Integro-differential Problem Research Articles

On the Fourier cosine-Kontorovich-Lebedev generalized convolution transforms

We deal with several classes of integral transformations of the form $$f(x) \to D\int_{\mathbb{R}_ + ^2 } {\frac{1} {u}} \left( {e^{ - u\cosh (x + v)} + e^{ - u\cosh (x - v)} } \right)h(u)f(v)dudv,$$ , where D is an operator. In case D is the identity operator, we obtain several operator properties on Lp(ℝ+) with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on L2(ℝ+) and define the inversion formula. Further, for an other class of differential operators of finite order, we apply these transformations to solve a class of integro-differential problems of generalized convolution type.

A STRONGLY ILL-POSED INTEGRO-DIFFERENTIAL PARABOLIC PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS

Via Carleman estimates we prove uniqueness and continuous dependence results for an identification and strongly ill-posed linear integro-differential parabolic problem with the Dirichlet boundary condition, but with no initial condition. The additional information consists in a boundary linear integral condition involving the normal derivative of the temperature on the whole of the lateral boundary of the space-time domain.

Double zero bifurcation of non-linear viscoelastic beams under conservative and non-conservative loads

The combined effects of conservative and non-conservative loads on the mechanical behavior of an unshearable and inextensional visco-elastic beam, close to bifurcation, are investigated. The equations of motion and boundary conditions are derived via a constrained variational principle, and the Lagrange multiplier successively condensed, to get integro-differential equations. These latter, with the mechanical boundary conditions appended, are put in an operator-form, amenable to perturbation analysis. A linear stability analysis is carried out in the space of the two loading parameter, displaying the existence of codimension-1 and codimension-2 bifurcations. The influence of both internal and external damping on this scenario is thoroughly investigated. A post-critical analysis is carried out around a double-zero bifurcation, by using an adapted version of the multiple scale method, based on fractional series expansions in the perturbation parameter. The integro-differential problem is directly attacked, so that any a priori discretization is avoided. Emphasis is given to the interaction between the two damping coefficients. This reveals the existence, also in the non-linear range, of a phenomenon of destabilization, so far known only in the linear range.

Laplace transform – Finite element method for non Fickian flows

In this paper we consider numerical methods for integro-differential problems based on time discretization via Laplace transformation. We focus our attention in models arising in the context of non Fickian solute transport phenomena in porous media. The mathematical models which describe the evolution of the solute concentrations are characterized by Volterra equations. We present and analyze an hybrid method which combines the Laplace transformation with respect to the time variable with the finite element discretization in the spatial variables. Numerical results illustrate the performance of the method.

Severely ill-posed linear parabolic integro-differential problems

Abstract. Via Carleman estimates we prove uniqueness and continuous dependence results for the temporal traces of solutions to three overdetermined linear parabolic ill-posed problems with no initial condition, the first being integro-differential, the latter two being with deviating arguments. The overdetermination is prescribed in an open subset of the (space-time) lateral boundary.

A Finite Difference Scheme for Pricing American Put Options under Kou's Jump-Diffusion Model

We present a stable finite difference scheme on a piecewise uniform mesh along with a penalty method for pricing American put options under Kou's jump-diffusion model. By adding a penalty term, the partial integrodifferential complementarity problem arising from pricing American put options under Kou's jump-diffusion model is transformed into a nonlinear parabolic integro-differential equation. Then a finite difference scheme is proposed to solve the penalized integrodifferential equation, which combines a central difference scheme on a piecewise uniform mesh with respect to the spatial variable with an implicit-explicit time stepping technique. This leads to the solution of problems with a tridiagonal M-matrix. It is proved that the difference scheme satisfies the early exercise constraint. Furthermore, it is proved that the scheme is oscillation-free and is second-order convergent with respect to the spatial variable. The numerical results support the theoretical results.

Nonlocal Integrodifferential Boundary Value Problem for Nonlinear Fractional Differential Equations on an Unbounded Domain

This paper investigates the existence of nonnegative solutions for nonlinear fractional differential equations with nonlocal fractional integrodifferential boundary conditions on an unbounded domain by means of Leray-Schauder nonlinear alternative theorem. An example is discussed for the illustration of the main work.

Adaptive Semidiscrete Finite Element Methods for Semilinear Parabolic Integrodifferential Optimal Control Problem with Control Constraint

The aim of this work is to study the semidiscrete finite element discretization for a class of semilinear parabolic integrodifferential optimal control problems. We derive a posteriori error estimates inL2(J;L2(Ω))-norm andL2(J;H1(Ω))-norm for both the control and coupled state approximations. Such estimates can be used to construct reliable adaptive finite element approximation for semilinear parabolic integrodifferential optimal control problem. Furthermore, we introduce an adaptive algorithm to guide the mesh refinement. Finally, a numerical example is given to demonstrate the theoretical results.

Numerical Solution of Nonlinear Fredholm Integro-Differential Equations Using Spectral Homotopy Analysis Method

Spectral homotopy analysis method (SHAM) as a modification of homotopy analysis method (HAM) is applied to obtain solution of high-order nonlinear Fredholm integro-differential problems. The existence and uniqueness of the solution and convergence of the proposed method are proved. Some examples are given to approve the efficiency and the accuracy of the proposed method. The SHAM results show that the proposed approach is quite reasonable when compared to homotopy analysis method, Lagrange interpolation solutions, and exact solutions.

An analytical study of the self-induced inviscid dynamics of two-dimensional uniform vortices

The self-induced dynamics of a uniform planar vortex in an isochoric, inviscid fluid is analytically investigated, by writing a new nonlinear integrodifferential problem which describes the time evolution of the Schwarz function of its boundary. In order to overcome the difficulties related to the nonlinear nature of the problem, an approximate solution of the above problem is proposed in terms of a hierarchy of linear integral equations. In particular, the solution at the first order is here investigated and applied to a class of uniform vortices, the kinematics of which is well known. A rich mathematical phenomenology has been found behind the approximate dynamics. In particular, the motion of certain branch points and the consequent changes in the algebraic structure of the Schwarz function appear to be its key features. The approximate solutions are finally compared with the contour dynamics simulations of the vortex motion. The agreement is satisfactory up to times of the order of a quarter of the eddy turnover time, while it becomes more and more qualitative at later times. Eventually, the physical meaning of the approximate solutions is lost, the corresponding vortex boundaries becoming non-simple curves.

Finite Maturity Optimal Stopping of Levy Processes with Running Cost, Stopping Cost and Terminal Gain

Motivated by problems in mathematical finance and insurance, this paper discusses optimal stopping problem in general setting. It considers discounted running cost and stopping cost in addition to terminal gain in the objective function, subject to be optimized over finite-time period. The underlying source of uncertainty is modeled by Levy processes. We derive early exercise premium representation for the value function based on a partial integro-differential free-boundary problem associated with the optimal stopping problem. The representation gives rise to a nonlinear integral equation for the optimal stopping boundary. The integral equation generalizes that of found in Kim (1990), Myneni (1992), Carr et al. (1990), Jacka (1991), Pham (1997), and Peskir (2005). The boundary can be characterized as a unique solution of the integral equation within the class of continuous decreasing function of time to maturity. We show that the continuity of the boundary holds when the stopping cost function is either time-independent or decreasing in time. Uniqueness of such solution holds when the running cost and stopping cost functions satisfy a differential inequality. By reformulating the free-boundary problem as a linear complementarity, the problem is solved iteratively by adapting the implicit-explicit method of Cont and Voltchkova (2005) and the Brennan-Schwartz (1977) algorithm that was implemented in Jaillet et al. (1990) and Almendral (2005) for the pricing of American put option. We give an example in optimal capital structure. We also verify numerically the recent results in Kyprianou and Surya (2007) that the smooth pasting condition may not hold for general Levy processes.

Approximate controllability for an integro-differential control problem

We study a controlled integro-differential parabolic problem with the integral part being given by a generalized convolution and the control acting on a subdomain. We prove the approximate controllability and the basic tool is a unique continuation result at initial time, for parabolic equations which, in turn, relies on fine Carleman estimates.

Identification problems in Banach spaces for linear first-order partial differential equations in one space dimension and applications

Abstract. We prove existence and uniqueness results for mild solutions to direct and inverse Cauchy problems related to linear first-order partial differential equations in Banach spaces. Some applications are given to ultraparabolic differential and integrodifferential problems.

Solutions to Integro-differential Problems Arising on Pricing Options in a Lévy Market

We study an integro-differential parabolic problem arising in Financial Mathematics. Under suitable conditions, we prove the existence of solutions for a multi-asset case in a general domain using the method of upper and lower solutions and a diagonal argument. We also model the jump in the related integro differential equation and give a solution procedure for that model assuming that the brownian motions are not correlated. For a bounded domain, this model for the jump gives an elegant expression of the solution in terms of hyper-spherical harmonics.

Optimal Control for Stochastic Volterra Equations with Completely Monotone Kernels

In this paper, we study a class of optimal control problems for stochastic Volterra equations in infinite dimensions. We are concerned with a class of stochastic Volterra integro-differential problem with completely monotone kernels, where we assume that the noise enters the system when we introduce a control. We provide a semigroup setting for the problem, by the state space setting; the applications to optimal control provide other interesting results and require a precise description of the properties of the generated semigroup. In our stochastic optimal control problems, the drift term of the equation has a linear growth in the control variable, the cost functional has a quadratic growth, and the control process belongs to the class of square integrable, adapted processes with no bound assumed on it. Our main results are the existence for the optimal feedback control, the identification of the optimal cost with the value $Y_0$ of the maximal solution $(Y,Z)$ of the backward stochastic differential equation, the existence of a weak solution to the so-called closed loop equation and, finally, the construction of an optimal feedback in terms of the process $Z$.

On some anisotropic, nonlocal, parabolic singular perturbations problems

This article is devoted to the study of the anisotropic singular perturbations theory for some quasilinear parabolic problems. Describing the asymptotic behaviour of the solutions of nonlocal problems yields to show the existence of solution to some integro-differential problems. The closeness between the solutions of these two kinds of problems is established.

Numerical solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Levy market

We study the numerical solutions for an integro-differential parabolic problem modeling a process with jumps and stochastic volatility in financial mathematics. We present two general algorithms to calculate numerical solutions. The algorithms are implemented in PDE2D, a general-purpose, partial differential equation solver.

An approximate solution for the static beam problem and nonlinear integro-differential equations

We consider a Kirchhoff type nonlinear static beam and an integro-differential convolution type problem, and investigate the effectiveness of the Optimal Homotopy Asymptotic Method (OHAM), in solving nonlinear integro-differential equations. We compare our solutions via the OHAM, with bench mark solutions obtained via a finite element method, to show the accuracy and effectiveness of the OHAM in each of these problems. We show that our solutions are accurate and the OHAM is a stable accurate method for the problems considered.

Dynamic behaviour of an axially moving plate undergoing small cylindrical deformation submerged in axially flowing ideal fluid

The out-of-plane dynamic response of a moving plate, travelling between two rollers at a constant velocity, is studied, taking into account the mutual interaction between the vibrating plate and the surrounding, axially flowing ideal fluid. Transverse displacement of the plate (assumed cylindrical) is described by an integro-differential equation that includes a local inertia term, Coriolis and centrifugal forces, the aerodynamic reaction of the external medium, the vertical projection of membrane tension, the bending resistance, and external perturbation forces. In the two-dimensional model thus set up, the aerodynamic reaction is found analytically as a functional of the cylindrical displacement, using the techniques of complex analysis. The resulting integro-differential problem is discretized in space with the Fourier–Galerkin method, and integrated in time with the diagonalization method. Examples are computed with physical parameters corresponding to air and some paper materials. The effects of the surrounding fluid on the critical velocity and first natural frequency are investigated, for stationary air, for an air mass moving with the plate, and for some arbitrary axial fluid velocities. The obtained results are applicable for both an ideal membrane and a plate with nonzero bending rigidity.

Singular integrodifferential inequalities and application to fractional differential problems

In this paper we are concerned with an integrodifferential problem which arises for instance when we study a Cauchy-type fractional differential equation. This problem involves a convolution of a kernel with a nonlinear function of the solution together with its derivatives up to order two. For ordinary (third order) differential equations the kernel is regular while in our case it is singular and nonintegrable. Combining a desingularization technique due to the second author with some other estimations, we find bounds for solutions of the problem with different nonlinearities. Our results are illustrated by an application to fractional differential equations. Mathematics subject classification (2010): 26D15, 26A51, 32F99, 41A17.