Narrow Results

New solutions to the Frank-Kamenetskii partial differential equation modeling a thermal explosion in a cylindrical vessel are obtained. The classical Lie group method is used to determine an approximate solution valid in a small interval around the axis of the cylinder. Non-classical symmetries are used to determine solutions valid after blow-up. These solutions have multiple singularities. Solutions are plotted and analyzed.

Group invariant analytical and numerical solutions for the evolution of a two-dimensional fracture with nonzero initial length in permeable rock and driven by an incompressible non-Newtonian fluid of power-law rheology are obtained. The effect of fluid leak-off on the evolution of the power-law fluid fracture is investigated.

In this paper, by introduction of pseudopotentials, the nonlocal symmetry is obtained for the Ablowitz–Kaup–Newell–Segur system, which is used to describe many physical phenomena in different applications. Together with some auxiliary variables, this kind of nonlocal symmetry can be localized to Lie point symmetry and the corresponding once finite symmetry transformation is calculated for both the original system and the prolonged system. Furthermore, the nth finite symmetry transformation represented in terms of determinant and exact solutions are derived.

In this work, one-point Lie symmetry method is applied to time fractional super KdV equation in order to obtain similarity variables and similarity transformations with Riemann–Liouville derivative. These transformations reduce the governing equation to an ordinary differential equation of fractional order. A new and effective conservation theorem based on Noether’s theorem is used to obtain conserved vectors. Then, we construct power series solutions for the reduced time fractional ordinary differential equation and prove that the solutions are convergent. Lastly, some interesting graphs are given to explain physical behaviors.

We consider symmetries and perturbed symmetries of canonical Hamiltonian equations of motion. Specifically we consider the case in which the Hamiltonian equations exhibit a Λ-symmetry under some Lie point vector field. After a brief survey of the relationships between standard symmetries and the existence of first integrals, we recall the definition and the properties of Λ-symmetries. We show that in the presence of a Λ-symmetry for the Hamiltonian equations, one can introduce the notion of "Λ-constant of motion". The presence of a Λ-symmetry leads also to a nice and useful reduction of the form of the equations. We then consider the case in which the Hamiltonian problem is deduced from a Λ-invariant Lagrangian. We illustrate how the Lagrangian Λ-invariance is transferred into the Hamiltonian context and show that the Hamiltonian equations are Λ-symmetric. We also compare the "partial" (Lagrangian) reduction of the Euler–Lagrange equations with the reduction which can be obtained for the Hamiltonian equations. Several examples illustrate and clarify the various situations.