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Engineers are researching solutions to resolve many of today's technical challenges. Numerical techniques are used to solve the mathematical models in engineering problems. Many of the mathematical models of engineering problems are expressed in terms of Boundary Value Problems, which are partial differential equations with boundary conditions. Two of the most popular techniques for solving partial differential equations are the Finite Element Method and the Finite Difference Method. In the last few decades another numerical technique has been increasingly used to solve mathematical models in engineering research, the B-spline Collocation Method. A collocation method involves satisfying a differential equation to some tolerance at a finite number of points, called collocation points. The B-spline Collocation Method does have a few distinct advantages over the Finite Element and Finite Difference Methods. The advantage over the Finite Difference Method is that the B-spline Collocation Method efficiently provides a piecewise-continuous, closed form solution. An advantage over the Finite Element Method is that the B-spline Collocation Method procedure is simpler and it is easy to apply to many problems involving partial differential equations. Although there are some advantages to using the B-spline Collocation Method, there are also disadvantages. The main disadvantage of the B-spline Collocation Method compared to the Finite Element Method is that the Finite Element Method is better for computation where complex geometries are involved. The B-spline Collocation Method is suitable for use with standard geometries, like rectangles. The B-spline Collocation Method has been used in fluid flow problems with a great deal of success, but has not been used to solve Mechanics of Materials type problems.
The current research involves developing, and extensively documenting, a comprehensive, step-by-step procedure for applying the B-spline Collocation Method to the solution of Boundary Value problems. The simplicity of this approximation technique makes it an ideal candidate for computer implementation. Therefore, a symbolic Matlab code was developed, that calculates and plots everything necessary to apply this technique to a wide variety of boundary value problems. In addition, the current research involves applying the B-spline Collocation Method to solve the mathematical model that arises in the deflection of a geometrically nonlinear, cantilevered beam. The solution is then compared to a known solution found in the literature.