Numerical solution of PDEs is rich and active field of modern applied mathematics. The steady growth of the subject is stimulated by everincreasing demands from the natural sciences, engineering and economics to provide accurate and reliable approximations to mathematical models involving partial differential equations (PDEs) whose exact solutions are either too complicated to determine in clos...

The paper considers a class of parametric elliptic partial differential equations (PDEs), where the coefficients and right-hand side function depend on infinitely many (uncertain) parameters. W...

We give a systematic method for discretising Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same method, applied to PDEs with constant dissipative structure, also preserves the correct monotonic decrease of energy. The method is illustrated by many examples. In the Hamiltonian case these include: the sine-Gordon, ...

solitons are ubiquitous and exist in almost every area from sky to bottom. for solitons to appear, the relevant equation of motion must be nonlinear. in the present study, we deal with the korteweg-devries (kdv), modied korteweg-de vries (mkdv) and regularised longwave (rlw) equations using homotopy perturbation method (hpm). the algorithm makes use of the hpm to determine the initial expansio...

I am going to start a series of lectures on nonlinear partial differential equations. Advancements made in the theory of nonlinear PDEs is one of the main achievements in XX century mathematics. Let me first make a remark on linear PDEs. In some sense one can say that there is a complete theory of linear PDEs, and perhaps the best source would be the four volumes of "The Analysis of Linear Part...

Mathematical morphology (MM) offers a wide range of operators to address various image processing problems. These operators can be defined in terms of algebraic (discrete) sets or as partial differential equations (PDEs). In this paper, we introduce a novel formulation of MM formalized as a framework of partial difference equations (PdEs) over weighted graphs of arbitrary topology. Then, we pre...

A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations Abstract This paper presents constructive a posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations (PDEs) on a bounded domain. This type of estimates plays an important role in the numerical verification of the...

We present fast, spatially dispersionless and unconditionally stable high-order solvers for Partial Differential Equations (PDEs) with variable coefficients in general smooth domains. Our solvers, which are based on (i) A certain “Fourier continuation” (FC) method for the resolution of the Gibbs phenomenon, together with (ii) A new, preconditioned, FC-based solver for two-point boundary value p...

I report on a contribution to the point symmetry classification problem for second-order partial differential equations (PDEs) in z(x, y), i.e. to an overview over all possible symmetry groups admitted by this class of equations. The article also contains a concise introduction into classical symmetry analysis. Sophus Lie (1842–1899) determined all continuous transformation groups of the 2D pla...