Inverse Source Problem for a Multiterm Time-Fractional Diffusion Equation with Nonhomogeneous Boundary Condition

This paper is devoted to identify a space-dependent source function in a multiterm time-fractional diffusion equation with nonhomogeneous boundary condition from a part of noisy boundary data. The well-posedness of a weak solution for the corresponding direct problem is proved by the variational method. We firstly investigate the uniqueness of an inverse initial problem by the analytic continuation technique and the Laplace transformation. Then, the uniqueness of the inverse source problem is derived by employing the fractional Duhamel principle. The inverse problem is solved by the Levenberg-Marquardt regularization method, and an approximate source function is found. Numerical examples are provided to show the effectiveness of the proposed method in one- and two-dimensional cases.

Multiple solutions for an inclusion quasilinear problem with nonhomogeneous boundary condition through Orlicz–Sobolev spaces

In this work, we study multiplicity of nontrivial solution for the following class of differential inclusion problems with nonhomogeneous Neumann condition through Orlicz–Sobolev spaces, [Formula: see text] where [Formula: see text] is a domain, [Formula: see text] and [Formula: see text] is the generalized gradient of [Formula: see text]. The main tools used are Variational Methods for Locally Lipschitz Functional and Critical Point Theory.

An Analytical Solution for Radiofrequency Ablation with a Cooled Cylindrical Electrode

We present an analytical solution to the electrothermal mathematical model of radiofrequency ablation of biological tissue using a cooled cylindrical electrode. The solution presented here makes use of the method of separation of variables to solve the problem. Green’s functions are used for the handling of nonhomogeneous terms, such as effect of electrical currents circulation and the nonhomogeneous boundary condition due to cooling at the electrode surface. The transcendental equation for determination of eigenvalues of this problem is solved using Newton’s method, and the integrals that appear in the solution of the problem are obtained by Simpson’s rule. The solution obtained here has the possibility of handling different functional dependencies of the source term and nonhomogeneous boundary condition. The solution provides a tool to understand the physics of the problem, as it shows how the solution depends on different parameters, to provide mathematical tools for the design of surgical procedures and to validate other modeling techniques, such as the numerical methods that are frequently used to solve the problem.

On the Second-Order Wave Radiation of an Oscillating Vertical Circular Cylinder in Finite-Depth Water

A vertical circular cylinder which is in periodic oscillatory motion with small amplitudes in finite depth is considered. The usual assumptions necessary for the potential flow stand valid in the present study. A classical perturbation procedure is employed to solve the nonlinear problem through the second-order. According to the solution method presented, the fluid domain is separated into interior and exterior regions in which boundary-value problems (BVP) are decomposed into two BVPs each having one nonhomogeneous boundary condition. A nonhomogeneous second-order free-surface condition is treated by means of a modified form of Weber's integral theorem. Eigenfunction expansions are used for homogeneous solutions. Thus, to conclude the solution, the exterior and interior solutions are then matched on the common boundary. Numerical results are given for a heaving vertical circular cylinder. Wave field analysis around a vertical cylinder shows that the second-order wave pattern is typically dominated by the second-order wave number related to the second-order dispersion relation. The procedure also satisfies the conditions at infinity through the second-order.

Dynamic Programming and a Distributed Parameter Maximum Principle

A proof of a distributed parameter maximum principle is given by using dynamic programming. An example problem involving a nonhomogeneous boundary condition is also treated by using the dynamic programming technique and by extending the definition of the differential operator. It is thus demonstrated that for linear systems the dynamic programming approach is just as powerful as the variational approach originally used to derive the maximum principle.