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1510 - Differential Equations in MATLAB
Differential Problems in MATLAB 1. What Equations Can MATLAB Handle? 2. Where Can I Find Tutorials or Additional Information? Frequently Asked Questions 3. What Changes in Syntax Exist for ODE Solvers? 4. How Do I Reduce the Order of an ODE? 5. How Do I Solve Time-Dependent ODEs? 6. How Do I Use a Fixed Time Step? 7. How Do I Use Stochastic Differential Equations? Examples 8. Systems of Equations 9. Boundary Value Problem (BVP): Channel Flow Stiffness 10. What Is Stiffness? 11. Implicit vs. Explicit Methods 12. Examples Options 13. How Can I Change Options When Solving a Differential Equation? 14. What Option Parameters Can Be Modified? 15. How Can Options Be Used as Functions? Differential-Algebraic Equations and their Index 16. How Can Differential Algebraic Equations Systems Be Solved in MATLAB?
Section 1: What Equations Can MATLAB Handle?
MATLAB provides functions for solving several classes of problems involving differential equations: 1. Initial Value Problems for Ordinary Differential Equations (ODEs) This is the most popular type of problems solved using MATLAB ODE solvers. Initial value problems are typically solved with ODE45 for the nonstiff case, and with ODE15S in the stiff case. (For an explanation of Stiffness, refer to the section "What is Stiffness".) 2. Initial Value Problems for Differential Algebraic Equations (DAEs) These are frequently encountered in areas where conservation laws dictate a constant relationship between some variables. MATLAB can solve DAE's of index 1 using ODE15S or ODE23T. (For an explanation of index, refer to the section DAEs and their index.) 3. Boundary Value Problems (BVPs) These consist of differential equations with conditions specified on both sides. While not encountered as frequently as IVP's, these are still a common problem in engineering applications. They can be solved using the function BVP4C. 4. Delay Differential Equations (DDEs) These differential equations involve delays in the independent variable. They are frequently encountered in a variety of applications such as biological and chemical modeling, and can be solved using the function DDE23. 5. Partial Differential Equations (PDEs) Initial-boundary-value problems for systems of parabolic and elliptic differential equations in one spatial dimension and time can be solved using PDEPE. The PDE Toolbox is available for those interested in solving more general classes of PDEs. For more information regarding general integration techniques using MATLAB, see Solution 8314 . For more information on the algorithms for the various solvers in MATLAB, see the following URLs: ODE functionality BVP functionality DDE functionality PDE functionallity
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The MathWorks - Support - Differential Equations in MATLAB
Section 2: Where Can I Find Tutorials or Additional Information? A series of papers and tutorials available on MATLAB Central, our newsgroup and file exchange site, further explain the algorithms and usage of the MATLAB solvers for each type of equations (ODE,DAE,BVP,DDE). The papers and tutorials come with a variety of examples for you to download for each type of equations. The tutorials are found under the category Mathematics | Differential Equations. Chapter 7 of Numerical Computing with MATLAB, by Cleve Moler, discusses ODEs in additional detail and includes many examples and sample problems.
Section 3: What Changes in Syntax Exist for ODE Solvers?
The preferred syntax when using ODE solvers in MATLAB 6.5 (R13) is...