Numerical Solution: Ordinary & Partial Differential Equation from Udemy

The course provides an introduction to the numerical solution of ordinary and partial differential equations and is at a level appropriate for undergraduate-level STEM students. Prior knowledge of numerical methods is helpful but not necessary as (most) prerequisite material is introduced on an as-needed basis. Knowledge of a scientific programming language is necessary for those wishing to write their own codes. All codes used to demonstrate methods and solve example problems (primarily in both Fortran and Python) are available for downloading, as are the class notes. For the ordinary differential equations, we will study numerical techniques to solve:

1) Initial value (or propagation) problems

2) Boundary value (or equilibrium) problems

3) Eigenvalue (or characteristic value) problems

In terms of partial differential equations, we will concentrate on finite-difference approaches to solve second-order partial differential equations.

These equations may be classified as elliptic, parabolic, or hyperbolic. The classification helps determine the best approach to obtain a numerical solution. We will focus on elliptic and parabolic partial differential equations.

The primary course sections are:

SECTION 2: ODE’s: INITIAL VALUE PROBLEMS

SECTION 3: ODE’s: BOUNDARY VALUE PROBLEMS

SECTION 4: ODE’s: EIGENVALUE PROBLEMS

SECTION 5: ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS

SECTION 6: PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

What's inside

Learning objective

Basic numerical solution techniques for solving ordinary and partial differential equations.

Syllabus

Understand how to classify 2nd order partial differential equations with 2 independent variables.

Students will see some practical applications of numerical methods.

This lecture shows a student how to set up a ParaView case for the flow about a cylinder.

Which of the following is first order accurate in time and unconditionally stable.

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Provides downloadable codes in both Fortran and Python, which allows learners to implement numerical solutions and compare different programming approaches

Explores finite-difference approaches, which are fundamental techniques for numerically solving differential equations in various engineering and scientific applications

Covers initial value, boundary value, and eigenvalue problems, which are essential topics in the study of ordinary differential equations

Requires knowledge of a scientific programming language, which may pose a barrier to entry for students without prior coding experience

Focuses on elliptic and parabolic partial differential equations, which may not cover the full spectrum of PDE types encountered in some fields

Includes a challenging solid-to-liquid phase change programming assignment, which develops skills applicable to computational physics and materials science

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Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Numerical Solution: Ordinary & Partial Differential Equation with these activities:

Review Calculus Fundamentals

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Strengthen your understanding of calculus concepts, which are essential for grasping the underlying principles of differential equations.

Browse courses on Differential Equations

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Review differentiation and integration rules.
Practice solving basic differential equations.
Work through example problems involving limits and continuity.

Brush Up on Linear Algebra

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Revisit linear algebra concepts, particularly eigenvalues and eigenvectors, as they are crucial for solving eigenvalue problems in ODEs.

Browse courses on Eigenvalues

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Review matrix operations and solving linear systems.
Study eigenvalue and eigenvector calculations.
Practice applying linear algebra to solve differential equations.

Read 'Numerical Methods for Engineers' by Steven Chapra and Raymond Canale

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Supplement your learning with a comprehensive textbook on numerical methods, providing a broader perspective and additional examples.

View Numerical Methods For Engineers on Amazon

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Read relevant chapters on ODEs and PDEs.
Work through example problems in the book.
Compare the book's approach to the course material.

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Implement Euler's Method

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Reinforce your understanding of numerical methods by implementing Euler's method for solving initial value problems.

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Write code to implement Euler's method.
Test your code with various initial value problems.
Compare your results with analytical solutions.

Read 'Partial Differential Equations: An Introduction' by Walter Strauss

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Gain a deeper theoretical understanding of PDEs by studying a classic textbook on the subject.

View Partial Differential Equations: An Introduction... on Amazon

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Read chapters on elliptic and parabolic equations.
Focus on the theoretical aspects of the equations.
Relate the theory to the numerical methods learned in the course.

Visualize Solutions to the Heat Equation

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Deepen your understanding of parabolic PDEs by creating visualizations of solutions to the heat equation using different numerical methods.

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Solve the heat equation using explicit and implicit methods.
Generate visualizations of the temperature distribution over time.
Compare the results of different methods.

Solve the Laplace Equation on a Complex Domain

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Apply your knowledge of elliptic PDEs by solving the Laplace equation on a non-rectangular domain using finite difference methods.

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Discretize the domain using a finite difference grid.
Implement an iterative solver (e.g., SOR) to solve the resulting linear system.
Visualize the solution and analyze its properties.

Career center

Learners who complete Numerical Solution: Ordinary & Partial Differential Equation will develop knowledge and skills that may be useful to these careers:

Mathematical Modeler

Mathematical modelers create and analyze mathematical representations of real-world systems to understand and predict their behavior. This course helps build a foundation for this profession. The course's coverage of ordinary and partial differential equations, along with numerical techniques for solving them, provides the essential tools for building mathematical models. Understanding the classification of PDEs and the application of finite-difference methods are directly applicable to modeling various phenomena. You should take this course for its comprehensive coverage of numerical solutions to differential equations, which are fundamental in mathematical modeling.

See salaries and explore the career path for Mathematical Modeler

Computational Scientist

As a computational scientist, you apply numerical methods and modeling techniques to solve complex scientific problems. This course helps build a foundation in the numerical solution of ordinary and partial differential equations, skills that are essential for simulating physical phenomena and analyzing scientific data. The course's focus on finite-difference methods and the classification of partial differential equations into elliptic, parabolic, and hyperbolic types directly applies to the development and implementation of computational models. You should take this course to help build a foundation in the numerical techniques necessary for this role.

See salaries and explore the career path for Computational Scientist

Climate Modeler

A climate modeler develops and uses computer models to simulate the Earth’s climate system, often requiring an advanced degree. This Numerical Solutions course helps prepare you for the tasks of a climate modeler, as it provides the numerical techniques necessary for solving the complex partial differential equations that govern atmospheric and oceanic processes. A deep understanding of finite-difference methods and the classification of PDEs is crucial for accurately simulating climate phenomena. The focus on elliptic and parabolic PDEs is particularly relevant for modeling heat transfer and fluid dynamics in the climate system. You should take this course to gain the expertise in numerical methods used for climate modeling.

See salaries and explore the career path for Climate Modeler

Simulation Engineer

A simulation engineer develops and uses computer models to simulate physical systems and processes. This course helps build a foundation for this career, as it provides an introduction to the numerical solution of ordinary and partial differential equations, crucial for creating accurate and efficient simulations. The study of finite-difference approaches and the classification of PDEs are highly relevant to designing and implementing simulation software. The course's emphasis on elliptic and parabolic partial differential equations perfectly aligns with many real-world engineering problems. You should take this course to gain expertise in numerical techniques essential for simulation engineering.

See salaries and explore the career path for Simulation Engineer

Reservoir Engineer

Reservoir engineers manage the development and production of oil and gas reservoirs. This course helps build a foundation for a career as a reservoir engineer, as it provides the numerical techniques necessary for simulating fluid flow in porous media, heat transfer in reservoirs, and multiphase flow processes. A strong grounding in finite-difference methods and the solution of partial differential equations is crucial for accurately simulating reservoir behavior. The course's focus on elliptic and parabolic PDEs is particularly relevant for modeling pressure distribution and fluid saturation changes in reservoirs. You should take this course to gain expertise in numerical methods used in reservoir simulation.

See salaries and explore the career path for Reservoir Engineer

Quantitative Analyst

A quantitative analyst develops and implements mathematical models for financial analysis and risk management, often requiring an advanced degree. This course helps build a foundation for designing these models, as it provides expertise in solving ordinary and partial differential equations numerically. The course’s focus on initial value problems, boundary value problems, and eigenvalue problems is directly applicable to pricing derivatives and managing risk. The emphasis on finite-difference methods is extremely helpful for implementing these models computationally. You should take this course to enhance your skills in numerical techniques essential for quantitative finance.

See salaries and explore the career path for Quantitative Analyst

Hydrologist

Hydrologists study the movement, distribution, and quality of water. This course helps build a foundation for a career in hydrology, as it provides the numerical techniques necessary for simulating groundwater flow, surface water dynamics, and contaminant transport. The material on finite-difference methods and the solution of partial differential equations is directly applicable to modeling hydrological processes. The focus on elliptic and parabolic PDEs is particularly relevant for simulating steady-state and transient flow conditions. You should take this course to gain expertise in numerical methods used in hydrology.

See salaries and explore the career path for Hydrologist

Aerospace Engineer

Aerospace engineers design, develop, and test aircraft and spacecraft. This course builds a foundation for a career in aerospace engineering, as it provides the numerical techniques necessary for simulating fluid dynamics, heat transfer, and structural analysis. The course's emphasis on finite-difference methods and the solution of partial differential equations is directly applicable to solving complex engineering problems in the aerospace industry. The focus on elliptic and parabolic PDEs is particularly relevant for computational fluid dynamics simulations, essential in aircraft design. You should take this course to gain expertise in numerical methods used to simulate such systems.

See salaries and explore the career path for Aerospace Engineer

Mechanical Engineer

Mechanical engineers design, develop, and test mechanical devices and systems. This course may be useful for mechanical engineers, especially those involved in simulation and analysis. The course's coverage of ordinary and partial differential equations, along with numerical methods for solving them, provides the tools needed to model and analyze mechanical systems. The focus on finite-difference approaches is relevant to simulating heat transfer, fluid flow, and structural mechanics. You should take this course to enhance your skills in numerical modeling and simulation, which are valuable in mechanical engineering.

See salaries and explore the career path for Mechanical Engineer

Research Scientist

A research scientist conducts experiments and develops theories to advance knowledge in a specific field. This course may be useful, as it introduces the numerical solution of ordinary and partial differential equations, which are often encountered in scientific research. The course's coverage of initial value, boundary value, and eigenvalue problems, as well as finite-difference methods, are relevant to the numerical modeling and simulation aspects of research. You should take this course to enhance your skills in solving complex mathematical problems that arise in scientific research.

See salaries and explore the career path for Research Scientist

Civil Engineer

Civil engineers design, construct, and maintain infrastructure projects. This course may be useful in a civil engineering career, as it introduces numerical methods for solving ordinary and partial differential equations, relevant in structural analysis, heat transfer, and fluid dynamics. The course's coverage of finite-difference methods and the classification of PDEs can be applied to modeling and simulation of civil engineering systems. You should take this course to improve your skills in numerical techniques for solving engineering problems.

See salaries and explore the career path for Civil Engineer

Data Scientist

A data scientist uses computational and mathematical techniques to extract insights and knowledge from data. This course may be useful as it focuses on the numerical solution of ordinary and partial differential equations, which are relevant in modeling and simulation. The understanding of numerical methods and techniques for solving initial value, boundary value, and eigenvalue problems can be applied to data analysis and predictive modeling. You should take this course to gain a deeper understanding of the mathematical foundations behind many data science algorithms.

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Biomedical Engineer

Biomedical engineers apply engineering principles to solve medical and healthcare-related problems. This course may be useful, assisting in the study of numerical solutions of ordinary and partial differential equations, which apply to modeling physiological systems, simulating drug delivery, and analyzing medical images. The course's coverage of finite-difference methods and the classification of PDEs is useful in biomedical modeling and simulation. You should take this course to improve your quantitative skills useful for biomedical engineering applications.

See salaries and explore the career path for Biomedical Engineer

Software Engineer

Software engineers design, develop, and test software applications. This course may be useful to software engineers, especially those working on scientific or engineering software. The course's focus on numerical methods for solving ordinary and partial differential equations provides the mathematical foundation needed to develop simulation and modeling tools. The availability of codes in Fortran and Python is particularly helpful for implementing these methods in software. You should take this course to enhance your knowledge of numerical techniques, which is beneficial in developing specialized software applications.

See salaries and explore the career path for Software Engineer

Financial Analyst

As a financial analyst, you analyze financial data, provide investment recommendations, and manage financial risk. This course may be useful as it helps build a foundation in the numerical solution of ordinary and partial differential equations, skills that are essential for pricing derivatives, modeling financial markets, and forecasting economic trends. The focus on initial value problems and parabolic partial differential equations directly applies to modeling stochastic processes and option pricing. The course can enhance your ability to develop and implement sophisticated financial models. You should take this course to help build a foundation in the numerical techniques necessary for quantitative finance.

See salaries and explore the career path for Financial Analyst

Numerical Solution

Ordinary & Partial Differential Equation

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