Homogeneous Partial Differential Equations Research Articles

Free vibrations of variablesection beams taking rotational and frictional forces into account

Introduction. Beams are widely used in the construction industry as load-bearing structures of bridges, overpasses, coverings, slabs, stairs, equipment platforms, etc. In order to fully utilize the bearing capacity of such structures and to reduce the rate of material consumption, beams of variable cross-section along the length can be used. During operation, such structural elements are subjected to various types of vibrations, which determines the relevance of studying various aspects of vibrational motion.Aim. To apply numerical methods for studying the influence of rotational inertial forces in the presence of viscous frictional forces on free vibrations of variable-section beams. This area of research presents interest, since the calculations are directly related to the determination of frequencies and forms of natural vibrations of structures.Materials and methods. Free vibrations were described by a homogeneous partial differential equation of hyperbolic type. The methods of variable separation and finite differences were used. A discrete domain in the form of a set of uniform grid nodes and a homogeneous system of algebraic equations were introduced. The system of equations in the matrix-vector form was used.Results. The spectra of natural frequencies, damping coefficients, and eigenforms of beam vibrations are determined. It is shown that the coefficient matrix has a banded and pentadiagonal form. The matrix elements are functions of the characteristic index. The damping coefficient and the frequency of free vibrations are determined from a system of two nonlinear equations. The solution of the system of equations is found using the method of coordinate descent. An example of calculation of a welded I-beam is considered. Five elements of the spectra of damping coefficients and natural frequencies are calculated.Conclusions. State-of-the-art MATLAB solvers allows numerical and graphical methods to be combined. In the solved examples, the advantages of these methods were successfully applied to determine the eigenvalues of matrices and eigenfunctions. The high validity and accuracy of the results obtained confirm the simplicity and versatility of the methodology for determining the characteristics of free vibrations of beams of variable cross-section.

On a boundary problem for the fourth order equation with the third derivative with respect to time

In this paper, we consider a boundary value problem in a rectangular domain for a fourth-order homogeneous partial differential equation containing the third derivative with respect to time. The uniqueness of the solution of the stated problem is proved by the method of energy integrals. Using the method of separation of variables, the solution of the considered problem is sought as a multiplication of two functions X (x) and Y (y). To determine X (x),we obtain a fourth-order ordinary differential equation with four boundary conditions at the segment boundary [0,p], and for a Y (y) – third-order ordinary differential equation with three boundary conditions at the boundary of the segment [0,q]. Imposing conditions on the given functions, we prove the existence theorem for a regular solution of the problem. The solution of the problem is constructed in the form of an infinite series, and the possibility of term-by-term differentiation of the series with respect to all variables is substantiated. When substantiating the uniform convergence, it is shown that the “small denominator” is different from zero.

Stability of cylindrical anisotropic composite shells under torsion in a three-dimensional formulation

The article presents a calculation of the stability of non-thin cylindrical anisotropic layered shells under the action of end torsional moments in a spatial formulation. The anisotropy of the used material is characterized by one plane of elastic symmetry of characteristics. This is caused by the mismatch between the main elastic directions of the composite fibrous orthotropic material and the axes of the curvilinear cylindrical coordinate system.
 A three-dimensional inhomogeneous system of partial differential equations describing the subcritical stress-strain state within the linear theory of elasticity is derived using the Hu-Washizu variational principle. Reducing the dimension of the problem under consideration from three-dimensional to one-dimensional is carried out by taking into account the axial symmetry of the deformation of the cylindrical shell and using the method of straight lines along the generatrix.
 Based on the modified Hu-Washizu variational principle, a three-dimensional system of homogeneous partial differential stability equations is derived within the framework of the spatial theory of elasticity. The reduction of a three-dimensional system to a one-dimensional one is carried out along the generatrix and in the circular direction - by expanding the components of stresses and displacements into double trigonometric series when applying the procedure of the Bubnov-Galorkin method, as well as taking into account the periodicity of the resolving functions.
 An algorithm has been developed, implemented in the form of application software packages for PCs. In it, in a single computational process using the numerical method of discrete orthogonalization in the direction normal to the middle surface of the shell, the establishment of the parameters of the subcritical stress-strain state and the solution on this basis of stability problems for non-thin anisotropic cylindrical shells under the influence of torsion are combined.
 The problem of the influence on the stability of an anisotropic cylindrical non-thin shell of an increase in the number of cross-reinforced layers depending on the angle of rotation of the main directions of elasticity of the material and the direction of application of torque is considered. The obtained results of stability calculations according to the proposed approach were compared with critical torsion loads calculated using an orthotropic model for calculating anisotropic shells. It is shown that for single-layer cylindrical shells the difference between the compared results reaches 69%. An increase in the number of cross-reinforced layers leads to a decrease in this discrepancy, and with seven to eight layers, the difference between the critical loads obtained using the described approach and the orthotropic model is within 5%. This result is consistent with those obtained using classical or refined theories of calculations of both thin and non-thin anisotropic cylindrical shells.

Exact solutions to the Navier–Stokes equations for describing inhomogeneous isobaric vertical vortex fluid flows in regions with permeable boundaries

A family of exact solutions to the Navier–Stokes equations is constructed to describe nonuniform two-dimensional fluid motions. The superposition of the main unidirectional flow with the secondary flow is considered. The secondary flow is determined by suction or injection through permeable boundaries. This class of exact solutions is obtained by multiplicative and additive separation of variables. The flow of a viscous incompressible fluid is described by a polynomial of the horizontal (longitudinal) coordinate. The polynomial coefficients are functions of the vertical (transverse) coordinate and time. They are determined by a chain of homogeneous and inhomogeneous parabolic partial differential equations with a convective term. In the case of a steady flow, it is described by a system of ordinary differential equations with constant coefficients. An algorithm for integrating a system of ordinary differential equations for studying the steady motion of a viscous fluid is presented. In this case, all the functions defining the velocity are quasipolynomials since the system of ordinary differential equations has an Euler-form exact solution.

Direct connection between Navier and spherical harmonic kernels in elasticity

<abstract> <p>Linear isotropic elasticity is an interesting branch of continuum mechanics, described by the fundamental laws of Hooke and Newton, which are combined in order to construct the governing generalized Navier equation of the displacement within any material. Implying time-independence and in the absence of external body forces, the latter is reduced to the corresponding form of a homogeneous second-order partial differential equation, whose solution is given via the Papkovich differential representation, which expresses the displacement field in terms of harmonic functions. On the other hand, spherical geometry provides the most widely used framework in real-life applications, concerning interior and exterior problems in elasticity. The present work aims to provide a little progress, by producing ready-to-use basic functions for linear isotropic elasticity in spherical coordinates. Hence, we calculate the Papkovich eigensolutions, generated by the spherical harmonic eigenfunctions, obtaining connections between Navier and spherical harmonic kernels. A set of useful results are provided at the end of the paper in the form of examples, regarding the evaluation of displacement field inside and outside a sphere.</p> </abstract>

AN INHOMOGENEOUS COMPLETE PDE SOLUTION AND THE IMPLEMENTATION OF BOUNDARY CONDITIONS THROUGH RANDOM WALKS

In a Partial Differential Equation (PDE), the inhomogeneous term describes different physical systems that contain sources or sinks such as: charge, matter or energy, while the term that does not have a derivative (zero order term) is related to diverse physical processes such as: Newton cooling, Lambert absorption or radioactive disintegration among others. There are stochastic methods such as random walks to solve equations, for example, Suxo (2011) formulated a theorem applied exclusively to homogeneous PDEs, without taking into account the zero order term. Therefore, in order to extend the study to several physical phenomena, in this work we reformulate that theorem taking into account the zero order term within an inhomogeneous PDE. In addition, we implement the study of the Dirichlet, Neumann and Mixed boundary conditions. Finally, we verify the effectiveness of the study by comparing the results obtained with analytical solutions of the Poisson, Fick and Fourier equations.

An integral equation method for the advection-diffusion equation on time-dependent domains in the plane

Boundary integral methods are attractive for solving homogeneous elliptic partial differential equations on complex geometries, since they can offer accurate solutions with a computational cost that is linear or close to linear in the number of discretization points on the boundary of the domain. However, these numerical methods are not straightforward to apply to time-dependent equations, which often arise in science and engineering. We address this problem with an integral equation-based solver for the advection-diffusion equation on moving and deforming geometries in two space dimensions. In this method, an adaptive high-order accurate time-stepping scheme based on semi-implicit spectral deferred correction is applied. One time-step then involves solving a sequence of non-homogeneous modified Helmholtz equations, a method known as elliptic marching. Our solution methodology utilizes several recently developed methods, including special purpose quadrature, a function extension technique and a spectral Ewald method for the modified Helmholtz kernel. Special care is also taken to handle the time-dependent geometries. The numerical method is tested through several numerical examples to demonstrate robustness, flexibility and accuracy.

Effect of the Arbitrary Coefficients on the convergence of numerical solution of General Second Order Linear Homogeneous Partial Differential Equation

Effect of the Arbitrary Coefficients on the convergence of numerical solution of General Second Order Linear Homogeneous Partial Differential Equation

Differential-Symbol Method of Solving the Dirichlet Problem in an Infinite Strip for a System of Differential Equations with Shifts

The class of quasipolynomials has been singled out as a class of existence and uniqueness of the solution of the Dirichlet problem in the infinite strip {(t, x) : t ∈ (0, τ), x ∈ ℝ}. The task consists in finding a solution of the system of two homogeneous partial differential equations second order in time variable and with shifts in the spatial variable. The solution satisfies inhomogeneous Dirichlet conditions at the strip boundary. A differential-symbol method of solution construction has been proposed. Examples of the method implementation have been presented.

Improved Analytical Solution for Air–Boosted Vacuum Consolidation of Saturated Soil Using Eigenfunction Expansion Method

Soft foundation consolidation engineering applications and experimental studies have proven the effectiveness of the air–boosted vacuum preloading method (AVP). The sand well consolidation theory, a typical axisymmetric consolidation, is adapted to analyze the whole consolidation process of the saturated soil by air–boosted vacuum preloading. The present analytical solution for air–boosted vacuum consolidation of the saturated soil is more suitable for application in a deep soft foundation. With the solving method, the general solution through separation of variables is used, which is mathematically used for the homogeneous partial differential equations. However, the partial differential equations for solving the consolidation of the AVP method are nonhomogeneous. Therefore, the eigenfunction expansion method for nonhomogeneous equations is proposed and validated in this study. Results showed that the improved analytical solution by the eigenfunction expansion method is more consistent with the numerical solution than that of the previous method using the general solution by separation of variables, which leads to a lower error ratio of less than 2%. The improved analytical solution can be used to predict the consolidation of deep foundations by air–boosted vacuum preloading effectively.

Investigation of Exact Solutions of Nonlinear Evolution Equations Using Unified Method

In this article, an analytical technique based on unified method is applied to investigate the exact solutions of non-linear homogeneous evolution partial differential equations. These partial differential equations are reduced to ordinary differential equations using different traveling wave transformations, and exact solutions in rational and polynomial forms are obtained. The obtained solutions are presented in the form of 2D and 3D graphics to study the behavior of the analytical solution by setting out the values of suitable parameters. The acquired results reveal that the unified method is a suitable approach for handling non-linear homogeneous evolution equations.

Skills of Music Creation Based on Homogeneous First-Order Linear Partial Differential Equations

Abstract The research of differential equation models in music creation and audio is a current research hotspot. This paper uses homogeneous first-order linear partial differential equations to analyze the relevant music signals in music creation. This paper focuses on analyzing the expression of music signals based on time series. This paper discusses its characteristic function and conducts partial differential equation modeling of music characteristic signals. The method successfully established the string vibration model and the time-series autoregressive model of the music signal. This paper becomes a new differential equation to solve the music signal processing method in music creation.

A Trefftz based meshfree local method for bending analysis of arbitrarily shaped laminated composite and isotropic plates

A simple meshfree method is developed in the context of Trefftz approaches to solve bending problem of arbitrarily shaped laminated composite and isotropic plates. Widely used deformation theories including classical plate theory, first-order and third-order shear deformation theory are adapted to the method, so the solution range contains thin to thick plates without any restrictions on geometry, genus, layer sequences or boundary conditions. The solution domain is discretized by a set of nodes at which the degrees of freedom are defined. A sub-domain (cloud) centered at each node and containing several adjacent nodes is considered, over which the corresponding Partial Differential Equation (PDE) is split into homogeneous and particular parts. The solution of the homogeneous PDE is formed using a series of exponential basis functions capable of exactly satisfying the PDE, thus putting the method in the context of Trefftz approaches. The boundary conditions are applied in a point-wise manner without the need for boundary meshes. No integration is enforced throughout the solution procedure. By investigating several sample problems compared the literature or the Finite Element Method (FEM), perfect agreement and high convergence rate have been appeared, while the required nodes in the present method is considerably fewer than FEM.

The localized method of fundamental solutions for 2D and 3D inhomogeneous problems

In this paper, the newly-developed localized method of fundamental solutions (LMFS) is extended to analyze multi-dimensional boundary value problems governed by inhomogeneous partial differential equations (PDEs). The LMFS can acquire highly accurate numerical results for the homogeneous PDEs with an incredible improvement of the computational speed. However, the LMFS cannot be directly used for inhomogeneous PDEs. Traditional two-steps scheme has difficulties in finding the particular solutions, and will lead to a loss of the accuracy and efficiency. In this paper, the recursive composite multiple reciprocity method (RC-MRM) is adopted to re-formulate the inhomogeneous PDEs to higher-order homogeneous PDEs with additional boundary conditions, which can be solved by the LMFS efficiently and accurately. The proposed combination of the RC-MRM and the LMFS can analyze inhomogeneous governing equation directly and avoid troublesome caused by the two-steps schemes. The details of the numerical discretization of the RC-MRM and the LMFS are elaborated. Some numerical examples are provided to demonstrate the accuracy and efficiency of the proposed scheme. Furthermore, some key factors of the LMFS are systematically investigated to show the merits of the proposed meshless scheme.

Monogenic functions with values in algebras of the second rank over the complex field and a generalized biharmonic equation with a triple characteristic

The statement that any two-dimensional algebra 𝔹* of the second rank with unity over the field of complex numbers contains such a basis {e1; e2} that 𝔹*-valued “analytic” functions Φ(xe1 + ye2) (x, y are real variables) satisfy such a fourth-order homogeneous partial differential equation with complex coefficients that its characteristic equation has a triple root is proved. A set of all triples (𝔹*; {e1; e2}; Φ) is described in the explicit form. A particular solution of this fourth-order partial differential equation is found by use of these “analytic” functions.

APLIKASI PERSAMAAN GELOMBANG UNTUK MENENTUKAN KARAKTERISTIK GELOMBANG SENAR GITAR YANG DIPETIK

Partial differential equations are often used to explain physical phenomena, one of which is the wave equation. One application of the wave equation is in plucked strings. This study describes the formation of a wave equation from guitar strings, determines the solution to the wave equation by using the variable separation method and certain boundary conditions and initial conditions, determines the amplitude of the wave, and simulates the movement of the wave based on the initial position of the plucked string. The result obtained is the wave equation of the guitar strings. When the string is plucked, the string will vibrate and produce a wave that can be formulated as a wave equation in the form of a homogeneous second order partial differential equation. The solution to this equation is in the form of a series. If given the initial conditions of plucking in the form of a function, then the amplitude of the wave is obtained. Simulations are given to see the movement of the amplitude and wave on the strings through three cases of the initial position of plucking the strings, namely: less than half, half, and more than half the length of the strings. The behavior of these amplitudes and waves is a characteristic or characteristic of the waves produced from a plucked guitar string

Some coefficient estimates on real kernel 𝛼-harmonic mappings

We call the solution of a kind of second order homogeneous partial differential equation as real kernel α − \alpha - harmonic mappings. For this class of mappings, we explore its Heinz type inequality. Furthermore, for a subclass of real kernel α − \alpha - harmonic mappings with real coefficients, we estimate their coefficients. At last, we study the extremal function of Schwartz type lemma for the class of real kernel α − \alpha - harmonic mappings.

Monogenic functions with values in algebras of the second rank over the complex field and a generalized biharmonic equation with a triple characteristic

The statement that any two-dimensional algebra $\mathbb{B}_{\ast}$ of the second rank with unity over the field of complex numbers contains such a basis $\{e_{1},e_{2}\}$ that $\mathbb{B}_{\ast}$-valued ''analytic'' functions $\Phi(xe_{1}+ye_{2})$ ($x$, $y$ are real variables) satisfy such a fourth-order homogeneous partial differential equation with complex coefficients that its characteristic equation has a triple root is proved. A set of all triples $\left( \mathbb{B}_{\ast},\{e_{1},e_{2}\},\Phi\right)$ is described in the explicit form. A particular solution of this fourth-order partial differential equation is found by use of these ''analytic'' functions.

Numerical Solution with Non-Polynomial Cubic Spline Technique of Order Four Homogeneous Parabolic Partial Differential Equations

An innovative technique of NPCS are being used in engineering, computer sciences and natural sciences field to solve PDEs and ODEs Problems. There are many problems not having exact solution or not much stable and convergent exact solution, to solve such problem one apply different approximation, iterative and many other methods. The developed technique is one of them and implemented on some homogeneous parabolic PDEs of different dimensions and getting results will compare with exact solution and one other existing method, by tabular and graphically as well. Graphs and Mathematical result are found by using MATHEMATICA.
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Numerical Solution of Fourth Order Homogeneous Parabolic Partial Differential Equations (PDEs) using Non-Polynomial Cubic Spline Method (NPCSM)

Non-polynomial cubic spline functions are already being used in the field of engineering, computer sciences, and natural sciences to solve ordinary differential equations (ODEs) and partial differential equations (PDEs). However, many of the above-mentioned problems do not have an exact, stable, or convergent exact solution. There are different approximations and methods that can be applied to solve these problems. This study implemented the purposed method on homogeneous parabolic PDEs having different dimensions. The results obtained were compared with the exact solution and results of other existing methods in tabular and graphical form. Mathematica was used to find the mathematical and graphical results.EMATICA.
 Keywords: Adomian decomposition method (ADM), non-polynomial cubic spline method (NPCSM), continuous approximation, finite difference approximations, fourth order homogeneous parabolic partial differential equations (PDEs)
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