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It is established that the new boundary element-free Galerkin method, namely the virtual boundary element-free Galerkin method (VBEFGM), using dual compactly supported radial basis function interpolation (CSRBFI) for 2D inhomogeneous heat conduction problems with variable heat sources, whose solutions are divided into homogeneous and particular solutions. The virtual loads of the homogeneous solutions and undetermined coefficients of particular solutions are interpolated by CSRBFI. Its formula is established by the Galerkin method. The homogeneous solutions are obtained by the virtual boundary element-free method (VBEFM). The recommended method for 2D inhomogeneous heat conduction problems is real meshfree without background integration elements and exhibits the merits of the meshfree method, Galerkin method, and boundary element method. Its calculation scheme is deduced in detail and easily programmed. The coefficient matrix has symmetrical characteristics. The implementation steps and implementation flow-chart are also shown in detail. Seven numerical examples, such as exponential functionally graded material, are calculated and contrasted with other numerical methods. The precision and stableness of the proposed method for 2D inhomogeneous heat conduction problems are verified.

In this study, a new approach is presented to determine the stress and displacement field of a shallow buried circular tunnel subject to gravity and arbitrary lateral stress based on the complex variable method. The soil mass and lining are conformally mapped onto two ring regions. A more effective method than the power series method is used to solve the undetermined coefficients of complex potential functions. The interaction mechanism between the lining and the soil mass is fully considered. The boundary conditions along the inner boundary of the lining and the ground surface and the stress and displacement continuity conditions along the excavation line are used in the analytical derivation. In the verification, the stress and displacement distributions obtained in the semi-analytical solution agree with the numerical results and satisfy the boundary and continuity conditions. Using examples, the stress distributions along the inner boundary of the lining and interface are presented to examine the influence of key parameters including the Young's modulus and buried depth. The effect of the lining in reducing stress concentration and restricting tunnel convergence is quantitatively analyzed.

In this research paper, we discuss systems of conformable linear differential equations. The conformable fundamental exponential matrix has been used to express the solution of the homogeneous and nonhomogeneous systems. The method of variation of parameters has been investigated to find the particular solution of the conformable system of nonhomogeneous linear differential equations. Several illustrative examples have been provided at the end of our study to validate all obtained results.

We investigate the interaction between a singular surface and a strong shock in the self-gravitating interstellar gas clouds with the assumption of spherical symmetry. Using the method of the Lie group of transformations, a particular solution of the flow variables and the cooling–heating function for an infinitely strong shock is obtained. This paper explores an application of the singular surface theory in the evolution of an acceleration wave front propagating through an unperturbed medium. We discuss the formation of an acceleration, considering the cases of compression and expansion waves. The influence of the cooling–heating function on a shock formation is explained. The results of a collision between a strong shock and an acceleration wave are discussed using the Lax evolutionary conditions.

The purpose of this study is to formulate a truly and effective boundary-type meshfree method for analysis of 2D transient heat conduction problems including moving laser heat sources. Being based on the method of fundamental solutions (MFS), the present work formulates new time-dependent formulations to solve the problem directly in the time-domain. The motion path and the intensity of the laser heat source over a moving area can be arbitrary functions of time and space. In the proposed formulations, the particular solution associated to the heat source function is evaluated using a simple approach without using any internal points or cells. By solving two example problems, the efficiency and accuracy of the proposed method are demonstrated. The analyses show that the presented boundary-type meshfree method is relatively more efficient than other domain-type techniques. From the analyses, it is also observed that very accurate results can be obtained by the proposed method employing large time-step size and small number of collocation points.

The paper concerns with a new iterative method for solving a monotone variational inclusion problem in a Hilbert space. The method is of the proximal contraction type incorporated with the regularization technique. Under the prediction stepsize conditions, we establish the strong convergence of the iterative sequences generated by the method to a particular solution of the problem satisfying a variational inequality problem. Finally, we give some numerical examples to illustrate the behavior of the new method in comparison with existing ones.

The higher order Pompeiu integral operator providing a particular solution to the inhomogeneous polyanalytic equation provides via the higher order Cauchy–Pompeiu representation an attempt for solv...

Using the method of undetermined coefficients and the theory of Pellian equation, we show that there exist infinitely many isosceles Heron triangles whose sides are certain polynomial values.

In this study, the method of fundamental solutions (MFS), which is a boundary-type meshfree method, is applied for solving two-dimensional stationary anisotropic thermoelastic problems. Because of the semi-analytic nature of the MFS, very accurate solutions can be obtained by this method. The solution of the problem is decomposed into homogeneous and particular solutions. The homogeneous solution is expressed in terms of the fundamental solutions of the anisotropic elastostatic problem. The particular solution corresponds to the effects of the temperature change in the domain of the problem. For cases with a quadratic distribution of the temperature change in the domain, the particular solution is derived in an explicit form. For cases with an arbitrary temperature change distribution, the thermal load is approximated by radial basis functions (RBFs), particular solutions of which are derived analytically. Three numerical examples in simply- and multiply-connected domains under plane stress and plane strain conditions are presented to verify the accuracy of the proposed method. The effects of some parameters, such as the number of source points and the magnitude of the location parameter of source points on the results are investigated. From the numerical results, it is observed that very accurate results can be obtained by the proposed MFS in problems with very complicated temperature change distribution. The numerical convergence tests performed in this study shows that the proposed MFS with a small number of source points can results in solutions that are comparable with the FEM solutions obtained using a large number of nodes.

In the mathematical description of various phenomena and processes that arise in mathematical physics, electrical engineering, economics, one has to deal with matrix differential equations. Therefore, these equations are relevant both for mathematicians and for specialists in other areas of natural science. Many studies are devoted to them, in which the solvability of matrix equations in various function spaces, boundary value problems for matrix differential equations, and other problems were investigated. In this article, a quasilinear matrix equation is considered, the coefficients of which can be represented in the form of absolutely and uniformly converging Fourier series with coefficients and frequency slowly varying in a certain sense. The problem is posed of obtaining sufficient conditions for the existence of particular solutions of a similar structure for the equation under consideration. For this purpose, the corresponding linear equation is considered first. It is written down in component-wise form, and, based on the assumptions made, the existence of the only particular solution of the specified structure is proved. Then, using the method of successive approximations and the principle of contracting mappings, the existence of a unique particular solution of the indicated structure for the original quasilinear equation are proved.

Purpose The purpose of this study is to improve the efficiency and accuracy of response surface method (RSM), as well as its robustness. Design/methodology/approach By introducing cut-high-dimensional representation model (HDMR), the delineation of cross terms and the constitution analysis of component function, a new adaptive RSM is presented for reliability calculation, where a sampling scheme is also proposed to help constructing response surface close to limit-state. Findings The proposed method has a more feasible process of evaluating undetermined coefficients of each component function than traditional RSM, and performs well in terms of balancing the efficiency and accuracy when compared to the traditional second-order polynomial RSM. Moreover, the proposed method is robust on the parameter in a wide range, indicating that it is able to obtain convergent result in a wide feasible domain of sample points. Originality/value This study constructed an adaptive bivariate cut-HDMR by introducing delineation of cross-terms and constitution of univariate component function; and a new sampling technique is proposed.

In this paper we gave a general formula for particular solution of the non-homogenous linear ordinary differential equation in a method of variation of parameters of kth order. We solved the linear...

This paper reports the first attempt to extend a novel localized boundary knot method to the 2D and 3D acoustic problems with complicated geometry. The solution of original partial differential equation with inhomogeneous term is firstly described as a sum of a particular solution and a homogeneous one. Secondly, the Chebyshev interpolation technique is applied to the approximation of particular solution. The interpolation approach adopts the Chebyshev-Gauss-Lobatto nodes to offer the spectral convergence and high accuracy. And then the numeircal solution of homogeneous equation is obtained by applying the localized boundary knot method with non-singular general solutions. As a localized meshless method, the localized boundary knot method in conjunction with the interpolation technique is very simple mathematically, accurate numerically, and particularly feasible for large-scale computation in a complicated geometry. Numerical experiments including 2D and 3D models verify the performance of the proposed method for solving inhomogeneous Helmholtz-type equations.

This study proposed an efficient numerical technique for coupled differential equations (CDEs) using the clique polynomials of the Complete graph. Recently, Graph theory has dragged the attention of many mathematicians due to its wide applications. Here, some problems have been considered to examine the proposed scheme proficiency. Some theorems on convergence are discussed. Here, the CDEs are rehabilitated into an algebraic equation system using the clique polynomials and collocation technique. The proposed scheme results are compared with the undetermined coefficients (UDCM) and variation of parameters method (VPM) solutions through tables and graphs. Obtained results reveal that the current approach is more accurate.

The impact of Joule heating due to electric double layer (EDL) and viscous dissipation on electroosmotic mixed convection flow in a vertical microchannel with asymmetric heat fluxes is established in this article. The Poisson-Boltzmann, momentum and energy equations representing the electric potential, velocity profile and temperature distribution in the microchannel are obtained in dimensionless forms. Using the Debye-Huckel linearization, exact solutions are obtained for electric potential, velocity profile and temperature distributions by method of undetermined coefficients in the absence of viscous dissipation and Joule heating while an inbuilt Matlab function called pdepe is employed to solve the coupled nonlinear momentum and energy equations in the presence of Joule heating and viscous dissipation. Results show that the presence of Joule heating and viscous dissipation lead to decrease in velocity profile and temperature distributions throughout the microchannel.

Different from previous viewpoints, multivariate polynomial matrix Diophantine equations are studied from the perspective of modules in this paper, that is, regarding the columns of matrices as elements in modules. A necessary and sufficient condition of the existence for the solution of equations is derived. Using powerful features and theoretical foundation of Grobner bases for modules, the problem for determining and computing the solution of matrix Diophantine equations can be solved. Meanwhile, the authors make use of the extension on modules for the GVW algorithm that is a signature-based Grobner basis algorithm as a powerful tool for the computation of Grobner basis for module and the representation coefficients problem directly related to the particular solution of equations. As a consequence, a complete algorithm for solving multivariate polynomial matrix Diophantine equations by the Grobner basis method is presented and has been implemented on the computer algebra system Maple.

This study analyzes and verifies the electromagnetic performance and thrust characteristics of linear magnetic gears using the subdomain-based analytical methods. By simplifying the analytical model, the governing equations of each domain are calculated based on the Maxwell equation. In addition, the undetermined coefficients of the governing equations are derived using four boundary conditions. In magnetic gears, where power transmission is important, the force characteristics are also important. The force characteristics were calculated using the undetermined coefficients and the Maxwell stress tensor.

In practical problems, which can be stated in the form of multi objective programming problem, we have usually a large set of efficient solutions. So, which solution from this set should be taken, appears as a natural question here. In the paper we propose some sustainable principles and simple numerical method for such choice. The method respects aspirations and priorities of decision makers and enables iterations for possible improvement of the solution. In this way decision makers can clearly understand why a particular solution is obtained and when and how it can be improved. The features of the method are explained by a practical example. An applications to bilevel programming problems are also presented, where the both side mapping, between the set of efficient solutions and the set of all possible priorities, is shown. It is illustrated in detail through several linear and nonlinear examples.

An analytical model extending classical thermal elasticity is presented. It allows to introduce a correction to the attenuation of the mechanical waves at the higher frequency range. A material data set taken from experimental studies can be used to identify the attenuation rate as a function of frequency. An example is provided. The particular solution of the developed equations system in the form of the traveling monochromatic wave is obtained.

This paper secures exact bright, dark and singular soliton solutions with parabolic–nonlocal combo nonlinearity in polarization-preserving optical fibers. The included perturbation terms are all of Hamiltonian type. Subsequently, an analytical bright 1-soliton solution is recovered, in the presence of higher-order dispersion effects by semi-inverse variational principle.

Graphical abstract, Highlights • The 3D variable-order time fractional variable-order time fractional diffusion is generated. • An efficient meshless method is proposed for numerical solution of the new problem. • The proposed approach is established upon the singular boundary method. • The method accuracy is examined by some numerical examples on various geometries. • The method can be extended for other types of variable-order fractional problems., Introduction This study describes a novel meshless technique for solving one of common problem within cell biology, computer graphics, image processing and fluid flow. The diffusion mechanism has extremely depended on the properties of the structure. Objectives The present paper studies why diffusion processes not following integer-order differential equations, and present novel meshless method for solving. diffusion problem on surface numerically. Methods The variable- order time fractional diffusion equation (VO-TFDE) is developed along with sense of the Caputo derivative for (0

In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the nonlinear partial differential equations. Its essence is in adding to the nonlinear PDE, a much simpler, as a rule ordinary, differential equation, compatible with the given one. Then any solution of the ODE is a particular solution of the PDE as well. However the main problem is to find this compatible ODE. Our generalization is that we look for an ordinary differential equation that is compatible not with the nonlinear partial differential equation itself, but with its linearization. Such a generalized invariant manifold is effectively sought. Moreover, it allows one to construct such important attributes of integrability theory as Lax pairs and recursion operators for integrable nonlinear equations. In this paper, we show that they provide a way to construct particular solutions to the equation as well.

We deal with the problem of cyclic scheduling tasks in the distributed manufacturing system with factories having the flow shop layout. The optimization aim is to minimize the cycle time. We have proposed the proprietary model and introduced some special properties of the problem, which allow us to find the cycle time for each given particular solution quickly. Since the formulated problem is NP-hard, we propose a metaheuristic algorithm based on the tabu search approach. The quality of the algorithm has been verified experimentally on common benchmarks and compared to other methods known in the literature, namely the scatter search and list scheduling. The problem also has a relation with the problem of balancing workloads of identical parallel production lines.

Solutions of 3D thermal analysis in inhomogeneous media with variable thermal source are divided into homogeneous and particular solutions by a novel boundary-type element-free method, namely virtual boundary element-free Galerkin method (VBEFGM). The homogeneous solution can be obtained by the virtual boundary element-free method (VBEFM). The virtual source function of the homogeneous solution and the unknown coefficient of the particular solution are constructed by the radial basis function interpolation (RBFI). And VBEFGM for 3D thermal analysis in inhomogeneous media with variable thermal source is real meshfree and does not require the background integral element. Its equation is obtained the Galerkin method of the weighted residual methods. The demonstration of ANSYS Parametric Design Language (APDL) can greatly facilitate the programming for the extraction of the coordinate information and the outer normal cosines on nodes. The calculation scheme and the specific weighted coefficients are deducted in detail for 3D thermal analysis in inhomogeneous media with variable thermal source by VBEFGM. The numerical calculational steps and flow chart are given. Three numerical examples are computed. The results are compared with other numerical methods and the exact solutions. The stability and the accuracy of the proposed method is proved by VBEFGM for 3D thermal analysis in inhomogeneous media with variable thermal source.

Due to certain difficulties in solving fourth-order partial differential equations (PDEs) using localized methods, the given differential equation is normally split into two decoupled second order PDEs. Such an approach is only feasible for Dirichlet and Laplace boundary conditions. In this paper the localized method of particular solutions is applied to fourth-order PDEs directly using polynomial basis functions. The effectiveness of the proposed algorithms is demonstrated by considering four numerical examples.

In this paper, two localized meshless methods based on polynomial basis functions are utilized to solve axisymmetric problems. In the first approach, we applied the localized method of particular solutions (LMPS) and the closed-form particular solution to simplify the two-stage approach using Chebyshev polynomial as the basis functions for solving axisymmetric problems. We also propose the modified local Pascal polynomial method (MLPM) to compare the results with LMPS. Since only the low order polynomial basis functions are used, no preconditioning treatment is required and the solution is quite stable. Four numerical examples are given to demonstrate the effectiveness of the proposed methods.

The incomplete version of the Macdonald function has various appellations in literature and earns a well-deserved reputation of being a computational challenge. This paper ties together the previously disjoint literature and presents the basic properties of the incomplete Macdonald function, such as recurrence and differential relations, series and asymptotic expansions. This paper also shows that the incomplete Macdonald function, as a simple closed-form expression, is a particular solution to a parabolic partial differential equation, which arises naturally in a wide variety of transient and diffusion-related phenomena. Ministry of Education (MOE) Published version This work was supported by the Singapore Ministry of Education Academic Research Fund Tier 1 (04MNP002133C160).

Підручник призначено для студентів спеціальності 104 «Фізика та астрономія» освітньої програми «Комп’ютерне моделювання фізичних процесів», які вивчають дисципліну «Диференціальні та інтегральні рівняння» англійською мовою. У підручнику розглянуто поняття диференціальних рівнянь першого порядку; теорія диференціальних рівнянь вищих порядків, включаючи лінійні диференціальні рівняння зі сталими коефіцієнтами та їх застосування до дослідження механічних коливань; системи диференціальних рівнянь; основні поняття теорії стійкості та інтегральних рівнянь. Кожна частина містить основні математичні поняття, пояснює нові математичні терміни та надає контрольні запитання та вправи, які дозволять студентам оволодіти навичками розв’язування практичних задач. Найважливіші теоретичні поняття пояснюються та ілюструються рисунками та прикладами, щоб допомогти краще зрозуміти теорію та показати деякі застосування. Підручник можна рекомендувати також студентам інших спеціальностей, які вивчають диференціальні та інтегральні рівняння. Він може бути використаний для очного, дистанційного або змішаного навчання. The textbook is designed for students of the specialty 104 Physics and Astronomy, educational program Computer Modeling of Physical Processes, who study the discipline Differential and Integral Equations in English. The textbook introduces the concepts of the first-order differential equation; the theory of higher-order differential equations, including the linear differential equations with constant coefficients and their application to exploring mechanical vibrations; systems of differential equations; the main concepts of stability theory and integral equations. Each part contains basic mathematical conceptions, explains new mathematical terms and is provided review questions and exercises, which will allow students to master the skills of solving practical problems. The most important concepts of the theory are explained and illustrated by figures and examples to help clarify the theory and show some applications. The textbook can be also recommended to students of other specialties, who study Differential and Integral Equations. The manual can be used to provide full-time, distance or mixed education.

In this paper we investigate the question of the existence of strong solution in finite time for the Korteweg system for small initial data provided that the initial momentum ρ 0 u 0 belongs to bmo −1 T (R N) for T > 0 and the initial density ρ 0 is in L ∞ (R N) with N ≥ 1 and far away from the vacuum. This result extends the so called Koch-Tataru theorem for the Korteweg system. It is also interesting to observe that any initial shock on the density is instantaneously regularized inasmuch as the density becomes Lipschitz for any ρ(t, ·) with t > 0. We also prove the existence of global strong solution for initial data (ρ 0 − 1, ρ 0 u 0) ∈ (B N 2 −1 2,∞ (R N) ∩ B N 2 2,∞ (R N)∩L ∞ (R N))×(B N 2 −1 2,∞ (R N)) N. This result allows in particular to extend the notion of Oseen solution (corresponding to particular solution of the incompressible Navier Stokes system in dimension N = 2) to the Korteweg system provided that the vorticity of the momentum ρ 0 u 0 is a Dirac mass αδ 0 with α sufficiently small. IHowever unlike the Navier Stokes equations the property of self similarity is not conserved for the Korteweg system since there is no invariance by scaling because the term of pressure.

In this paper, asymmetric equations for an elliptical boundary layer in the vicinity of the conditional front of Rayleigh surface waves, which occurs in shells of revolution under shock end impacts of normal type are constructed. The technique of asymptotic derivation of these equations, based on the use of the symbolic Lurie method and the introduction of special coordinates that distinguish a small frontal zone required to reduce the original problem to an equivalent problem for an infinite shell by isolating a particular solution. The considered boundary layer complements the full description of the considered type of stress-strain state (SSS) in all sections of the phase plane. It also uses a quasi-static boundary layer of the Saint-Venant type in a small vicinity of the butt end, a parabolic boundary layer according to the two-dimensional Kirchhoff–Love theory, a quasi-plane short-wave component, and a hyperbolic boundary layer in a small neighborhood of the shear wave front. In conclusion, an example of constructing an elliptical boundary layer under shock action on the butt end of a cylindrical shell is considered.

In the one-dimensional domain it is studied the unique solvability of a class of integro-differential equations with logarithmic singularity in the kernels. The solution of the considering integro-differential equations with logarithmic singularity in the kernels on dependent of roots of corresponding characteristic equations is obtained in an explicit form. The first part of paper deals with the introduction of the Mellin transformation as a holomorphic function in the complex plane. The definition of this integral transform is given and its main properties are described in details. The second part deals with the application of the Mellin transform to the solution of one-dimensional integro-differential equations with the logarithmic singularity in the kernels. Here, in order to obtain particular solution of nonhomogeneous equations, we applied the analogies of Duhamel’s theorem. It is shown that, if the kernel of studied equation has first order logarithmic singularity of polynomial’s form, then to this equation corresponds characteristic (algebraic) equation of the third order. In this case the solution is found by means of degree of logarithmic and trigonometric functions. If the integro-differential equation has $$n$$ -th order logarithmic singularity of polynomial’s form, then to this equation corresponds $$n$$ -th order characteristic equation. At three main cases the solution is found in an explicit form. Also, it is established that the degree of logarithmic singularity acts to number of linearly independent solutions of given classes of integro-differential equations.

In this paper we will present a formulation for solving elastodynamic problems in 2D using the method of fundamental solutions (MFS). The governing equation for the displacement in the elastodynamic problem is expressed by the Navier equation with an additional non-homogeneous inertial term, which involves the second derivative of displacement with respect to time. The inertial term is approximated by the Houbolt finite difference formula. In each time step, the solution for the displacement is separated into a homogeneous solution and a particular solution. The homogeneous solution is constructed in terms of the fundamental solutions of the elastostatic problem. The particular solution corresponds to the non-homogeneous inertial term, which is obtained by approximating the inertial term with radial basis functions (RBFs). In addition, a formulation based on the MFS for free vibration analysis of 2D bodies is presented in this paper. By solving several numerical examples, we have demonstrated that the proposed methods are robust and accurate. The convergence of the results with respect to the number of collocation points, the number of internal points, the value of the RBF parameter, and the time step size is also studied.

This paper develops an accurate computational method based on the shifted Chebyshev cardinal functions (CCFs) for a new class of systems of nonlinear variable-order fractional quadratic integral equations (QIEs). In this way, a new operational matrix (OM) of variable-order fractional integration is obtained for these cardinal functions. In the proposed method, the unknown functions of a system of nonlinear variable-order fractional QIEs are approximated by the shifted CCFs with undetermined coefficients. Then, these approximations are substituted into the system. Next, the OM of variable-order fractional integration and the cardinal property of the shifted CCFs are utilized to reduce the system into an equivalent system of nonlinear algebraic equations. Finally, by solving this algebraic system an approximate solution for the problem is obtained. The main idea behind this approach is to reduce such problems to solving systems of nonlinear algebraic equations, which greatly simplifies the problem. Convergence of the presented method is investigated theoretically and numerically. Furthermore, the proposed approach is numerically evaluated by solving some test problems.

This paper presents a meshless local Petrov–Galerkin method based on the particular solution (MLPG-PS) to solve a time-fractional inverse problem with unknown boundary condition. The particular sol...

The paper considers the problem of finding common solutions of a system of pseudomonotone equilibrium problems and fixed point problems for quasi-nonexpansive mappings. The problem covers various mathematical models of convex feasibility problems and the problems whose constraints are expressed by the intersection of fixed point sets of mappings. The main purpose of the paper is to design and improve computations over each step and weaken several assumptions imposed on bifunctions and mappings. Two parallel algorithms for finding of a particular solution of the problem are proposed in Hilbert spaces where each subproblem in the family can be computed simultaneously. The first one is a modified hybrid method which combines three methods including the generalized gradient-like projection method, the Mann’s iteration and the hybrid (outer approximation) method. This algorithm improves the hybrid extragradient method at each computational step where only one optimization problem is solved for each equilibrium subproblem in the family and the hybrid step does not deal with the feasible set of the considered problem. The strong convergence of the algorithm comes from the hybrid method under the Lipschitz-type condition of bifunctions. The second algorithm is a viscosity-like method with a linesearch procedure that aims to avoid the Lipschitz-type condition imposed on bifunctions. With the incorporated viscosity technique, the algorithm also provides strong convergence. Several numerical experiments are performed to illustrate the efficiency of the proposed algorithms and also to compare them with known parallel hybrid extragradient methods.

A phase enlargement of semi-Markov systems that does not require determining stationary distribution of the embedded Markov chain is considered. Phase enlargement is an equivalent replacement of a semi-Markov system with a common phase state space by a system with a discrete state space. Finding the stationary distribution of an embedded Markov chain for a system with a continuous phase state space is one of the most time-consuming and not always solvable stage, since in some cases it leads to a solution of integral equations with kernels containing sum and difference of variables. For such equations there is only a particular solution and there are no general solutions to date. For this purpose a lemma on a type of a distribution function of the difference of two random variables, provided that the first variable is greater than the subtracted variable, is used. It is shown that the type of the distribution function of difference of two random variables under the indicated condition depends on one constant, which is determined by a numerical method of solving the equation presented in the lemma. Based on the lemma, a theorem on the difference of a random variable and a complicated recovery flow is built up. The use of this method is demonstrated by the example of modeling a technical system consisting of two series-connected process cells, provided that both cells cannot fail simultaneously. The distribution functions of the system residence times in enlarged states, as well as in a subset of working and non-working states, are determined. The simulation results are compared by the considered and classical method proposed by V. Korolyuk, showed the complete coincidence of the sought quantities.

The determination of a particular solution for the systems of linear differential equations with constant coefficients that have on the column of free terms functions such as e a · x ·P(x) ·cos n x or e b · x ·Q(x)·sin n x, or/and e b · x ·Q(x) sin n x, n ∈ N, is based on the expansion of cos n x and sin n x, and, on the other hand, on how a particular solution for a column of free terms with functions such as e a · x ·P(x) ·cos n x or e b · x ·Q(x)·sin n x, n ∈ N, looks like. We can also write the way a particular solution looks like when we have a combination of two or more functions on the form e a · x ·P(x) ·cos n x or e b · x ·Q(x)·sin n x, n ∈ N in the column of free terms.

A particular solution for the n-th order linear differential equations with constant coefficients that are free of term such as P(x)· e a · x ·cos n x or/and Q(x)·e b · x · sin n x, n∈ N, can be determined based on two elements: the way in which cos n x and sin n x can develop, and, on the other hand on the way a particular solution for the free terms P(x) ·e a · x ·cos n x or/and Q(x)·e b · x ·sin m x, n ∈N is sought. We can, of course, write the way a particular solution looks also in the case we have a combination of the two terms or more terms of this kind.

Solitons are structurally stable solitary waves that propagate in a nonlinear medium. In this paper, solitons will be considered as the basis for solving many classical nonlinear equations of motion. Some classical solutions that were modeled through the application of Wolfram Mathematica System and MATLAB programming language. In this paper some soliton solutions will also be compared and some types of solitons were modeled.The dynamics of solitons was studied in consideration of solutions of some equations, such as the Korteweg – de Vries equation and as a particular solution for the nonlinear Schrodinger equation provided that the nonlinearity parameter in the equation. We concluded by showing solitons in more detail which are often used in practice as a simpler methods for explaining complex phenomena and solving non-classical equations.

Purpose Based on univariate dimension-reduction model, this study aims to propose an adaptive anisotropic response surface method (ARSM) and its high-order revision (HARSM) to balance the accuracy and efficiency for response surface method (RSM). Design/methodology/approach First, judgment criteria for the constitution of a univariate function are derived mathematically, together with the practical implementation. Second, by combining separate polynomial approximation of each component function of univariate dimension-reduction model with its constitution analysis, the anisotropic ARSM is proposed. Third, the high-order revision for component functions is introduced to improve the accuracy of ARSM, namely, HARSM, in which the revision is also anisotropic. Finally, several examples are investigated to verify the accuracy, efficiency and convergence of the proposed methods, and the influence of parameters on the proposed methods is also performed. Findings The criteria for constitution analysis are appropriate and practical. Obtaining the undetermined coefficients for every component functions is easier than the existing RSMs. The existence of special component functions is useful to improve the efficiency of the ARSM. HARSM is helpful for improving accuracy significantly and it is more robust than ARSM and the existing quadratic polynomial RSMs and linear RSM. ARSM and HARSM can achieve appropriate balance between precision and efficiency. Originality/value The constitution of univariate function can be determined adaptively and the nonlinearity of different variables in the response surface can be treated in an anisotropic way.

Mathematically, the complex reactions can be simplified through different linear/non-linear algebraic techniques that are the easiest way to balance the complex models. Furthermore, the complex chemical equations can be balanced with the linear algebraic technique based on matrix-inversion applications. That is the other way for the researchers and chemists to balance the complex chemical equation. In this article, we provide information about coefficients that balance the chemical equations through null-space methodology and the steady-state approximation by applying linear algebra technique. The purpose is not only to identify the undetermined coefficients but also to compute the kernel matrix or null spaces. Furthermore, the steady state for a two-step reaction mechanism is established.

In this study, the singular boundary method is applied to solve time-dependent pseudo-parabolic equations in two space dimensions with initial and Dirichlet-type boundary conditions. A splitting procedure is used to split the solution of the inhomogeneous governing equation into a homogeneous solution and a particular solution. This work presents the numerical operation for calculating the particular solution and homogeneous solution. Several numerical examples are provided to show the accuracy and efficiency of the method. Furthermore, the analysis of stability and convergence is presented.