A Neumann problem for a diffusion equation with n-dimensional fractional Laplacian

We study an initial-boundary value problem for a n-dimensional stochastic diffusion equation with fractional Laplacian on $\mathbb{R}_{+}^{n}$ R + n . In order to prove existence and uniqueness, we generalize the Fokas method to construct the Green function for the associated linear problem and then we apply a fixed point argument. Also, we present an example where the explicit solutions are given.

The Modified Helmholtz Equation on a Regular Hexagon—The Symmetric Dirichlet Problem

Using the unified transform, also known as the Fokas method, we analyse the modified Helmholtz equation in the regular hexagon with symmetric Dirichlet boundary conditions; namely, the boundary value problem where the trace of the solution is given by the same function on each side of the hexagon. We show that if this function is odd, then this problem can be solved in closed form; numerical verification is also provided.

Fokas method for linear boundary value problems involving mixed spatial derivatives

We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.

Unified Transform method for the Schr¨odinger Equation on a Simple Metric Graph

Integral-representation of solutions of the initial-boundary value problems for the Schr¨odinger equation on simple metric graphs was obtained with the use of the Fokas method. This method uses special gen- eralization of the Fourier transform that is referred to as the unified transform. Obtained representation of solutions of the problem for open and closed simple star graphs allows one to identify transmitted, reflected and trapped waves at the graph branching point

Stochastic diffusion equation with fractional Laplacian on the first quadrant

In this work, we consider an initial boundary-value problem for a stochastic evolution equation with fractional Laplacian and white noise on the first quadrant. To construct the integral representation of solutions we adapt the main ideas of the Fokas method and by using Picard scheme we prove its existence and uniqueness. Moreover, Monte Carlo methods are implemented to find numerical solutions for particular examples.

The massive Thirring system in the quarter plane

The unified transform method (UTM) or Fokas method for analyzing initial-boundary value (IBV) problems provides an important generalization of the inverse scattering transform (IST) method for analyzing initial value problems. In comparison with the IST, a major difficulty of the implementation of the UTM, in general, is the involvement of unknown boundary values. In this paper we analyze the IBV problem for the massive Thirring model in the quarter plane, assuming that the initial and boundary data belong to the Schwartz class. We show that for this integrable model, the UTM is as effective as the IST method: Riemann-Hilbert problems we formulated for such a problem have explicit (x, t)-dependence and depend only on the given initial and boundary values; they do not involve additional unknown boundary values.

A Hybrid Method for Solving Inhomogeneous Elliptic PDEs Based on Fokas Method

In this paper we propose a hybrid method for solving inhomogeneous elliptic PDEs based on the unified transform. This approach relies on the derivation of the global relation, containing certain integral transforms of the given boundary data as well as of the unknown boundary values. Herewith, the approximate global relation for the Poisson equation is solved numerically using a collocation method on the complex λ-plane, based on Legendre expansions. The corresponding numerical results are presented using closed-form expressions and numerical approximations for different types of boundary and source data, indicating the applicability of the considered approach. Additionally, the full solution is computed in a recursive manner by splitting the domain into smaller concentric polygons, and by using a spatial-stepping scheme followed by an interpolation step. Furthermore, numerical results are also given for the solution of the Poisson and the inhomogeneous Helmholtz equations on several convex polygons. Additional results are provided for the case of nonconvex polygons as well as for the case of a problem with discontinuities across an interface. The proposed approach provides a framework for solving inhomogeneous elliptic PDEs using the unified transform.