Vol 19, No 4 (2015)

The paper deals with the integral equations of the second kind with a sumdifference kernel. These equations describe a series of physical processes in a medium with a reflective boundary. It has noted some difficulties at applying the methods of harmonic analysis, mechanical quadrature, and other approaches to approximate solution of such equations. The kernel average method is developed for numerical-analytical solution of considered equation in non singular case. The kernel average method has some similarity with known strip method. It was applied for solution of Wiener-Hopf integral equation in earlier work of the author. The kernel average method reduces the initial equation to the linear algebraic system with Toeplitz-plus-Hankel matrix. An estimate for accuracy is obtained in the various functional spaces. In the case of large dimension of the obtained algebraic system the known methods of linear algebra are not efficient. The proposed method for solving this system essentially uses convolution structure of the system. It combines the method of non-linear factorization equations and discrete analogue of the special factorization method developed earlier by the author to the integral equations.

We study the first boundary value problem for the elliptic-hyperbolic type equation with two perpendicular lines of change of type and spectral parameter. We prove the existence and uniqueness of the solution. In the proof of the uniqueness of solution we use the completeness of biorthogonal system in space $L_2$ . When building a solution as the sum of a series there is a problem of small denominators. We obtained estimates of the denominators of the separation from zero.

We study the problem of determining the kernel of the integral term in the one-dimensional integro-differential equation of heat conduction from the known solution of the Cauchy problem for this equation. First, the original problem is replaced by the equivalent problem where an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of the kernel. Next, assuming that there are two solutions $ k_1 (x, t) $ and $ k_2 (x, t), $ integro-differential equations, Cauchy and additional conditions for the difference of solutions of the Cauchy problem corresponding to the functions $ k_1 (x, t), $ $ k_2 (x, t)$ are obtained. Further research is being conducted for the difference $k_1 (x, t) - k_2 (x, t) $ of solutions of the problem and using the techniques of integral equations estimates it is shown that if the unknown kernel $ k (x, t) $ can be represented as $ k_j (x, t) = \sum_ {i = 0} ^ N a_i (x) b_i (t)$, $ j = 1, 2, $ then $ k_1 (x, t ) \equiv k_2 (x, t). $ Thus, the theorem on the uniqueness of the solution of the problem is proved.

It is shown that a simple postulate ``The displacement field of the vacuum is a normalized electric field'', is equivalent to three parametric representation of the displacement field of the vacuum: $$ u(x;t) = P(x) \cos k(x)t + Q(x) \sin k(x)t. $$ Here $t$ is time; $k(x)$ -- frequency vibrations at the point of three-dimensional Euclidean space; $P(x), Q(x)$ -- a pair of stationary orthonormal vector fields; $(k,P, Q)$ -- parameter list of the displacement field. In this case, the normalization factor has dimension $T^{-2}$. The speed of the displacement field $$ v(x;t) = \frac{\partial u(x;t)}{\partial t} = k(x)(Q(x) \cos k(x)t - P(x) \sin k(x)t). $$ The electric field corresponding to this distribution of the displacement field of vacuum, is given by the formula $$ E(x;t) = -\frac{\partial v(x;t)}{\partial t} = k^2(x)u(x;t). $$ Moreover, the magnetic induction $$ B(x;t) = \mathop{\mathrm{rot }} v(x; t). $$ These constructions are used in the determination of local and global solutions of Maxwell's equations describing the dynamics of electromagnetic fields.

It is shown that the second order partial differential equations system defined by author is the most general system. It is possible to get all systems, solutions of which are hypergeometric functions of two variables from a Horn list and biorthogonal systems of Hermite and Appel polynomials. In this case the main apparatus of biorthogonal polynomials of two variables study is special functions of two variables. The resulting system of hypergeometric type allows us to use unified approach for the construction of biorthogonal systems of polynomials. All possible singular curves of the studied system are set. The existence of regular solutions is set by Frobenius-Latysheva method.

We consider the questions of one value solvability of the inverse problem for a nonlinear partial Fredholm type integro-differential equation of the fourth order with degenerate kernel. The method of degenerate kernel is developed for the case of inverse problem for the considering partial Fredholm type integro-differential equation of the fourth order. After denoting the Fredholm type integro-differential equation is reduced to a system of integral equations. By the aid of differentiating the system of integral equations reduced to the system of differential equations. When a certain imposed condition is fulfilled, the system of differential equations is changed to the system of algebraic equations. For the regular values of spectral parameterthe system of algebraic equations is solved by the Kramer metod. Using the given additional condition the nonlinear Volterra type integral equation of second kind with respect to main unknowing function and the nonlinear Volterra special type integral equation of first kind with respect to restore function are obtained. We use the method of successive approximations combined with the method of compressing maps. Further the restore function is defined. This paper developes the theory of Fredholm integro-differential equations with degenerate kernel.

A two-layer mathematical model of a human femur neck reinforced implants of different design for modeling stress-strain state which occurs during a surgical procedure to prevent femur neck fractures by the forced introduction of metallic implants is proposed. Engineered implant designs are provided. Methods and software for geometric modeling of femur embedded with the implants are developed. New boundary value problems to evaluate kinetics in creep conditions of the stress-strain state of reinforced and non-reinforced femoral neck during prolonged static loads corresponding to human foot traffic are formulated. Effective elastic properties of cortical and cancellous bone, power and kinematic boundary value problems. A phenomenological creep model for compact bone tissue is constructed. The technique of identifying the parameters is developed. A check of its adequacy to experimental data is carried out. Based on the finite element method the numerical method for solving the provided boundary value problems at macro level of continuum mechanics is developed. A lot of variative calculations allowed developing recommendations for the rational positioning of the implant in order to minimize stress concentrations. The performed analysis showed that there is a significant relaxation of stresses in the most loaded areas due to creep. Relaxation is more intense in reinforced femoral neck than in the unreinforced. Thus the tension in the most loaded femoral neck area due to creep is reduced by 49 % with respect to the intensity of the initial time of loading for femur which is reinforced by the spoke-spoke-type implant when loading duration is 1 year under natural loads corresponding to human foot traffic. It was found that the time component (long-term fixed load) does not impair the positive effect of reducing the stress concentration due to a femoral neck reinforcement which is a positive fact from the medical practice point of view.