Boundary Value Problem For System Research Articles

Countably many positive solutions for iterative system of boundary value problems on time scales

This paper establishes the existence of countably many positive solutions for a second order iterative system of two-point boundary value problems by using Holder’s inequality and Guo–Krasnoselskii’s fixed point theorem for operator on a cone.

PERIODIC ATEB-FUNCTIONS AND THE VAN DER POL METHOD FOR CONSTRUCTING SOLUTIONS OF TWO-DIMENSIONAL NONLINEAR OSCILLATIONS MODELS OF ELASTIC BODIES

In the process of operation, the simplest elements (hereinafter elastic bodies) of machines and mechanisms under the influence of external and internal factors carry out complex oscillations ‒ a combination of longitudinal, bending and torsion combinations in various combinations. In general, mathematical models of the process of such complex phenomena in elastic bodies, even for one-dimensional calculation models, are boundary value problems for systems of partial differential equations. A two-dimensional mathematical model of oscillatory processes in a nonlinear elastic body is considered. A method of constructing an analytical solution of the corresponding boundary-value problems for nonlinear partial differential equations is proposed, which is based on the use of Ateba functions, the Van der Pol method, ideas of asymptotic integration, and the principle of single-frequency oscillations. For "undisturbed" analogues of the model equations, single-frequency solutions were obtained in an explicit form, and for "perturbed" ‒ analytical dependences of the basic parameters of the oscillation process on a small perturbation. The dependence of the main frequency of oscillations on the amplitude and non-linearity parameter of elastic properties in the case of single-frequency oscillations of "unperturbed motion" is established. An asymptotic approximation of the solution of the autonomous "perturbed" problem is constructed. Graphs of changes in amplitude and frequency of oscillations depending on the values of the system parameters are given.

Rihi Mesh Placement Algorithm In Collocation Methods

Various adaptive mesh selection strategies for solving single higher order two-points boundary value problems (BVPs) by using collocation methods are intensively investigated for along time and they are now well established. In this work we concern with numerical investigations of adaptive mesh selection algorithms using the criterion function rihi for solving first order system of BVPs and developing some algorithms. The algorithms perform quite nicely and appear competitive with De Boor algorithm.

Many Positive Solutions for Iterative Systems of Boundary Value Problems with Navier Boundary Conditions on Time Scales

Many Positive Solutions for Iterative Systems of Boundary Value Problems with Navier Boundary Conditions on Time Scales

Global error analysis of discontinuous Galerkin methods for systems of boundary value problems

Global error analysis of discontinuous Galerkin methods for systems of boundary value problems

Thermal synthesize of shear thinning nanofluid in small flexible vessel type channel with low seepage Reynold number and Au nanoparticles

The present study presents an analysis of Casson nanofluid flow in small expanding/contracting vessel with permeable walls. Further, we have presented the coupling of expanding/contracting parameter with low seepage Reynolds number. Similarity transformation is used to convert the governing partial differential equation into set of coupled nonlinear ordinary differential equations which are solved numerically using R–K method after converting the system of boundary value problem into initial value problem with the help of a shooting technique numerically. The behavior of emerging parameters are presented graphically for velocity, temperature, and nanoparticles fraction. Most important effect of physical parameter skin friction, Nusselt and Sherwood number are presented in table form. It is found that, reduced Nusselt number is decreasing function of Prendtl number and reduce Sherwood number is the increasing Brownian motion parameter Nb and Thermophoresis parameter Nt.

On a version of hybrid existence result for a system of nonlinear equations

Abstract By combining monotonicity theory related to the parametric version of the Browder-Minty theorem with fixed point arguments, hybrid existence results for a system of two operator equations are obtained. Applications are given to a system of boundary value problems with mixed nonlocal and Dirichlet conditions.

Superconvergence Analysis of Discontinuous Galerkin Methods for Systems of Second-Order Boundary Value Problems

In this paper, we present an innovative approach to solve a system of boundary value problems (BVPs), using the newly developed discontinuous Galerkin (DG) method, which eliminates the need for auxiliary variables. This work is the first in a series of papers on DG methods applied to partial differential equations (PDEs). By consecutively applying the DG method to each space variable of the PDE using the method of lines, we transform the problem into a system of ordinary differential equations (ODEs). We investigate the convergence criteria of the DG method on systems of ODEs and generalize the error analysis to PDEs. Our analysis demonstrates that the DG error’s leading term is determined by a combination of specific Jacobi polynomials in each element. Thus, we prove that DG solutions are superconvergent at the roots of these polynomials, with an order of convergence of O(hp+2).

Solvability of an infinite system of three point BVP in c0 and ℓ1 spaces using measure of noncompactness

Various existing works in the literature have addressed the solvability of different infinite systems of boundary value problems in different types of Banach spaces. In most of these works, the authors have restricted themselves in systems of differential equations with boundary conditions defined at two points. In this work, we have addressed the solvability of an infinite system of third-order differential equations with boundary conditions defined at three points in the sequence spaces [Formula: see text] and [Formula: see text]. We have also illustrated the results for both sequence spaces with suitable examples.

Heat transfer of a Carreau fluid in a thin elastic film over an unsteady stretching sheet

An analysis is carried out to investigate the influence of the magnetic field over Carreau fluid flow and heat transfer in a liquid film on an unsteady stretching surface. The governing equations are converted to a system of boundary value problem (BVP) by using the similarity transforms. The differential equations are solved numerically using the modified Laplace decomposition method (MLDM) and the obtained results are compared with the existing schemes. An excellent agreement between them illustrates the accuracy of MLDM to the suggested model. The influence of the dimensionless governing parameters like magnetic field number, Prandtl number, unsteadiness parameter, Weissenberg parameter, and radiative number on velocity and temperature profiles is discussed in tabular form and shown through illustrative graphs.

Solving Fuzzy System of Boundary Value Problems by Homotopy Perturbation Method with Green’s Function

In this research, we successfully demonstrated the use of the homotopy perturbation method with Green's function to find approximate solutions for the fuzzy system of boundary value problems. Our results showcase the effectiveness of this method in providing accurate and reliable solutions. Our results showcase the effectiveness of this method in providing accurate and reliable solutions. The consistent way to reduce the size of the computation gives to reach the exact solution is one of the best methods adopted to determine the behavior of the solution directly in order to determine the approximate solution analytically, Finally, the problems that have been addressed confirmed the validity of the method applied analytically in this research using comparison with some numerical problems

An algorithm based on homotopy perturbation theory and its mathematical analysis for singular nonlinear system of boundary value problems

In this paper, a promising technique based on the homotopy perturbation method for solving singular nonlinear system of boundary value problems (BVPs) is employed. The original singular system of BVPs is transformed into an equivalent system of integral equations to overcome the singular behavior at , and then, the recursive series for the solution is established. The mathematical formulation is further supported by conducting the convergence analysis of the recursive schemes. The accuracy of the proposed method is tested by comparing against the current variational iteration method using many standard examples. It is shown that the proposed method is more accurate and efficient than the existing method such as Chebyshev operational matrix method and variational iteration method.

Denumerably many Positive Solutions for Iterative System of Boundary Value Problems with N-Singularities on Time Scales

In this paper we consider a iterative system of two-point boundary value problems with integral boundary conditions having n singularities and involve an increasing homeomorphism, positive homomorphism operator. By applying Hölder’s inequality and Krasnoselskii’s cone fixed point theorem in a Banach space, we derive sufficient conditions for the existence of denumerably many positive solutions. Finally we provide an example to check validity of our obtained results.

Positive radial solutions for a boundary value problem associated to a system of elliptic equations with semipositone nonlinearities

<abstract><p>In this paper we use the fixed point index theory to study the existence of positive radial solutions for a system of boundary value problems with semipositone second order elliptic equations. Some appropriate concave and convex functions are utilized to characterize coupling behaviors of our nonlinearities.</p></abstract>

A mathematical model of an inductively coupled radio-frequency discharge at low pressure as nonlinear partial eigenproblem

A new approach for modeling steady state inductively coupled radio frequency discharges at low pressure is described. A simple one-dimensional model is considered, which includes Maxwell’s equations and the electron balance equation with boundary conditions of the third kind. It is shown that the system of boundary value problems is a two-parameter partial eigenvalue problem. The smallest eigenvalue of the problem is the boundary value of the magnetic field strength. The second parameter of the problem is the concentration of electrons at the center of the plasma bunch. The developed approach makes it possible to calculate the inductor current required to maintain a steady state of the discharge. The results of calculations of the dependence of the inductor current, electron density, electric and magnetic fields on pressure are presented.

Algorithms based on a 3-derivative method for singular differential equations including equations with blow-up solutions

We propose block algorithms based on a 3-derivative Nyström-type method (TDNM) for integrating singular second-order ordinary differential equations (ODEs), including parabolic and hyperbolic partial differential equations (PDEs) with blow-up solutions. The block algorithms are implemented in a block-by-block mode for initial value problems (IVPs) and block unification mode for boundary value problems (BVPs). In particular, the algorithms are applied to PDEs by first reducing them into ODEs via the method of lines, whereby the space variable is discretized to yield a system of initial value problems, or the time variable is discretized to yield a system of boundary value problems. It is shown that the numerical solution given by the block-by-block and block unification algorithms are superior to those available in the literature. It is also demonstrated that the implementation of the variable step version of the TDNM, which is generally recommended for IVPs, has no advantage over the block-by-block algorithm.

Linear stability of a falling film over a heated slippery plane.

A detailed parametric study on the linear stability analysis of a three-dimensional thin liquid film flowing down a uniformly heated slippery inclined plane is carried out for disturbances of arbitrary wavenumbers, where the liquid film satisfies Newton's law of cooling at the film surface. A coupled system of boundary value problems is formulated in terms of the amplitudes of perturbation normal velocity and perturbation temperature, respectively. Analytical solution of the boundary value problems demonstrates the existence of three dominant modes, the so-called H mode, S mode, and P mode, where the S mode and P mode emerge due to the thermocapillary effect. It is found that the onset of instabilities for the H mode, S mode, and P mode reduces in the presence of wall slip and leads to a destabilizing influence. Numerical solution based on the Chebyshev spectral collocation method unveils that the finite wavenumber H-mode instability can be stabilized, but the S-mode instability and the finite wavenumber P-mode instability can be destabilized by increasing the value of the Marangoni number. On the other hand, the Biot number shows a dual role in the H-mode and S-mode instabilities. But the P-mode instability can be made stable with the increasing value of the Biot number and the decreasing values of the Marangoni number and the Prandtl number. Furthermore, the H-mode and S-mode instabilities become weaker, but the P-mode instability becomes stronger, with the increasing value of the spanwise wavenumber. In addition, the shear mode emerges in the numerical simulation when the Reynolds number is large, which can be destabilized slightly with the increasing value of the Marangoni number; however, it can be stabilized with the increasing value of the slip length and introducing the spanwise wavenumber to the infinitesimal perturbation.

Forced convection flow over a moving porous and non-uniform cylinder of variable thickness

Forced convection flow of viscous fluid over a moving, porous, and heated cylinder of variable thickness is studied in this paper. The non-uniform cylinder is placed vertically in a quiescent fluid. All the field quantities (the normal and axial velocities and temperature), defined at the wall or surface of cylinder, are variable and may have non-linear form. An appropriate set of transformations is developed on the analogies of systematic classical symmetries for a system of partial differential equations (PDE’s), whereas, they are equipped with the representative of each velocity component, temperature function, and similarity variable. However, the PDE’s along with the boundary conditions (BC’s) are reduced into a system of boundary value problem (BVP) of ordinary differential equations (ODE’s) by using these new and unseen similarity variables. The system of non-linear coupled ODE’s, combined with the BC’s is solved, whereas, the numerical results are obtained for different values of the existing governing parameters. All the field quantities, skin friction, rate of heat transfer at the surface of the cylinder are evaluated with the help of [Formula: see text] package of MATLAB and the results are shown in different graphs. The present simulation generalizes all types of forced convection flow problems, over a porous and heated cylinder of variable (uniform) radius, when it is stretched (shrunk) with linear and non-linear (uniform) velocity. However, the simulated problem gives rise to a new set of problems and that they are valid for non-linear (linear and uniform) injection (suction) velocity.

A Theoretical Analysis of a Fractional Multi-Dimensional System of Boundary Value Problems on the Methylpropane Graph via Fixed Point Technique

Few studies have investigated the existence and uniqueness of solutions for fractional differential equations on star graphs until now. The published papers on the topic are based on the assumption of existence of one junction node and some boundary nodes as the origin on a star graph. These structures are special cases and do not cover more general non-star graph structures. In this paper, we state a labeling method for graph vertices, and then we prove the existence results for solutions to a new family of fractional boundary value problems (FBVPs) on the methylpropane graph. We design the chemical compound of the methylpropane graph with vertices specified by 0 or 1, and on every edge of the graph, we consider fractional differential equations. We prove the existence of solutions for the proposed FBVPs by means of the Krasnoselskii’s and Scheafer’s fixed point theorems, and further, we study the Ulam–Hyers type stability for the given multi-dimensional system. Finally, we provide an illustrative example to examine our results.

Existence of fixed point results in orthogonal extended b-metric spaces with application

<abstract><p>In this paper, we investigate the conditions for the existence of fixed-point for generalized contractions in the orthogonal extended b-metric spaces endowed with an arbitrary binary relation. We establish some unique fixed-point theorems. The obtained results generalize and improve many earlier fixed point results. We also provide some nontrivial examples to corroborate our results. As an application, we investigate solution for the system of boundary value problem.</p></abstract>