Volume 17, Issue 5

We consider numerical methods for solving inverse problems for time fractional diffusion equation (TFDE) with the variable generalized diffusion coefficient q ( x ). Inverse problems are to find q ( x ) and the order α of the time derivative according to an additional information about a solution of TFDE. The weighted difference scheme is constructed via integro-interpolation method and the generalized factorization method is developed; results of stability analysis of the difference scheme are presented and properties of numerical solutions of forward problems are investigated. Inverse problems for TFDE with the variable coefficient are formulated as residual function minimization problems and properties of corresponding residual functions are discussed. The Levenberg–Marquardt algorithm for residual function minimization is used and numerical results are presented.

We consider the multidimensional nonrelativistic Newton equation in a static electromagnetic field where V ∈ C 2 (, ℝ), B ( x ) is the n × n real antisymmetric matrix with elements B i,k ( x ), B i,k ∈ C 1 (, ℝ) (and B satisfies the closure condition), and ≤ β | j 1 | (1 + | x |) –( α + | j 1 |) for x ∈ , 1 ≤ | j 1 | ≤ 2, 0 ≤ | j 2 | ≤ 1, | j 2 | = | j 1 | – 1, i, k = 1, . . . , n and some α > 1. We give estimates and asymptotics for scattering solutions and scattering data for the equation (∗) for the case of small angle scattering. We show that at high energies the velocity valued component of the scattering operator uniquely determines the X-ray transforms P ∇ V and PB i,k (on sufficiently rich sets of straight lines). Applying results on inversion of the X-ray transform P we obtain that for n ≥ 2 the velocity valued component of the scattering operator at high energies uniquely determines (∇ V, B ). We also consider the problem of recovering (∇ V, B ) from our high energies asymptotics found for the configuration valued component of the scattering operator. Results of the present work were obtained by developing the inverse scattering approach of Novikov [Ark. Mat. 37: 141–169, 1999] for (∗) with B ≡ 0 and of Jollivet [J. Math. Phys. 47: 062902, 2006] for the relativistic version of (∗). We emphasize that there is an interesting difference in asymptotics for scattering solutions and scattering data for (∗) on the one hand and for its relativistic version on the other.

We are concerned with a degenerate in time first-order identification problem related to a closed operator in a Banach space. The degeneracy with respect to time — due to scalar time coefficients — is assumed to be integrable . For both direct and inverse problem we exhibit explicit representations of the solutions in terms of the linear operator A and function ƒ (cf. equation (1.1)), when the latter possess specific properties. Some applications to partial differential equations are given.

In this paper, we introduce a method for recovering implied volatility from European option price. Under some assumptions, an integral equation can be obtained from Dupire equation, and we show the local uniqueness and stability of implied volatility. In the end of this paper, we give some numerical examples to show that the new treatment is effective.

The aerodynamic optimization problem of turbomachinery blades is addressed by means of an inverse problem and its adjoint equations. The blade geometry is simplified by considering it a thickless flow surface. The shape of the blade leading edge and the load distribution over the blades are the design variables. The flow is modeled by the compressible Euler equations in an axisymmetric framework. A set of discrete adjoint equations are employed to determine the sensitivities of the blade aerodynamic characteristics with respect to the above mentioned design parameters. Finally, the sensitivities are fed to a gradient based algorithm to optimize performance.