Complex Differential Equations Research Articles

On the Complex Difference Equation of Hypergeometric Type on Non-uniform Lattices

In this article, we obtain a new fundamental theorems for Nikiforov–Uvarov–Suslov complex difference equation of hypergeometric type by the method of Euler integral transformation, its expression is different from Suslov’s Theorem. We also establish the adjoint equation for Nikiforov–Uvarov–Suslov difference equation of hypergeometric type on non-uniform lattices, and prove it to be a difference equation of hypergeometric type on non-uniform lattices as well. The particular solutions of the adjoint equation are then obtained. As an appliction of these particular solutions, we use them to obtain the particular solutions for the original difference equation of hypergeometric type on non-uniform lattices and other important results.

An Investigation of Solving Third-Order Nonlinear Ordinary Differential Equation in Complex Domain by Generalising Prelle–Singer Method

A method to solve a family of third-order nonlinear ordinary complex differential equations (NLOCDEs) —nonlinear ODEs in the complex plane—by generalizing Prelle–Singer has been developed. The approach that the authors generalized is a procedure of obtaining a solution to a kind of second-order nonlinear ODEs in the real line. Some theoretical work has been illustrated and applied to several examples. Also, an extended technique of generating second and third motion integrals in the complex domain has been introduced, which is conceptually an analog to the motion in the real line. Moreover, the procedures of the method mentioned above have been verified.

Quasi-static Flow Model for Predicting the Extreme Values of Air Pocket Pressure in Draining and Filling Operations in Single Water Installations

Inertial models have been used by researchers to simulate the draining and filling processes in water pipelines, based on the evolution of the main hydraulic and thermodynamic variables. These models use complex differential equations, which are solved using advanced numerical codes. In this study, a quasi-static flow model is developed to study these operations in hydraulic installations. The quasi-static flow model represents a simplified formulation compared with inertial flow models, in which its numerical resolution is easier because only algebraic equations must be addressed. Experimental measurements of air pocket pressure patterns were conducted in a 4.36 m long single pipeline with an internal diameter of 42 mm. Comparisons between measured and computed air pocket pressure oscillations indicate how the quasi-static flow model can predict extreme values of air pocket pressure for experimental runs, demonstrating the possibility of selecting stiffness and pipe classes in actual pipelines using this model. Two case studies were analysed to determine the behaviour of the quasi-static flow model in large water pipelines.

Cracks Interaction in a Pre-Stressed and Pre-Polarized Piezoelectric Material

ABSTRACTWe formulate and solve the mathematical problem for antiplane cracks in a pre-stressed and pre-polarized piezoelectric material with static initial fields, assuming the initially deformed configuration of the body is locally stable. Using the boundary conditions of antiplane cracks, we get the Riemann-Hilbert problems. Nonhomogeneous linear complex differential equations having the unknown complex potential are obtained. For constant value of the applied incremental forces can be obtained the complex potentials, incremental displacement and stress fields corresponding to the third mode of the classical fracture. The problem of interaction of two collinear, unequal cracks in a pre-stressed and pre-polarized piezoelectric material, is also studied.

Linear differential equations with solutions lying in weighted Fock spaces

Sufficient conditions for coefficients of the non-homogeneous linear complex differential equation are found such that all solutions belong to some weighted Fock spaces, where are entire functions, . Furthermore, sufficient conditions for the coefficient such that all solutions of the special second order equation belong to some weighted Fock spaces are given by the Bergman reproducing kernel, where is an entire function.

$\alpha$-Differentiable functions in complex plane

В комплексной плоскости вводится взвешенная дробная производная порядка $\alpha$. Относительно многозначной функции $ z ^ {1- \alpha} $ получены дробные уравнения Коши-Римана, которые при $ \alpha = 1 $ совпадают с классическими уравнениями Коши-Римана. Для некоторых функций в комплексной плоскости рассмотрены свойства, относящиеся к комплексной взвешенной дробной производной. Обсуждаются два комплексных дифференциальных уравнения специальной формы. Для некоторых значений $\alpha$ приводятся римановы поверхности их решений и сравниваются их графики.

On inversion in the case of the fundamentally finite integrable Vekua complex differential equation

The class of so called fundamentally finite integrable Vekua CDE is defined using the fixed point of the inversion and where one solution is equal to the coefficient of the equation. Then the different manifestations of inversion in relation to the general solution, an arbitrary analytical function inside and the core of the coefficient are examined. It shows that all the major problems of the Vekua equation theories, including boundary value problems can be interpreted and solved using the principle of inversion. The main significance of the fundamentally finite integrable Vekua equation is that the real and imaginary part of the solution can be separated, which in many mechanical and technique problems have certain physical meanings.

Solution of complex differential equations by using reduced differential transform method

Solution of complex differential equations by using reduced differential transform method

Математична модель динаміки рухливих об’єктів на основі кватерніонів

One of the main tasks of managing groups of dynamic moving objects is to ensure their consistent movement in space. It is advisable to optimize the movements (motion) in space using mathematical models i.e. systems of equations of motion. The movement of any moving object can be represented as a combination of reciprocating and rotational motion, and the velocity of the object as a combination of reciprocating and rotational speed. Recently, a great deal of attention has been paid in the literature to the description of rotational motion by quaternions. When you need to model the coordinated motion of many objects, models are transformed into complex and cumbersome differential equations that are not always easy to solve. The article developed a mathematical model and showed that quaternions can simulate both rotational and reciprocating motion of an object. The movement itself is represented as a transformation of orthogonal bases, and the displacement model is reduced to a quaternion multiplication operation. The performance of the model is tested on the simulation of the movement of aircraft.

Analytic solutions of the generalized water wave dynamical equations based on time-space symmetric differential operator

It is well known that there is a deep connection between the symmetric and traveling wave solutions. It has been shown that all symmetric waves are traveling waves. In this paper, we establish new analytic solution collections of nonlinear conformable time-fractional water wave dynamical equation in a complex domain. For this purpose, we construct a new definition of a symmetric conformable differential operator (SCDO). The operator has a symmetric representation in the open unit disk. By using SCDO, we generalize a class of water wave dynamical equation type time-space fractional complex Ginzburg–Landau equation. The results show that the obtainable approaches are powerful, dependable and prepared to apply to all classes of complex differential equations.

Determination of the blow up point for complex nonautonomous ODE with cubic nonlinearity

We present a method for determination of the blow up point for a complex nonautonomous ordinary differential equation with a cubic nonlinearity. This equation describes stationary nonlinear modes for two important NLS-based models: (i) the Gross–Pitaevskii equation with complex potential and repulsive nonlinearity and (ii) the Lugiato–Lefever equation in the case of normal dispersion. We derive and justify an asymptotic expansion in the vicinity of the blow up point. This expansion is employed for the construction of a numeric procedure for the computation of the coordinate of the blow up point. We illustrate applications of the proposed procedure by two examples, where exact analytical solutions are available. It is shown that it allows one to find the blow up point with high accuracy. The method may be efficiently used for the search of soliton solutions for vector and scalar NLS-type equations within the strategy of ’filtering out’ of blow up solutions (Alfimov et al., Physica D, 394, 39 (2019)).

A New Geometric Vector Optimization of Predictive Direct Power Control

This article proposes a new vector optimization method suitable for predictive direct power control (P-DPC). Based on the geometric property of parallelograms and diamonds, a simple generic equation is established to form a set of evenly-distributed vectors in the converter voltage space through an iteration process. The optimum vector is selected from the vector set via the detection of the optimum sector and vector sequence number using a simple algebraic function rather than solving complex differential equations common in multivector P-DPC. The application time of the selected vector is optimized to further improve the control performance. It is demonstrated that the proposed method significantly reduces the computation burden, whilst achieving a remarkable steady-state and transient response, especially at the low sampling frequency. A set of simulation studies is conducted to compare the proposed method with its popular counterparts. In addition, experimental results are provided to verify the effectiveness of the new method.

Meromorphic exact solutions of the generalized (3+1)‐dimensional Kadomtsev‐Petviashvili equations

In this work, the generalized (3+1)‐dimensional Kadomtsev‐Petviashvili equation and its new form have been systematically investigated by using the complex method. The method is based on complex analysis and complex differential equations. And we get plentiful meromorphic exact solutions of these equations, which include rational solutions, exponential function solutions, and elliptic function solutions. The dynamic behaviors of these solutions are also shown by some graphs.

Logarithmic difference lemma in several complex variables and partial difference equations

In this paper, we mainly propose improvements of the logarithmic difference lemma for meromorphic functions in several complex variables, and then investigate meromorphic solutions of partial difference equations from the viewpoint of Nevanlinna theory.

Dependence of solutions and eigenvalues of third order linear measure differential equations on measures

This paper deals with a complex third order linear measure differential equation $${\rm{id}}{(y')^ \bullet} + 2{\rm{i}}q(x)y'{\rm{d}}x + y({\rm{id}}q(x) + {\rm{d}}p(x)) = \lambda y{\rm{d}}x$$ on a bounded interval with boundary conditions presenting a mixed aspect of the Dirichlet and the periodic problems. The dependence of eigenvalues on the coefficients p and q is investigated. We prove that the n-th eigenvalue is continuous in p and q when the norm topology of total variation and the weak* topology are considered. Moreover, the Frechet differentiability of the n-th eigenvalue in p and q with the norm topology of total variation is also considered. To deduce these conclusions, we investigate the dependence of solutions of the above equation on the coefficients p and q with different topologies and establish the counting lemma of eigenvalues according to the estimates of solutions.

SOLUTION OF THE MAIN DIFFERENTIAL EQUATION OF DEFORMATIONS OF THE CASING IN A CURVED WELL

In a curvilinear well, the casing functions as a long continuous rod. It is installed on the supports-centralizers and replicates the complex profile of the well, as a result of which it receives large deformations. To describe them, a system of differential equilibrium equations of internal and external forces and moments was composed, which was supplemented to a closed form with a differential equation of curvature. It is non-uniform, because it takes into account the own distributed weight of the rod. Two ways are proposed to solve the problem: by the method of mathematical compression of the system equations into a complex inhomogeneous differential equation or by projecting the equilibrium equations of forces on the global (vertical-horizontal) and on the local (tangent-normal) coordinate systems. It is shown that the first integral of the system can also be found from the equilibrium equations of a portion of a curved rod of finite length. This integral has the form of a second-order inhomogeneous differential equation with variable coefficients and is the main equation that describes the deformation of a long elastic rod under the action of the longitudinal and transverse components of the forces of distributed weight. The main requirement of the technology is the installation of a pipes column on the centering supports, the purpose of which is to ensure the coaxiality of the pipes and the borehole walls and the creation between them a cement ring of the same thickness and strength. Accounting for this requirement allowed us to linearize the main equation. Its solution is the clue to the formulas of deflections, angular slopes, internal bending moments and transverse forces in the rod with the arbitrary arrangement of supports and boundary conditions in their intersections. The solution of the main differential equation of angular deformations of a long bar is found in the form of a linear combination of Airy and Scorer’s functions and in the form of three linearly independent polynomial series in the sum with a partial solution. The obtained formulas of flexure and power parameters allow us to calculate stress and strain in the pipes column during the process of casing the borehole of an arbitrary profile which increases the reliability and durability of the well.

LEGION-based image segmentation by means of spiking neural networks using normalized synaptic weights implemented on a compact scalable neuromorphic architecture

LEGION (Locally Excitatory, Globally Inhibitory Oscillator Network) topology has demonstrated good capabilities in scene segmentation applications. However, the implementation of LEGION algorithm requires machines with high performance to process a set of complex differential equations limiting its use in practical real-time applications. Recently, several authors have proposed alternative methods based on spiking neural networks (SNN) to create oscillatory neural networks with low computational complexity and highly feasible to be implemented on digital hardware to perform adaptive segmentation of images. Nevertheless, existing SNN with LEGION configuration focus on the membrane model leaving aside the behavior of the synapses although they play an important role in the synchronization of several segments by self-adapting their weights. In this work, we propose a SNN-LEGION configuration along with normalized weight of the synapses to self-adapt the SNN network to synchronize several segments of any size and shape at the same time. The proposed SNN-LEGION method involves a global inhibitor, which is in charge of performing the segmentation process between different objects with different sizes and shapes on time. To validate the proposal, the SNN-LEGION method is implemented on an optimized scalable neuromorphic architecture. Our preliminary results demonstrate that the proposed normalization process of the synaptic weights along with the SNN-LEGION configuration keep the capacity of the LEGION network to separate the segments on time, which can be useful in video processing applications such as vision processing systems for mobile robots, offering lower computational complexity and area consumption compared with previously reported solutions.

Borel directions of solutions of a second order linear complex differential equation

Borel directions of solutions of a second order linear complex differential equation

A Functional Data Method for Causal Dynamic Network Modeling of Task-Related fMRI.

Functional MRI (fMRI) is a popular approach to investigate brain connections and activations when human subjects perform tasks. Because fMRI measures the indirect and convoluted signals of brain activities at a lower temporal resolution, complex differential equation modeling methods (e.g., Dynamic Causal Modeling) are usually employed to infer the neuronal processes and to fit the resulting fMRI signals. However, this modeling strategy is computationally expensive and remains to be mostly a confirmatory or hypothesis-driven approach. One major statistical challenge here is to infer, in a data-driven fashion, the underlying differential equation models from fMRI data. In this paper, we propose a causal dynamic network (CDN) method to estimate brain activations and connections simultaneously. Our method links the observed fMRI data with the latent neuronal states modeled by an ordinary differential equation (ODE) model. Using the basis function expansion approach in functional data analysis, we develop an optimization-based criterion that combines data-fitting errors and ODE fitting errors. We also develop and implement a block coordinate-descent algorithm to compute the ODE parameters efficiently. We illustrate the numerical advantages of our approach using data from realistic simulations and two task-related fMRI experiments. Compared with various effective connectivity methods, our method achieves higher estimation accuracy while improving the computational speed by from tens to thousands of times. Though our method is developed for task-related fMRI, we also demonstrate the potential applicability of our method (with a simple modification) to resting-state fMRI, by analyzing both simulated and real data from medium-sized networks.

Growth and Uniqueness Related to Complex Differential and Difference Equations

In this paper, we consider the growth of entire solutions of differential–difference equations of certain types. We also consider the growth of meromorphic solutions of complex linear difference equations of certain types and derive a result, which is a supplement of previous theorems given by Chen (J Math Anal Appl 373:235–241, 2011). Our additional purpose is to give a difference analogues of Bruck conjecture, which is a joint theorem of some previous theorems.