Boundary Value Problem For System Research Articles

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>

Existence results for a system of boundary value problems for hybrid fractional differential equations

Existence results for a system of boundary value problems for hybrid fractional differential equations

Relative Lagrangian Formulation of Finite Thermoelasticity

The motion of a body can be expressed relative to the present configuration of the body, known as the relative motion description, besides the classical Lagrangian and the Eulerian descriptions. When the time increment from the present state is small enough, the nonlinear constitutive equations can be linearized relative to the present state so that the resulting system of boundary value problems becomes linear. This formulation is based on the well-known ``small-on-large'' idea, and can be implemented for solving problems with large deformation in successive incremental manner. In fact, the proposed method is a process of repeated applications of the well-known “small deformation superposed on finite deformation” in the literature. This article presents these ideas applied to thermoelastic materials with a brief comment on the exploitation of entropy principle in general. Some applications of such a formulation in numerical simulations are briefly reviewed and a numerical result is shown.

Common Fixed-Point Theorems in the Partial b -Metric Spaces and an Application to the System of Boundary Value Problems

In this paper, we investigate the conditions for the existence of the common fixed points of generalized contractions in the partial b -metric spaces endowed with an arbitrary binary relation. We establish some unique common fixed-point theorems. The obtained results may generalize and improve earlier fixed-point results. We provide examples to illustrate our findings. As an application, we discuss the common solution to the system of boundary value problems.

Differential-algebraic boundary-value problems with the variable rank of leading-coefficient matrix

Conditions for the solvability of the linear boundary-value problem for systems of differential-algebraic equations with the variable rank of the leading-coefficient matrix and the corresponding solution construction procedure have been found.