NUMERICAL SOLUTIONS OF SOME PARTIAL DIFFERENTIAL EQUATIONS USING B-SPLINE

Abstract

Collocation method is an emerging popular technique to solve initial and boundary value problems. It was developed to seek an approximate solution of di erential equation in the form of linear combination of basis functions. The idea is to choose a nite dimensional space of candidate solution (usually, polynomials up to a certain degree) and a number of points in the domain (called collocation points), and to select that solution which satis es the given equation at the collocation points. Di erential quadrature method (DQM) is a higher order numerical discretization technique for solving di erential equations. DQM can provide the solution with a higher level of accuracy and with less computational e ort. It has been also pointed out that the DQM is basically equivalent to the collocation (pseudo-spectral) method, in fact, DQM directly compute the functional value at the grid points rather than spectral variables. In this method, determination of the weighting coe cients is the key procedure which is of paramount importance. One of the advantage of this method is that it satis es varieties of boundary conditions and require much less formulation and programming e ort. Moreover, the mathematical techniques involved in the method are also not so sophisticated. And therefore, they are more explicit and simple for some practical applications and especially advantageous for nonlinear problems. So the DQM could be easily learned and successfully applied in the varieties of problems originated in the applied sciences. In this research, we have developed collocation and di erential quadrature methods with B-spline functions to solve linear/nonlinear partial di erential equations (PDEs). The use of cubic B-spline basis functions in getting the numerical solutions of some partial di erential equations is shown to provide an easy and simple algorithm. Strong stability preserving Runge-Kutta (SSP-RK) methods of di erent stages and order are also combined with these methods. In case of nonlinear PDEs, the numerical solutions i ii can be obtained without using any transformation and linearization process of the equation. Therefore, the equations are solved more easily and elegantly using the developed techniques. These methods are simple and easy to use in comparison to other existing methods, e.g. nite element, nite volume and spectral methods, etc. All the chapters include several examples to demonstrate the applicability and e ciency of the presented methods. The chapter wise summary of the thesis is as follows: Chapter 1 is preface which contains some relevant de nitions, introduction to numerical techniques like the nite di erence method, nite element method and existing literature review. B-spline functions of various degree are drawn out from the recursive formula. Some of the signi cant properties of B-spline functions are also discussed. Afterwards a brief introduction on B-spline functions, it contributes an introduction to collocation method, di erential quadrature method and their execution process to solve linear/ nonlinear PDEs. Strong stability preserving Runge-Kutta methods of various stages and orders with their signi cant attributes are also brie y talked about. The formulae for computing error norms and order of convergence are also discussed. Chapter 2 deals with the numerical solutions of nonlinear Klein-Gordon equation and coupled Klein-Gordon-Schr odinger equation with Dirichlet and Neumann boundary conditions. One dimensional Klein-Gordon equation is given by utt + uxx + g(u) = f(x; t); x 2 (a; b); t > 0; with appropriate initial and boundary conditions. The parameter < 0 is a known real constant, f(x; t) is known analytic function and g(u) is a nonlinear force which may takes many forms such as: sin u; sinh u; sin u + sin 2u; sinh u + sinh 2u. The nonlinear Klein-Gordon equation describes a variety of physical phenomena such as dislocations, ferroelectric and ferromagnetic domain walls, DNA dynamics, and Josephson junctions. Yukawa-coupled Klein-Gordon-Schr odinger (KGS) equation is given by i t = 􀀀 1 2 xx 􀀀 ; tt = xx 􀀀 + j j2; x 2 R; t > 0; iii with suitable initial and boundary conditions. Here (x,t) is a complex function represents a scalar neutron eld and (x,t) is a real function represents a scalar neutral meson eld. The model describes the interaction between conservative complex neutron eld and neutral meson Yukawa in quantum eld theory and plays an important role in quantum physics. Numerical solutions of both the equations are obtained using cubic B-spline collocation method. Modi ed cubic B-spline basis functions are used to handle the Dirichlet boundary conditions. The equations are decomposed into a system of partial di erential equations, which is further converted to an amenable system of ordinary di erential equations (ODEs). The obtained system of ODEs is solved by SSP-RK54 scheme. Numerical results are presented for six examples, to show the accuracy and utility of proposed approach. The approximate solutions of both the equations are computed without using any transformation and linearization process. The computed results are of better accuracy than earlier results available in the literature. The execution of this method is very easy and cost-e ective. A portion of this chapter has been published in International Journal of Computer Mathematics (2014). Chapter 3 addresses the modi ed cubic B-spline collocation method to nd the numerical solution of nonlinear sine-Gordon equation with Dirichlet boundary conditions. One dimensional sine-Gordon equation turn out in many di erent applications such as propagation of uxion in Josephson junctions, di erential geometry, stability of uid motion, nonlinear physics and applied sciences. We consider one-dimensional nonlinear sine-Gordon equation utt = uxx 􀀀 sin(u); x 2 (a; b); t > 0; with suitable initial and boundary conditions. The method is based on collocation of modi ed cubic B-splines over nite elements so that the continuity of the dependent variable and its rst two derivatives throughout the solution range is preserved. The sine-Gordon equation is converted into a system of partial di erential equations. Using modi ed cubic B-spline collocation method, we iv obtain a system of rst order ordinary di erential equations. Finally obtained system of ODEs is solved by SSP-RK54 scheme. The particular feature of SSP-RK scheme is that, it inherently perpetuates certain stability properties and maximum norm stability. It also controls spurious oscillations and non-linear instability during simulation. In terms of computational cost, SSP-RK schemes have drawn the same cost as traditional ODE solvers. To demonstrate the accuracy and usefulness of present scheme, four numerical examples are presented. The obtain results are of better precision and competent accuracy than the results available in the earlier works. The order of convergence of the scheme is also computed and found to be approaching two. A part of this chapter has been published in International Journal of Partial Differential Equations (2014). Chapter 4 is concerned with the numerical solution of one dimensional hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions, using cubic B-spline collocation method. The one dimensional hyperbolic telegraph equation is given by utt(x; t) + 2 ut(x; t) + 2u(x; t) = uxx(x; t) + f(x; t); x 2 (a; b); t > 0; where and are known real constants. For > 0, = 0 it represents a damped wave equation and for > > 0 it is called as telegraph equation. The method is based on collocation of cubic B-spline basis functions over nite elements. Modi ed cubic B-spline basis functions are used to handle the Dirichlet boundary conditions. The use of B-spline basis functions for spatial variable and its derivatives, results in an amenable system of di erential equations. The resulting system of equations is solved by SSP-RK54 scheme. Stability of scheme is discussed using matrix stability analysis and found unconditionally stable. The e cacy of approach is con rmed with four numerical experiments and the numerical results are found to be very good in comparison with the existing solutions found in the literature. The advantage of this scheme is that, it can be conveniently use to solve the complex problems and also capable of reducing the size of computational work. First part of this chapter has been published in Applied Mathematics and Computation (2013). Second part of this chapter has been published in International v Journal of Computational Mathematics (2014). In chapter 5 we have proposed an e cient di erential quadrature method, to nd the numerical solution of two dimensional hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions. The hyperbolic partial di erential equations have signi cant role in formulating fundamental equations in atomic physics and are also very useful in understanding various phenomena in applied sciences like engineering, industry, aerospace as well as in chemistry and biology too. Consider utt(x; y; t) + 2 ut(x; y; t) + 2u(x; y; t) =uxx(x; y; t) + uyy(x; y; t) + f(x; y; t); (x; y; t) 2 [a; b] [c; d] (0; T]; with appropriate initial and boundary conditions. Here , are known real constants. For > 0, = 0, it represents a damped wave equation and for > 0, > 0, it is called telegraph equation. In order to nd the numerical solutions, modi ed cubic B-spline basis functions based di erential quadrature method is developed. The equation is converted into a system of partial di erential equations and further reduced into a system of ordinary di erential equations using DQM. SSP-RK43 scheme is used to solve the obtained system of ODEs. By employing DQM, accurate solutions can be obtained using fewer grid points in spatial domain. The stability of the scheme is studied using matrix stability analysis and found to be unconditionally stable. The e cacy of proposed approach is con rmed with seven numerical experiments, where comparisons are made with some earlier works. It is observed that the obtained results are acceptable and are in good agreement with earlier studies. However, we obtain these results in much less CPU time. The method is very simple, e cient and produces very accurate numerical results in considerably smaller number of nodes and hence saves computational e ort. A part of this chapter has been published in Applied Mathematics and Computation (2014). vi Chapter 6 discusses the application of modi ed cubic B-spline di erential quadrature method to nd the numerical solutions of some nonlinear wave equations in one and two dimensions with Dirichlet boundary conditions. We consider utt = uxx + f(x; t; u; ux; ut); x 2 (a; b); t > 0; utt = uxx + uyy + f(x; y; t; u; ux; uy; ut); (x; y; t) 2 [a; b] [c; d] (0; T]; with suitable initial and boundary conditions. Here f is some nonlinear expression in terms of u, ux, ut, uy. Nonlinear wave equations are arise in many physical and engineering applications such as continuum physics, mixed models of transonic ows, uid dynamics and many other elds of science and engineering. To obtain the numerical solutions, above equations are decomposed into a system of partial di erential equations. Modi ed cubic B-spline basis functions based di erential quadrature method is used for space discretization to obtain a system of nonlinear rst order ordinary di erential equations. The resulting system of equations is solved using SSP-RK43 scheme. In numerical testing, the method is implemented on Vander pole type nonlinear wave equation, Dissipative nonlinear wave equation and Telegraph equation. The obtained numerical results are found to be very good in comparison with the existing solutions found in the literature. The numerical solutions of nonlinear equations are computed without linearizing the nonlinear term. The order of convergence of method is also computed and found to be two. A part of this chapter has been published in the proceeding of 3rd International Conference on Advances in Computing, Communications and Informatics (ICACCI 2014)(IEEE Xplore). Chapter 7 presents the numerical solution of two dimensional nonlinear coupled Burgers' equation @u @t + u @u @x + v @u @y = 1 R

@2u @x2 + @2u @y2

@v @t + u @v @x + v @v @y = 1 R

@2v @x2 + @2v @y2

(x; y; t) 2 [a; b] [c; d] (0; T]; where R is Reynolds number. This system models a large number of physical phenomena such as tra c ow, ow of a shock wave traveling in a viscous uid, phenomena of vii turbulence, interaction between the non-linear convection process and the di usive viscous process, sedimentation of two kinds of particles in uid suspensions under the e ect of gravity. Modi ed cubic B-spline di erential quadrature method (MCB-DQM) is used to discretized the spatial derivatives of coupled Burgers' equation and reduces it into a system of rst order ordinary di erential equations. The obtained system of equations is solved by SSP-RK43 scheme. The accuracy of the approach is tested on ve test problems and computed results are compared with some earlier works. The results indicate that MCB-DQM combined with SSP-RK scheme gives more accurate results than earlier works with less computational cost. Numerical results are computed for higher Reynolds number up to R = 1500. The strong points of the method are in ease to apply and less computational e ort. Finally, in chapter 8 conclusions are drawn based on the present study and future research work is suggested, in this direction.