Undetermined Coefficients Research Articles

Effect of Treatment Parameter on Oscillatory Flow of Blood Through an Atherosclerotic Artery with Heat Transfer

This research work has been carried out to investigate the influence of treatment parameter on flow of blood in a stenosed artery in the presence of magnetic field with heat transfer. The momentum equation governing the flow field has been solved by scaling it to dimensionless structure with the aid of some dimensionless parameters. The equations have been analytically solved using modified Bessel equation and by the method of undetermined coefficients in order to obtain the temperature profile and velocity profile of the blood flow. The characteristics of the flow have been derived for a certain set of values RT ; Da; \theta; Gr; Re; Pr; \omega; \delta involved in the model analysis and are presented graphically with the help of software Mathematica. Moreover the velocity of the blood is adopting a wavy pattern as the values of the parameters vary. The study can be useful in providing a perception of the treatment caused by the superfluous consumption of fatty foods hence decreasing the risk of cancer, hypertension and many heart related diseases.

Mathematical Modeling of LDL-C and Blood Flow through an Inclined Channel with Heat in the Presence of Magnetic Field

This investigation revealed that LDL-C and blood flow through an inclined channel with heat in the presence of magnetic field. In this study,mathematical models were formulated for LDL-C and blood flow and energy transfer as a coupled system of partial differential equations (PDEs), the PDEs were scaled using the dimensionless quantities to dimensionless partial differential equations, they are further reduced to an ordinary differential equations (ODEs) using the perturbation method involving the oscillatory term, the governing equations are solved directly using the method of undetermined coefficient. The velocity and temperature functions are obtained with some governing parameters involved, and simulation codes were developed using Mathematica to study the effect of entering parameters on the profiles, and are found to be effective in controlling the flow profiles. It is observed that the governing parameters influenced the flow profiles; also the angle of inclination also influences the flow profile.
 HIGHLIGHTS
 
 Heart attack occurs when blood flow decreases or blood stopped from entering the heart, thereby causing damage to the heart muscle
 Atherosclerosis is an inflammatory disease resulting in the pathological alteration of the intima and media of arteries such as e.g. the aorta
 Use mathematical models to investigate the pulsating flow of blood in an atherosclerotic artery in an inclined fashion
 Carrying out numerical simulation using Mathematica to investigate the impact of the model parameters on the velocity and temperature distribution of the blood. The analysis is relevant in the sense that it helps determine how best clinicians can administer treatment and care
 
 GRAPHICAL ABSTRACT

Accurate fracture analysis of multi-material V-notched Reissner plates under bending or twisting

An accurate fracture analysis of multi-material V-notched Reissner plates under bending or twisting is performed by a new coupling approach of isogeometric analysis (IGA) and eigenfunction expansion method. In this approach, the overall plate is divided into two regions: a singular region near the notch tip (near field) and a regular region far away from the notch tip (far field). The overall model is discretized by a T-spline-based IGA. Asymptotic solutions of singular stress fields are applied to the near field, and therefore, the large number of fundamental unknowns of control points is reduced into a small set of undetermined coefficients. The unknowns in the far field remain unchanged. Consequently, the computational cost is significantly reduced, and explicit expressions of singular stress components in the vicinity of notch tip are obtained without post processing. Comparisons are presented to demonstrate the accuracy and convergence of the present approach. Effects of key influencing factors on the singularity order and singular items of stresses are investigated also.

Breather wave and double-periodic soliton solutions for a (2+1)-dimensional generalized Hirota–Satsuma–Ito equation

Abstract In this work, a (2+1)-dimensional generalized Hirota–Satsuma–Ito equation realized to represent the propagation of unidirectional shallow water waves is investigated. We first study the breather wave solutions based on the three-wave method and the bilinear form. Second, the double-periodic soliton solutions are obtained via an undetermined coefficient method, which have not been seen in other literature. We present some illustrative figures to discuss the dynamic properties of the derived waves.

Exact solutions of fractional oscillation systems with pure delay

In this paper we study the exact solutions of a class of fractional delay differential equations. We consider the fractional derivative of the order between 1 and 2 in the sense of Caputo. In the first part, we introduce two novel matrix functions, namely, the generalized cosine-type and sine-type delay Mittag-Leffler matrix functions. Then we obtain the explicit solutions for the linear homogeneous equations subjecting to corresponding initial conditions, by means of undetermined coefficients. In the second part, we first obtain a particular solution by means of the Laplace transform for the inhomogeneous equations with null initial conditions. Then we give an analytical representation of the general solution of the inhomogeneous equations through the sum of its particular solution and the general solution of the corresponding homogeneous equation.

Third Order Convergent Finite Difference Method for the Third Order Boundary Value Problem in ODEs

We propose a third order convergent finite-difference method for the approximate solution of the boundary value problems. We developed our numerical technique by employing Taylor series expansion and method of undetermined coefficients. The convergence property of the proposed finite difference method discussed. To demonstrate the computational accuracy and effectiveness of the proposed method numerical results presented.

An exact solution for vibro-acoustic response of MEE composite plate

To the best of our knowledge, this is the first endeavor to present an exact solution to predict vibro-acoustic characteristics of Magneto-electro-elastic (MEE) composite plate. The transverse and in-plane fields are considered based on thin-plate conditions. The variation of electric and magnetic potentials is determined according to electromagnetic boundary conditions and the Maxwell equation. The stress resultants and mass inertias are used in Hamilton’s principle to generate the governing equation. Here the mathematical formulation is developed using third-order shear deformation theory. Also in this work, the dynamic displacement responses are provided by finding five undetermined mode coefficients relevant to u, v, w, ϕx, and ϕy to predict forced vibration response. The forced vibration response obtained based on the developed governing equation is used to calculate the acoustic characteristics using the Rayleigh integral. The effect of magnetic and electric potential is shown in the acoustic responses. From the results, it is understood that the acoustic responses are highly influenced by the applied magnetic and electric potential. The radiation efficiency of the MEE plate did not show any variations in the lower frequency and it shows the variation near the resonant frequencies on the application of electric and magnetic potential.

Construction of dynamic temperature field model of ball screw based on superposition of positive and negative temperature fields

In order to study the influence of internal heat source of ball screw feed transmission system on the thermal error of ball screw, the theoretical modeling and experimental research are carried out on the temperature field and temperature rise characteristics of ball screw in this paper. Firstly, on the basis of analyzing heat generation mechanism and heat transfer of ball screw, the positive temperature field of ideal heat transfer of heat source and the negative temperature field caused by heat transfer are derived. In addition, the mathematical model of dynamic temperature field of ball screw feed system about time and position is established according to the superposition principle of temperature field, and the temperature field and thermal error of x-axis ball screw in horizontal machining center are measured. Secondly based on the experimental temperature field data, a complete finite element model of the temperature field of the ball screw feeding system is constructed, and temperature at any point of the system and the undetermined coefficient of the numerical model of temperature field under this working condition are determined by the finite element model. Finally, the thermal error distribution of the ball screw is calculated by the numerical model, the theoretical and experimental error distributions are compared, and the correctness of the theoretical model of the temperature field under this working condition is verified.

Novel perturbation solutions for deep-buried non-circular tunnels under biaxial in situ stress field based on Mohr-Coulomb criterion

Based on Mohr-Coulomb criterion, the generalized perturbation solutions for elastic-brittle-plastic rock surrounding deep-buried non-circular tunnels are presented in this study. The asymptotic expansions and a novel equivalent elliptical method adaptable for arbitrary shaped tunnel are provided for the perturbation analysis. Concerning the plastic analysis, the recurrence formulas and a simplified semi-analytical approach are developed for solving arbitrary-order approximations of stress functions conveniently. Subsequently, a novel undetermined coefficient method incorporating the stress continuity on elastoplastic interface is presented for determining the plastic radius and stress functions in elastic zone. On the basis of the preceding solutions, the extended solutions with equivalent elliptical method are developed for further expanding the applicable range of solutions. Besides, the perturbation analysis based on generalized Hoek-Brown criterion can be implemented in terms of the equivalent angles of friction and cohesive strengths. The whole computing process is provided in flow chart for the convenience of programming. The reliability of the solutions has been verified by comparing with the calculating results of numerical simulation. Significant improvements can be noticed for the predicted plastic radius compared with the results by the existing equivalent circular method. From a theoretical point of view, the developed perturbation solutions with equivalent section method complete the framework of the elastoplastic analysis for non-circular tunnels. From the perspective of practicality, the proposed solutions can provide preliminary insight for the rapid stability evaluation, plastic radius prediction, and tunnel outline design in practical engineering.

A novel approach for the system of coupled differential equations using clique polynomials of graph

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 method (UDCM) and variation of parameters method (VPM) solutions through tables and graphs. Obtained results reveal that the current approach is more accurate.

Free Response and Musical Pitch of Shape Memory Alloy Wires Under Voltage Loading

An experimental setup, on which the shape memory effect is based, is developed to measure the free response frequency and musical pitches of SMA wires. In this paper, the voltage, force, and sound signals of SMA wires are determined simultaneously under two kinds of experiments, constant temperature uniaxial tensile tests and fixed-length thermal cycle tests. Based on constitutive equations and thermodynamic equations, an empirical formula for the driving voltage and the steady-state stress of SMA wires is proposed, and the undetermined coefficients at radii of 0.050 mm and 0.070 mm are determined by experiments. The musical pitches of SMA wires under a constant voltage are obtained by combining the empirical formula with the string vibration equation. The theoretical results are compared with the actual musical pitches measured by the sound signals, and the two have a high degree of coincidence. Just for an SMA wire with a radius of 0.070 mm, the free response frequency range reaches 640 Hz to 1120 Hz, which covers 6 whole musical tones in total. Thus the SMA wire can be used as a string to play simple music.

Research on the Modified Mathematical Prediction Model of Impeller Cover Side Cavity Liquid Pressure for Centrifugal Pumps

Accurate calculation of pressure distribution in the impeller cover side cavity is the key to predict the axial force of centrifugal pump. The existing calculation models almost does not involve the prediction of cavity pressure when the radial clearances of different sealing rings are matched with the diameter of different balance holes. On the basis of the original prediction model of pump cavity pressure, a mathematical model of pressure distribution of impeller cover side cavity with different radial clearance of sealing ring and the diameter of balance hole was established by introducing potential head correction coefficient and flow proportional coefficient. In order to improve the calculation accuracy of rotation coefficient for rear pump cavity, the balance aperture length ratio and the rotation undetermined coefficient were introduced in the calculation equation of original rotation coefficient. A test bed for pressure and leakage was designed and established, and the pressure of impeller cover side cavity and balance hole leakage was systematically tested when the radial clearance of sealing ring and the diameter of balance hole were different. Experimental results showed that the radial clearance of rear sealing ring and the diameter of balance hole had different effects on the radial pressure gradient of pump cavity. The diameter of balance hole had little effects on the pressure of the front pump cavity. When the clearance of the front and rear sealing rings were the same, the pressure of rear pump cavity was generally higher than that of the front pump cavity. For the equilibrium chamber liquid, increasing the diameter of balance hole could relieve the pressure, and increasing the radial clearance of sealing ring could increase the pressure. Combined with the test data, the potential head correction coefficient, the rotation undetermined coefficient and the flow proportional coefficient of different specific areas were calibrated, with a specific solution equation. In this study, the reliability of the proposed pressure mathematical model for impeller cover side cavity was verified by three cases. The results showed that the theoretical prediction value was more consistent with the actual measured value, proving that the proposed mathematical model had high accuracy and universality.

Ground Maneuvering Target Focusing via High-Order Phase Correction in High-Squint Synthetic Aperture Radar

Moving target imaging in high-squint synthetic aperture radar (SAR) shows great potential for reconnaissance and surveillance tasks. For the desired resolution, high-squint SAR has a long-time coherent processing interval (CPI). In this case, the maneuvering motion of the moving target usually causes high-order phase terms in the echoed data, which cannot be neglected for precise focusing. Many ground moving target imaging (GMTIm) algorithms have been proposed in the literature, but some high-order phase terms remain uncompensated in high-squint SAR. For this problem, a high-order phase correction-based GMTIm (HPC-GMTIm) method is proposed in this paper. We assumed that the target of interest has a constant velocity in the subaperture CPI, but maneuvering motion parameters for the whole CPI. Within the short subaperture CPI, the target signal can be simplified as a three-order phase expression, and the instantaneous Doppler frequency (DF) was estimated by some time–frequency analysis tools, including the Hough transform and the fractional Fourier transform. For the whole CPI, the subaperture, the instantaneous DF was combined to form a total least-squares problem, outputting the undetermined phase coefficients. Using the proposed local-to-global processing chain, all high-order phase terms can be estimated and corrected, which outperforms existing methods. The effectiveness of the HPC-GMTIm method is demonstrated by real measured high-squint SAR data.

Scattering of a scalene trapezoidal hill with a shallow cavity to SH waves

Abstract Both surface ground motion and cavity stress concentration have always been considered in the designs of earthquake engineering. In this paper, a theoretical approach is used to study the scattering problem of circular holes under a scalene trapezoidal hill. The wave displacement function was obtained by solving the Helmholtz equation that meets the zero-stress boundary conditions by the variable separation method and the image method. Based on the complex function, the multipolar coordinate method and the region-matching technique, algebraic equations were established at auxiliary boundaries and free boundary conditions in the complex domain. Auxiliary circles were used to solve the singularity of the reflex angle at the trapezoidal corner. Then, according to the sample statistics, instead of the Fourier expansion method, the least-squares method was used to solve the undetermined coefficient of the algebraic equations by discrete boundaries. Frequency responses for some parameters were calculated and discussed. The numerical results demonstrate that the continuity of the auxiliary boundaries and the accuracy of the zero-stress boundary are good; the displacement of the free surface and the stress of the circular hole are related to the shape of the trapezoid, the position of the circular hole, the direction of the incident wave and the frequency content of the excitation. Finally, time-domain responses were calculated by inverse fast Fourier transform based on the frequency domain theory, and the results have revealed the wave propagation mechanism in the complicated structure.

Research on a model-based burst pressure prediction method for pipelines with corrosion defects

ABSTRACT The leakage of pipelines will cause serious harm to the human living environment. To meet the safety assessment requirement for pipelines, a pipeline burst pressure prediction method based on a parametric model is proposed. The parameters of the model include defect size, pipeline size and material properties. First, the relationship between the defect size and the burst pressure was studied by finite element analysis, and the parametric model was deduced according to the analysis results. Second, the undetermined coefficients of the model were obtained using the improved genetic algorithm. Last, comparisons of the test burst pressures with those predicted by the proposed model and three standard assessment methods were carried out. The results show that the predicted model is more reliable than the existing pipeline assessment codes.

Prediction and analysis of slurry shield TBM disc cutter wear and its application in cutter change time

Disc cutter is mainly used in cutting through hard rock. Disc cutter wear is inevitable during tunnel boring machine (TBM) tunneling, and excessive wear can lead to a decrease in tunneling efficiency and increase construction risks. Therefore, the importance of predicting the disc cutter wear accurately and arranging the cutter change plan can never be overemphasized. In this study, disc cutter wear data and replacement records for three shield chamber openings were analyzed, which were collected from a water conveyance tunnel project in Guangdong, China. It was found that the normal wear failure accounted for approximately 76.19%, while the abnormal wear failure accounted for 23.81%, and the range of tools to be replaced due to normal wear continues expanded as the tunneling distance increased. Based on the observed wear data, it is found that the wear rate of edge tools is much lower than that of center tools, which does not conform to the normal wear rules. Subsequently, a new disc cutter wear prediction model was developed considering energy conversion and uneven thrust distribution for explaining this problem. The determination method of the energy conversion coefficient k and thrust distribution coefficient b was proposed, and they had been obtained from the wear data in the first replacement. On the other hand, the proposed model has higher accuracy in both cutter wear and cutter change range prediction compared with the existing models. However, the prediction accuracy of the proposed model will decrease with the increase of the tunneling distance, so the undetermined coefficients k and b need to be updated in a timely manner.

Conflict and agreement in the exact solutions of Schrodinger equation in a two and restricted three-body systems

Using the undetermined coefficients method, we obtained the exact solution of Schrodinger equation of a three-body system, H2+ ion, in a restricted form. We found that its ground state energy when the nuclear distance becomes zero can be the same as that of a He+ ion. However, its radial wavefunction is e-r while the corresponding wavefunction is e-2r for a He+ ion, where r is the electron-nucleus distance. We tentatively discussed clues leading to the interesting difference in a hope to find exact solutions of Schrodinger equation for systems of three or more bodies.

Large amplitude forced vibrations of Timoshenko beams using coupled displacement field method

Forced vibration behaviour of a Timoshenko beam, with a concentrated mass at the centre, subjected to a concentrated harmonic force is investigated using the coupled displacement field method. The expressions for the transverse displacement were deduced from the assumed admissible function having a single term undetermined coefficient for a total rotation of the beam using the coupled displacement field method. The equation of motion for the forced vibration condition is derived using the principle of conservation of total energy. The Newmark method is employed to solve the equation of motion to obtain the dynamic response of the Timoshenko beam. The dynamic forced vibration response for the two boundary conditions, simply supported and clamped-clamped, is computed by applying the mode superposition technique for the obtained coupled displacement field equations. The authors also carried out the finite element analysis using ANSYS 21 software to validate the results obtained by the coupled displacement field method, and the results are also validated with the existing literature wherever possible. The obtained results are very closely matching with the FEM and other published results.

Similarities in a First Differential Equations Course

Summary The key techniques in a traditional first course in differential equations are linked by reexpressing them in terms of differential operators and change-of-basis ideas, including integrating factors, variation of parameters, and the method of undetermined coefficients. The main tool is to convert linear operators from sums to compositions.

Time-domain minimum residual method combined with energy balance for nonlinear conservative systems

This paper presents a new semi-analytical method that combines the time-domain minimum residual method with energy balance for nonlinear conservative systems. Due to work done only by the conservative force, the total energy of the nonlinear conservative system will remain constant. The response frequency is also the function of the initial conditions and it is difficult to achieve the semi-analytical solutions accurately. The proposed method aims to quickly obtain the higher-order approximate semi-analytical periodic or quasi-periodic solutions of both weakly and strongly nonlinear conservative systems. To this end, the approximate analytical solution is expressed as a group of trigonometric series with undetermined coefficients firstly. Next, by deriving the assumed trigonometric series and substituting the state variables back into the original nonlinear conservative equation, the goal of solving the semi-analytical solution is transformed into an optimization problem that minimizes the residual objective function in a period. Finally, the energy constraints are introduced into the optimization process, and the enhanced response sensitivity approach is called to resolve the nonlinear objective function iteratively. A conservative Duffing system and a nonlinear symmetrically conservative two-mass system are taken as two numerical examples to illustrate the performance of the proposed method.