Numerical Study on Single and Multi-Dimensional Boundary Value Problems by the Method of Weighted Residual

The Galerkin method of weighted residual (MWR) for computing natural frequencies of some physical problems such as the Helmholtz equation and some second-order boundary value problems has been revised in this article. The use of one and two-dimensional characteristic Bernstein polynomials as the basis functions have been presented by the Galerkin MWR. The vibration of non-homogeneous membranes with Dirichlet type boundary conditions is also studied here. The useful properties of Bernstein polynomials, its derivatives and function approximations have also been illustrated. Besides, the efficiency and applicability of the proposed technique have been demonstrated through some numerical experiments.


Introduction
Mathematical model derived from multi-dimensional differential equations play a crucial role in modeling a variety of scientific and engineering application problems. The study of vibration of membrane of different geometries has been the subject of interest to scientists and engineers since long back owing to their wide applications in every field of modern technology. Different shapes or geometries of membranes are the fundamental constituents in many scientific states. Many researchers concentrated on membrane structures and studied about the vibration theory of membrane employing numerous analytical or numerical techniques. However, to the best of our knowledge, none had attempted to provide approximate eigenvalues of boundary value problems employing Bernstein-Galerkin method of weighted residual (MWR).
Some physical problems arise in science and engineering are modeled by the elliptic, parabolic and hyperbolic partial differential equations. The process of solving certain linear evolution equations such as the heat or wave equations have received increased interest from several fields of science and engineering, either on bounded domains. Certain class of linear evolution equations for instance the heat or wave equations have a second order linear differential eigenvalue problem with two boundary conditions but no unique solution exists.
A code for second order problems has been implemented and special attention has been paid to the approximation of the boundary conditions in the twodimensional case.
A new numerical method based on Bernstein polynomials expansion is proposed for solving one dimensional (1-D) elliptic interface problems [1]. At present, formulations exploiting Galerkin method of weighted residual are constructed to determine the coefficient matrix derived from linear system of equations. Helmholtz equation appears in vibrating-string problems and in finding the temperature distribution in a rod. Calculations of eigenvalues of Helmholtz equation using boundary method are presented in [2]. Numerical technique utilizing Bernstein polynomials basis to give the approximate solution of a parabolic partial differential equation illustrated in [3]. The large deflection theory of membrane is applied [4] to derive the governing vibration equations of orthotropic membrane. The Bernstein polynomials are used to find numerical solution of elliptic boundary value problems with a point collocation method [5].
The authors solved the problem and obtained the power series formula of nonlinear vibration frequency of rectangular membrane with four edges fixed. Bernstein Ritz-Galerkin method for solving an initial boundary value problem for two-dimensional (2-D) wave equation has been studied by [6]. A spectral element model for the transverse vibration of a membrane has been developed by [7] using the boundary splitting method and the waveguide FEM-based spectral super element method (SSEM). Exact solutions are well known for the free vibration of a membrane with simple geometry such as elliptical [8], circular [9], general convex polygonal membranes [10], rectangular membranes [11]- [17].
Nevertheless, closed-form solutions rarely exist which leads many researchers to have numerical solution applying several techniques in view to minimize the error with less computational difficulties. Free vibrations of membranes have been studied in the literature by applying the method of superposition using wave-type base functions [10], the optimized Galerkin-Kantorovich method [11], classical Rayleigh-Ritz method [12], finite difference method [13], the Hybrid method [14], the discrete singular advance convolution method [15], the Kan-torovich method [16]. Furthermore, the external and internal excitation methods [17], are also investigated. Two-dimensional (2-D) Helmholtz equation is solved for non-homogeneous membrane exploiting Sinc collocation method [18].
Double ultraspherical spectral Galerkin method with Chebychev polynomials is applied to solve elliptic partial differential equations (PDEs) [19]. Chebychev spectral quasi-inverse matrix diagonalization Galerkin method [20] is utilized to solve multidimensional PDE's. Finite difference methods have been offered for the numerical solution of the one-dimensional parabolic equation in the study [21]. Eigenvalue analysis for second order boundary value problems arises in several engineering application areas has been investigated using analytical or numerical techniques [22]. Bernstein basis has been exploited to solve ordinary and partial differential to solve eigenvalues as well as boundary value problems employing numerous techniques are in [23]- [28].
In this study, we have presented Bernstein polynomials based Galerkin method of weighted residual (MWR) technique that offers accurate solutions, are put up with in terms of truncated series of smooth polynomial functions. The key advantages of applying the Galerkin MWR are its superior accuracy which can be achieved even with a few numbers of polynomials compared to the other techniques reported in the proposed work.
The significant benefit of the present method is that the trial Bernstein basis function can be modified effortlessly to any desired form so as to satisfy the Dirichlet and derivative boundary conditions for both one and two-dimensional problems.
Our goal is to construct an appropriate method with Dirichlet and derivative type boundary conditions and to reduce the errors as well as the computational difficulties. The main reason why the Galerkin MWR is chosen is its flexibility and simple implementation. The notable properties of Bernstein polynomials vanishing at its endpoints over the finite interval inspire us to execute it in the Galerkin technique. Furthermore, the smallest eigenvalue which characterizes potentially the most visual structures of the dynamical systems can be computed very accurately applying the said technique which converges to large significant digits and fairly close to the analytical results.
Some analytical methods involve a reduced amount of memory and arithmetic, but usually they are unable to fix the maximal dimension for the spectrum. Perturbation methods limit the extent of its application due to disadvantages in its theory in its existing form and all types of techniques required small parameter and consequently require some skill to implement. Finite difference methods (FDM) and finite elements methods (FEM) are well-known discretization techniques that comprise extensive calculation and huge memory storage in machine for storing matrices having larger dimensions. Additionally, the precision rapidly worsens for the upper eigenvalues. Furthermore, FDM gives value at particular points ineffective to evaluate the values at the desired points between two grid points and takes more computational cost for getting higher accuracy. The

Two Dimensional Bernstein Polynomials
There are ( ) For suitability, we usually set ( ) Some interesting and useful properties of 2-D Bernstein polynomials are demonstrated in [30].

The Bernstein Approximation Theorem 1 [27]
Every continuous function f defined on [0, 1] can be uniformly approximated as closely as desired by a polynomial function. For any 0 ε > , there exists a positive integer N such that for all Hence given any power-form polynomial of degree N, it can be uniquely converted into a Bernstein polynomial of degree n for n N ≥ .
which implies the uniform convergence to the function f . Let

Derivatives of Multivariable Bernstein Polynomials
We can define p-th order partial derivative in x-axis direction in a following Also q-th order partial derivative in y-axis direction in a following way Finally derivative of order p q + is defined as follows

Description of 2-D Helmholtz Problem
We consider the governing transient heat transfer on a two dimensional region Ω [31] [32].
with the boundary conditions (b. c's) 11 22, n x y n u u u u q a n a n q x y and the initial condition where t specifies for the time and c, 11 a , 22 a , 0 a , 0 u , f, û , ˆn q are regarded as functions of positions and/or time.
The corresponding homogeneous form of the above Equation (11) is Approximate solution of Equation (12) is given as (12a) Thus Equation (12) becomes 11 22 0 where 0 c a λ α = + and boundary conditions take the form Here the eigenvalues λ 's and eigenfunctions U's satisfy the time independent form of Equation (13). Then Equation (13) where ( )

Galerkin Formulation for One Dimensional Heat Equation
Here we applied a modified weighted residual technique which involves only first derivative instead of second derivative terms in the residual equation. This can be performed applying integration by parts in which the operational matrices for integration and the product is utilized to convert the given differential eigenvalue problems to a system of algebraic equations.
One dimensional parabolic partial differential equation with the boundary and initial conditions which governs the transient heat transfer in one-dimensional system (i.e., a plane wall), here u denotes the temperature, k the thermal conductivity, ρ the density, A the cross-sectional area, c the specific heat, q the heat generation per unit length. Here ( ) Let us assume the approximate solution of Equation (16) The solution of Equation (15) can be written as The Galerkin weighted residual equation can be written as On simplification the by using the conditions the above residual Equation (17) where dda u t = refers to the time derivative.
Hence the matrix form of Equation (18) where d d The solution of Equation (18) can be given as Hence the matrix form of (18) [ Solving (19) we can determine λ and nonzero ( )

Brief Description of Hyperbolic Equations
The transverse motion of a membrane, is governed by a partial differential equation of the form as given in (14), where, ( ) , , u x y t must be determined such that it satisfies Equation (19) in a region Ω together with specified boundary and initial conditions. The problem of finding the solution as given by Equation (14) holds for homogeneous boundary and initial conditions and 0 f = is called an eigenvalue problem.

Solution of Two-Dimensional Wave Equation in Rectangular Membrane
We consider the equation for the vibration of a tightly stretched membrane as the membrane of a drum. We assume that the membrane is of uniform tension and the tension per unit length is same in all directions at every point and m be the mass of per unit of area of the membrane.
The fact that we are keeping the edges of the membrane fixed is expressed by the boundary conditions We must also specify how the membrane is initially deformed and set into motion. This is done via the initial conditions ( ) The solution of Equation (21) These are the solutions of the wave Equation (22)

Vibration of Distributed Inhomogeneous System in a Rectangular Domain
We consider free vibrations of a uniformly stretched inhomogeneous membrane of rectangular shape with clamped edges in an elastic medium. We consider the following eigenvalue problem where the boundary conditions are The Green's theorem states that if A is a region in the xy plane bounded by a closed curve Γ then we suppose: , U x y F x y i G x y j = + is a continuous vector field defined on a region A in Ω. Moreover, suppose F and G have continuous partial derivatives and that the boundary Γ is a simple closed curve C with positive orientation. Then Using Equations (26a), (27) We finally obtain   We consider a two dimensional boundary value problem.

Error and Convergence of Galerkin MWR
is non-self-adjoint unless 0 i C = . For convergence of the Galerkin method, we assume there exists a constant 0 ς such that for any point in V and real numbers i l , The coefficients , i j B , and their first derivatives are continuous in Ω and the coefficients C and F are continuous in Ω . We assume the problem has a unique solution. The proof of the theorem is illustrated in and the references therein [33]. Here Ω is a finite bounded domain and S Ω = ∪ Ω .
where the operator 0 L is positive bounded below for the set of functions which vanishes on S. The convergence proofs are done in terms of the operator 0 L and the remaining terms in the differential equation are bounded in terms of this operator and its inverse. Theorem2: Assume the problem (11) is unique for the boundary conditions Let us consider the two-dimensional the following eigenvalue problem

Numerical Experiments
In  To compute the eigenvalues depending on parameter e which determines the boundaries of the parametric instability [22], we need to construct periodic solutions of that: Since the function ( ) , q t e is even with respect to t, ( ) ( ) , , q t e q t e − = in (the conditions of perioiodicity in (37a) are equivalent to the conditions of the first kind given as (37b) and the second kind given as here the vibration of elastic Crankshaft with concentrated load yield the following equation with periodic co-efficient and modulation depth e illustrated in [22] is considered.
We consider a plane wall, initially at a uniform temperature [32].  [32] in a square region with boundary and initial conditions is considered.
Homogeneous form of (41) for 0 f = is American Journal of Computational Mathematics The derivative boundary conditions will be applied integrand of the residual equation. Hence for executing only the Dirichlet boundary conditions in Equation (41a), we modify the polynomial basis as The exact solution of (42) is ( ) ( ) ). The exact natural frequencies of the rectangular membrane are obtained by considering the full domain The fact that we are keeping the edges of the membrane fixed is expressed by the boundary conditions For implementing boundary conditions in Equations (43a), (43b) we modify the polynomial basis as

Result and Discussions
In Table 1, we computed even and odd eigen solutions for different modulation depth e ranges from 0.1 to 0.9. Analytical eigen-solutions using perturbation method are listed in [22]. Our computed relative errors for the smallest modes are less than 5% which shows that the present technique produced significantly small errors. From Table 2, the smallest eigenvalues attains the accuracy up to 10 −16 and error increases rapidly for higher eigenvalues than the lower values which is better than boundary method. As we increase the grid points or nodes from n = 15 to n = 30, the error decays fast for all the eigenvalues and consistent accuracy is obtained up to 10 −13 . We observed that increasing of nodes reveal the stable behaviour of all the ei-genvalues for n = 30. It is also observed that our present approach accomplishes accurate results and is compatible to the existence new boundary approach for one dimensional Helmholtz equation. Comparison of relative errors obtained for Gal. Relative errors between our proposed Galerkin MWR and FEM [32] are compared which are displayed in Figure 1. In Table 3(a) and Table 3(b) we computed seven eigen frequencies for the two sets of boundary conditions exploiting current formulation and comparison to the exact and finite element results are shown with relative errors. From these tables, we observed that the smallest error decreases up to 7 10 − which is much smaller than that of FEM [32]. Eigen frequencies using different degree of polynomial are displayed in Table 4 of example 4. We noticed that the smallest order of the accuracies for the largest eigenvalues are 10 −2 , 10 −3 , 10 −4 respectively with 4, 8, 10, 12 Bernstein polynomials and the first three eigenvalues converge to the exact results for n = 13 illustrated in Table 4. This confirms that the accuracy increases with the increased degree of polynomials. Our current method produces small errors in percentages (%) than those attained by FEM [31]. Plot of the relative errors between exact and approximate eigenvalues are depicted in Figure 2 for different values of n. Also, the relative errors obtained by Galerkin MWR for different degree of Bernstein polynomials are depicted in Figure 3. Table 5 shows that the maximum relative error for the smallest and the largest eigenvalues using present method are 10 −6 and 10 −7 respectively, whereas applying FEM [32], the errors obtained as 10 −3 , respectively. From the results in tabular form we conclude that the Galerkin MWR is much accurate, efficient and compatible with other techniques.

Conclusions
For the vibration of crankshaft computed results utilizing Galerkin MWR are very close to the analytical results [22] and error increased as the modulation depth closer to one. In the case of 1-D heat equation for the plane wall [32] and 2-D heat equation [31], the smallest errors by present method attains superior accuracy than finite element methods using linear and quadrilateral elements worked out in those studies. Eigenvalues achieved from 1D Helmholtz equation for smaller eigenvalues converge faster as the degree of Bernstein polynomials increases and attains many accurate results. For homogeneous rectangular membrane, relative errors in the present method are smaller than that of finite element method [32]. This proves that our present method is much more efficient than various numerical methods available in the literature and well suited which has much applicability in physical and engineering models.

Acknowledgements
The first author is very much thankful to Muhammad Sajjad Hossain, Assistant