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Mathematical Modeling and Computational Calculus Vol II - The Finite Difference Method
Now a Major Motion Picture .... well, how about a YouTube sequence of 20 VIDEOS, look for Mathematical Modeling and Computational Calculus I. In MMCC I we get lots of great results using Euler's method, which computes the solutions to ordinary differential equations after making the substitution df/dt = (f(t+dt) f(t))/dt, so that f(t+dt) = f(t) + dt×df/dt. Folks, that s all the calculus there is in MMCC I. Euler's method applied to partial differential equations is the finite difference method (FDM), and in MMCC II the FDM is used to analyze the following
Heat transfer
The wave equation
Stress and strain in materials
Fluid dynamics
Electro-magnetic radiation and Maxwell's equations Since the time of Newton the basic paradigm of mathematical physics and engineering has been the
1. Understand the physical laws governing the phenomena being studied.
2. Develop a differential equation model of the process.
3. Solve the differential equations. Ay, there's the rub.
4. Analyze the results The problem has always been step 3, as most differential equations do not have analytic solutions. We bypass all the difficulties of analytic calculus by using computational calculus, in our case the FDM, just as it's done in the real world. We follow the basic paradigm for each project in the book, starting with the physical laws, and deriving the differential equation model using baby steps, striving for intuitive clarity and transparency. The finite difference method substitutions are made for the derivatives in the model, giving a set of arithmetic expressions that are used to calculate results. A MATLAB/OCTAVE/FREEMAT program is written to implement the computations. The program is the thing. In the program everything in the model is made explicit and translated into additions, subtractions, multiplications, and divisions. There is no mystery in the program. It enables the student to see exactly how each component of the model affects other components and how it contributes to the total solution. In addition, by varying model and system parameters, the process can be studied for a variety of model characteristics, initial conditions and disturbance functions. The program gives the student a sense of complete mastery of the physical process and the mathematical analysis.
Heat transfer
The wave equation
Stress and strain in materials
Fluid dynamics
Electro-magnetic radiation and Maxwell's equations Since the time of Newton the basic paradigm of mathematical physics and engineering has been the
1. Understand the physical laws governing the phenomena being studied.
2. Develop a differential equation model of the process.
3. Solve the differential equations. Ay, there's the rub.
4. Analyze the results The problem has always been step 3, as most differential equations do not have analytic solutions. We bypass all the difficulties of analytic calculus by using computational calculus, in our case the FDM, just as it's done in the real world. We follow the basic paradigm for each project in the book, starting with the physical laws, and deriving the differential equation model using baby steps, striving for intuitive clarity and transparency. The finite difference method substitutions are made for the derivatives in the model, giving a set of arithmetic expressions that are used to calculate results. A MATLAB/OCTAVE/FREEMAT program is written to implement the computations. The program is the thing. In the program everything in the model is made explicit and translated into additions, subtractions, multiplications, and divisions. There is no mystery in the program. It enables the student to see exactly how each component of the model affects other components and how it contributes to the total solution. In addition, by varying model and system parameters, the process can be studied for a variety of model characteristics, initial conditions and disturbance functions. The program gives the student a sense of complete mastery of the physical process and the mathematical analysis.
156 pages, Spiral-bound
Published July 15, 2015
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