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If $A$ is a tridiagonal matrix, then the equations $AX=I$ and $XA=I$ defining the inverse $X$ of $A$ are in fact the second order recurrence relations for the elements in each row and column of $X$. Thus, the recursive algorithms should be…
In this paper we address the return to equilibrium problem for an axisymmetric floating structure in shallow water. First we show that the equation for the solid motion can be reduced to a delay differential equation involving an…
We present a second-order upwind numerical scheme for equations of relativistic hydrodynamics with a source term. A new non-linear Riemann solver is constructed. Solution of a Riemann problem on a cells boundary is based on exact relations…
We consider models of relativistic matter containing sharp interfaces across which the matter model changes. These models will be relevant for neutron stars with crusts, phase transitions, or for viscous boundaries where the length scale is…
We construct the solution of the Riemann problem for the shallow water equations with discontinuous topography. The system under consideration is non-strictly hyperbolic and does not admit a fully conservative form, and we establish the…
The central idea of this article is to present a systematic approach to construct some recurrence relations for the solutions of the second-order linear difference equation of hypergeometric-type defined on the quadratic-type lattices. We…
A reduced-order model algorithm, called ALP, is proposed to solve nonlinear evolution partial differential equations. It is based on approximations of generalized Lax pairs. Contrary to other reduced-order methods, like Proper Orthogonal…
The new form of the composite operator generalizing the Cooper pairs for a BCS superconductor is introduced. The approach is similar to the derivation of the composite operator of the odd - frequency superconductors. The examples of the…
We construct renormalised models of regularity structures by using a recursive formulation for the structure group and for the renormalisation group. This construction covers all the examples of singular SPDEs which have been treated so far…
The shallow-water system is a standard model for long waves in shallow water. The system is hyperbolic and, for a large class of initial data, solutions develop steep gradients leading to shock formation in finite time. Since such…
Based on the gradient-holonomic algorithm we analyze the integrability property of the generalized hydrodynamical Riemann type equation $%D_{t}^{N}u=0$ for arbitrary $N\in \mathbb{Z}_{+}.$ The infinite hierarchies of polynomial and…
A detailed description is given for the construction of the deformation of the N=2 supersymmetric $\alpha=1$ KdV-equation, leading to the recursion operator for symmetries and the zero-th Hamiltonian structure; the solution to a…
In this paper, we seek to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. We first present a conditional result on the construction of nontrivial global solutions to a general system of…
In this work, we obtain decay bounds for a class of ID dispersive equations that includes the linearized water wave. These decay bounds display a surprising growth factor, which we show is sharp, The proofs rely on careful analysis of…
In this work, we study the generalized shallow water wave equation to obtain novel solitary wave solutions. The application of this non-linear model can be found in tidal waves, weather simulations, tsunami prediction, river and irrigation…
Certain linear matrix operators arise naturally in systems analysis and design problems involving cascade interconnections of linear time-invariant systems, including problems of stabilization, estimation, and model order reduction. We…
In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the Hamiltonian formalism to nonabelian difference…
In this paper, a supersymmetric extension of a system of hydrodynamic type equations involving Riemann invariants is formulated in terms of a superspace and superfield formalism. The symmetry properties of both the classical and…
We describe a general method of constructing nonlocal recursion operators for symmetries of PDEs. As an example, the cotangent equation to the 3D rdDym equation $u_{yt} = u_xu_{xy} - u_yu_{xx}$ for which two mutually inverse operators are…
Active Flux is a third order accurate numerical method which evolves cell averages and point values at cell interfaces independently. It naturally uses a continuous reconstruction, but is stable when applied to hyperbolic problems. In this…