Travelling peakon and solitary wave solutions of modified Fornberg–Whitham equations with nonhomogeneous boundary conditions

In this study, the numerical solutions of the modified Fornberg–Whitham (mFW) equation, which describes immigration of the solitary wave and peakon waves with discontinuous first derivative at the peak, have been obtained by the collocation finite element method using quintic trigonometric B-spline bases. Although there are solutions of this equation by semi-analytical and analytical methods in the literature, there are very few studies on the solution of the equation by numerical methods. Any linearization technique has not been used while applying the method. The stability analysis of the applied method is examined by the von-Neumann Fourier series method. To show the performance of the method, we have considered three test problems with nonhomogeneous boundary conditions having analytical solutions. The error norms L 2 and L ∞ are calculated to demonstrate the accuracy and efficiency of the presented numerical scheme.

Theoretical and Numerical Aspect of Fractional Differential Equations with Purely Integral Conditions

In this paper, we are interested in the study of a Caputo time fractional advection–diffusion equation with nonhomogeneous boundary conditions of integral types ∫01vx,tdx and ∫01xnvx,tdx. The existence and uniqueness of the given problem’s solution is proved using the method of the energy inequalities known as the “a priori estimate” method relying on the range density of the operator generated by the considered problem. The approximate solution for this problem with these new kinds of boundary conditions is established by using a combination of the finite difference method and the numerical integration. Finally, we give some numerical tests to illustrate the usefulness of the obtained results.

A fractional q-difference equation eigenvalue problem with p(t)-Laplacian operator

This article is devoted to studying a nonhomogeneous boundary value problem involving Stieltjes integral for a more general form of the fractional q-difference equation with p(t)-Laplacian operator. Here p(t)-Laplacian operator is nonstandard growth, which has been used more widely than the constant growth operator. By using fixed point theorems of φ – (h, e)-concave operators some conditions, which guarantee the existence of a unique positive solution, are derived. Moreover, we can construct an iterative scheme to approximate the unique solution. At last, two examples are given to illustrate the validity of our theoretical results.

A Nonhomogeneous Boundary-Valued Problem for the coupled KDV system

In this paper, we study the initial-boundary-value problem (IBVP) for coupled Korteweg-de Vries equations posed on a finite interval with nonhomogeneous boundary conditions. We overcome the requirement for stronger smooth boundary conditions in the traditional method via the Laplace transform. Our approach uses the strong Kato smoothing property and the contraction mapping principle.

ON NONHOMOGENEOUS BOUNDARY VALUE PROBLEM FOR THE STATIONARY NAVIER-STOKES EQUATIONS IN A SYMMETRIC CUSP DOMAIN

The nonhomogeneous boundary value problem for the stationary NavierStokes equations in 2D symmetric multiply connected domain with a cusp point on the boundary is studied. It is assumed that there is a source or sink in the cusp point. A symmetric solenoidal extension of the boundary value satisfying the LerayHopf inequality is constructed. Using this extension, the nonhomogeneous boundary value problem is reduced to homogeneous one and the existence of at least one weak symmetric solution is proved. No restrictions are assumed on the size of fluxes of the boundary value.

Renormalized solutions for some nonlinear nonhomogeneous elliptic problems with Neumann boundary conditions and right hand side measure

Our aim in this paper is to study the existence of renormalized solution for a class of nonlinear p(x)-Laplace problems with Neumann nonhomogeneous boundary conditions and diuse Radon measure data which does not charge the sets of zero p(.)-capacity