),\kern1em 0<x<1.\end{array}} $$$$ Theoretically, there is a finite constant $$$ {T}_0>0 $$$ such that we have a unique local solution $$$$ u\in {C}^2\left(\left[0{T}_0\right);{H}^2\left(0,1\right)\cap {H}_0^1\left(0,1\right)\right) $$$$ satisfying [58]. $$$$ {I}_1(t)={\left\Vert u\Big(\cdot, t\Big)\right\Vert}_{L^2\left(0,1\right)}\to +\infty, \kern1em \mathrm{as}\kern0.5em t\to {T}_0, $$$$ and …”

Numerical solutions of the improved boussinesq equation by the galerkin quadratic B-spline finite element method

“…In this example, we consider the blow-up solution of Eq. (2) also discussed in [11,18,19]. The problem is taken into account on the interval x ∈ [0, 1] and with the initial conditions As it is seen from the figure that although there is a small reduction between the times 1.4 and 1.6, this reduction becomes more visible between the times 1.6 and 1.7.…”