Süleyman Ulusoy

An inverse source problem for a one-dimensional space–time fractional diffusion equation

Fractional(nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues and they are used to model anomalous diffusion, especially in physics. This paper deals with a nonlocal inverse source problem for a one-dimensional space–time fractional diffusion equation where and . At first we define and analyze the direct problem for the space–time fractional diffusion equation. Later we define the inverse source problem. Furthermore, we set up an operator equation and derive the relation between the solutions of the operator equation and the inverse source problem. We also prove some important properties of the operator . By using these properties and analytic Fredholm theorem, we prove that the inverse source problem is well posed, i.e. can be determined uniquely and depends continuously on additional data .

Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues and they are used to model anomalous diffusion, especially in physics. This paper is devoted to a nonlocal inverse problem related to the space-time fractional equation . The existence of the solution for the inverse problem is proved by using quasi-solution method which is based on minimizing an error functional between the output data and the additional data. In this context, an input–output mapping is defined and continuity of the mapping is established. The uniqueness of the solution for the inverse problem is also proved by using eigenfunction expansion of the solution and some basic properties of fractional Laplacian. A numerical method based on discretization of the minimization problem, steepest descent method and least squares approach is proposed for the solution of the inverse problem. The numerical method determines the exponents of the fractional time and space derivatives simultaneously. Numerical examples with noise-free and noisy data illustrate applicability and high accuracy of the proposed method.

An inverse problem for a nonlinear diffusion equation with time-fractional derivative

Abstract A nonlinear time-fractional inverse coefficient problem is considered. The unknown coefficient depends on the solution. It is proved that the direct problem has a unique solution. Afterwards the continuous dependence of the solution of the corresponding direct problem on the coefficient is proved. Then the existence of a quasi-solution of the inverse problem is obtained in the appropriate class of admissible coefficients.

The asymptotics of heavily burdened viscoelastic rods

This paper treats the spatial motion of a deformable nonlinearly viscoelastic rod carrying a heavy rigid body. The ratio of the inertia of the rod to that of the attached rigid body is characterized by a small parameter ε. The boundary conditions on the rod where it is attached to the rigid body are the ordinary differential equations of motion for the rigid body subject to the contact loads exerted on the rigid body by the rod. The entire system is thus governed by a quasilinear parabolic-hyperbolic system of partial differential equations coupled to the ordinary differential equations for the rigid body, with ε appearing in the coefficients of the acceleration terms of the rod. This paper gives a rigorous asymptotic expansion of the solutions of initial-boundary-value problems for this system, consisting of a regular expansion and an initial-layer expansion. The leading term of the regular expansion satisfies the reduced problem, obtained by setting ε = 0 in the governing equations. The reduced problem is governed by a curious set of quasilinear functional-differential equations, the solutions of which exhibit a rich and interesting behavior. (In the absence of dissipation, which is needed for the justification of the asymptotic expansion, the leading term of the regular expansion satisfies a steady-state problem parametrized by time, which enters through the boundary conditions.) The remaining terms of the regular expansion satisfy linear problems. The leading term of the initial-layer expansion satisfies a quasilinear parabolic system, and the remaining terms satisfy linear parabolic systems. Thus the asymptotic expansion leads to greatly simplified equations. The regular and initial-layer corrections to the Received January 2, 2012. 2010 Mathematics Subject Classification. Primary 35B25, 35C20, 35G31, 35K35, 35K70, 74C20, 74D10,74K10. E-mail address: ssa@math.umd.edu E-mail address: suleyman.ulusoy@zirve.edu.tr c ©2012 Brown University Reverts to public domain 28 years from publication 437 License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf 438 STUART S. ANTMAN AND SÜLEYMAN ULUSOY solution of the reduced problem show that it exhibits the main features of the solution to the whole system. The justification of the asymptotic expansion consists in estimating the error. For this purpose, a Faedo-Galerkin method is used to obtain sharp estimates for the exponential decay in time of the terms of the initial-layer expansion (satisfying parabolic systems). (This method is far more efficient than the repeated use of the Maximum Principle à la S.N. Bernstein (see Wiegner, Math. Z. 188 (1984) 3–22) for treating the analogous scalar problem by Yip et al., J. Math. Pures Appl. 81 (2002) 283–309. The Maximum Principle is not applicable to our parabolic systems. Even for such scalar problems, the Faedo-Galerkin method as used here is far simpler and more efficient.) The main focus of this paper is on the derivation of these estimates. A significant part of the analysis is devoted to handling technical difficulties due to the peculiarities of the geometrically exact equations governing the spatial motion of viscoelastic rods with a general class of nonlinear constitutive equations of strain-rate type invariant under rigid motions.

Scaling and scale invariance of conservation laws in Reynolds transport theorem framework

Scale invariance is the case where the solution of a physical process at a specified time-space scale can be linearly related to the solution of the processes at another time-space scale. Recent studies investigated the scale invariance conditions of hydrodynamic processes by applying the one-parameter Lie scaling transformations to the governing equations of the processes. Scale invariance of a physical process is usually achieved under certain conditions on the scaling ratios of the variables and parameters involved in the process. The foundational axioms of hydrodynamics are the conservation laws, namely, conservation of mass, conservation of linear momentum, and conservation of energy from continuum mechanics. They are formulated using the Reynolds transport theorem. Conventionally, Reynolds transport theorem formulates the conservation equations in integral form. Yet, differential form of the conservation equations can also be derived for an infinitesimal control volume. In the formulation of the governing equation of a process, one or more than one of the conservation laws and, some times, a constitutive relation are combined together. Differential forms of the conservation equations are used in the governing partial differential equation of the processes. Therefore, differential conservation equations constitute the fundamentals of the governing equations of the hydrodynamic processes. Applying the one-parameter Lie scaling transformation to the conservation laws in the Reynolds transport theorem framework instead of applying to the governing partial differential equations may lead to more fundamental conclusions on the scaling and scale invariance of the hydrodynamic processes. This study will investigate the scaling behavior and scale invariance conditions of the hydrodynamic processes by applying the one-parameter Lie scaling transformation to the conservation laws in the Reynolds transport theorem framework.

Quasi-solution approach for a two dimensional nonlinear inverse diffusion problem

This paper is related to a class of inverse problems for identification of the coefficient in a square porous medium. The unknown coefficient depends on the solution and belongs the class of admissible coefficients. Using continuous dependence of solutions on the coefficient convergence in the corresponding direct problems, the existence of the quasi-solution of the inverse problem is obtained in the appropriate class of admissible coefficients.

Structural stability for the Morris-Lecar neuron model

This paper is concerned with the biological neuron model of Morris-Lecar system of equations (Morris and Lecar, 1981) 17. We prove the existence and uniqueness of strong solutions. In addition, we prove the continuous dependence of solutions to the leak conductance parameter.