Vol 18, No 1 (2014)

We present integral transforms $\widetilde {\mathcal L}\left\{f(t);x\right\}$ and $\widetilde {\mathcal L}_{\gamma_1,\gamma_2,\gamma} \left\{f(t);x\right\}$, generalizing the classical Laplace transform. The $(\tau, \beta)$-generalized confluent hypergeometric functions are the kernels of these integral transforms. At certain values of the parameters these transforms coincides with the famous classical Laplace transform. The inverse formula for the transforms is given. The convolution theorem for transform $\widetilde {\mathcal L}\left\{f(t);x\right\}$ is proven. Volterra integral equations of the first kind with core containing the generalized confluent hypergeometric function ${\mathstrut}_1\Phi{\mathstrut}_1^{\tau,\beta}(a;c;z)$ are considered. The above equation is solved by the method of integral transforms. The treatment of integral transforms is applied to get the desired solution of the integral equation. The solution is obtained in explicit form.

For a degenerate hyperbolic equation in characteristic region (lune) a boundary-value problem with operators of fractional integro-differentiation is studied. The solution of this equation on the characteristics is related point-to-point to the solution and its derivative on the degeneration line. The uniqueness theorem is proved by the modified Tricomi method with inequality-type constraints on the known functions. Question of the problem solution's existence is reduced to the solvability of a singular integral equation with Cauchy kernel of the normal type.

The solvability of various problems for partial differential equations of the third order is researched in many papers. But, partial Fredholm integro-differential equations of the third order are studied comparatively less. Integro-differential equations have traits in their one-valued solvability. The questions of solvability of linear inverse problems for partial differential equations are studied by many authors. We consider a nonlinear inverse problem, where the restore function appears in the equation nonlinearly and with delay. This equation with respect to the restore function is Fredholm implicit functional integral equation. The one- valued solvability of the nonlinear inverse problem for a partial Fredholm integro-differential equation of the third order is studied. First, the method of degenerate kernel designed for Fredholm integral equations is modified to the case of partial Fredholm integro-differential equations of the third order. The nonlinear Volterra integral equation of the first kind is obtained while solving the nonlinear inverse problem with respect to the restore function. This equation by the special non-classical integral transformation is reduced to a nonlinear Volterra integral equation of the second kind. Since the restore function, which entered into the integrodifferential equation, is nonlinear and has delay time, we need an additional initial value condition with respect to restore function. This initial value condition ensures the uniqueness of solution of a nonlinear Volterra integral equation of the first kind and determines the value of the unknown restore function at the initial set. Further the method of successive approximations is used, combined with the method of contracting mapping.

We have developed a method of probabilistic reliability evaluation of microheterogeneous thick-walled pipe, based on the already received solution of the stochastic creep boundary value problem. The rheological properties of the material were described using random function of one variable (radius $r$). Damage parameter $0< \omega(t) <1$ was introduced here to study the process of degradation of the material during creep stage. Also the power law of the rate of $\omega(t)$ change on the equivalent stress σe , determined by Sdobyrev criterion, is assigned. The reliability evaluation is made by the mean integral value of the equivalent stress. We have found a random time before destruction and its distribution function, which was approximated by lognormal law. The problem of the probability of failure-free operation was calculated for a thick-walled microheterogeneous pipe with the specified parameters. The obtained results allow to evaluate reliability of stochastically inhomogeneous axisymmetric structural elements if necessary statistical data are obtained from the experiment.

The asymptotical solution to the problem of the mixed loading of the cracked specimen under plane stress conditions in materials with fractional-linear constitutive relations of steady -state creep is given. The stresses and creep strain rates in the vicinity of the mixed mode crack tip are obtained. The type of mixed loading is specified by the mixity parameter which is varying from 0 (this type of loading corresponds to pure shear) to 1 (the loading corresponds to tensile loading). The analytical presentation of the stress and the creep strain rate fields is found for all values of the mixity parameter. It is shown that the stress field consists of different regions inside which the stress components are determined by different formulae. The boundaries of the regions are found numerically. The comparison of the analytical solution with the numeric solution obtained for the power-law material for large values of the exponent n is given.

The methods for parametric identification of fractional differential operators with $\alpha \on (1, 2)$ degree according to instantaneous values of experimental observations based on the Barrett differential equation example are suggested. The methods are based on construction of the linear parametrical discrete model for fractional differential equation. The coefficients of the model are associated with the required parameters of differentiation equation of fractional order. Different approaches to the determination of the relationships between the parameters of the differential equation and the discrete model coefficients are considered. Connection expressions for coefficients of linear parametrical discrete model and Cauchy type problem parameters to be identified are obtained. The algorithm of the method which let us reduce the problem to computation of mean-square estimates for coefficients of linear parametrical discrete model is described. Numerical investigations have been done; furthermore, their results let us conclude high efficiency of the methods.

The article presents a study of difference schemes in time, which accuracy can be arbitrarily high. We present difference schemes in time for solving the Navier-Stokes equations, where series expansions are used to find the singularities of solutions of the Euler equations. These methods are generalized in this article for the arbitrary order schemes and for solving the Burgers equation and the Navier-Stokes equations for an incompressible fluid. The impact of the scheme on the calculation accuracy is examined. First, the method is applied to the test case associated with the Burgers equation, and then the problem of three-dimensional incompressible flow finding by solving the Navier-Stokes equations is considered. It is shown that the finite-difference scheme used to calculate the time derivatives is the main source of deviations of the approximate solution from the exact solution.

We formulate a discrete optimal control problem, which has not been considered earlier, which arises in the design of oil and gas networks. For this problem we set four theorems so that you can have a process, the optimal process and the optimum value. Necessary and sufficient conditions we give in Theorem 1. Under these conditions, by Theorem 1, we get not empty attainability intervals. For each interval, we choose the grid-a subset of its points, where by an arbitrary point of interval, we find the nearest point on the left. By means of such approximations, we define the Bellman functions on the grids. Using Bellman functions in Theorem 2 we give the process and we evaluate its deviation from the optimal process. In Theorem 2, we guarantee, that the given process is optimal when the attainability intervals and their grids coincide. In other cases, to get the optimal process, we use Theorem 3 and Theorem 4. In Theorem 3 we set that the process given in Theorem 2, is minimal in the lexicographical order which we introduce using Bellman functions. In Theorem 3 we give procedure that builds, if possible, in this order, the next process, skipping only the processes that are not optimal. We find the optimal process and the optimal value by Theorem 4, starting from the process given in Theorem 2, using one or more calls of the procedure given in Theorem 3.