Laplace Transforms and Convolutions Bernd Schr oder logo1 Bernd - PowerPoint PPT Presentation

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Laplace Transforms and Convolutions Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Everything Remains As It Was logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Original DE & IVP logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) L Original ✲ DE & IVP logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ Laplace transform of the solution logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ L − 1 Laplace transform ✛ of the solution logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization Everything Remains As It Was No matter what functions arise, the idea for solving differential equations with Laplace transforms stays the same. Time Domain ( t ) Transform domain ( s ) L Original Algebraic equation for ✲ DE & IVP the Laplace transform Algebraic solution, partial fractions ❄ L − 1 Laplace transform Solution ✛ of the solution logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization The Inverse Laplace Transform of a Product logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization The Inverse Laplace Transform of a Product 1. Solving initial value problems ay ′′ + by ′ + cy = f with Laplace transforms leads to a transform Y = F · R ( s )+ ··· . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization The Inverse Laplace Transform of a Product 1. Solving initial value problems ay ′′ + by ′ + cy = f with Laplace transforms leads to a transform Y = F · R ( s )+ ··· . 2. If the Laplace transform F of f is not easily computed or if the inverse transform of the product is hard, it would be nice to have a direct formula for the inverse transform of a product. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization The Inverse Laplace Transform of a Product 1. Solving initial value problems ay ′′ + by ′ + cy = f with Laplace transforms leads to a transform Y = F · R ( s )+ ··· . 2. If the Laplace transform F of f is not easily computed or if the inverse transform of the product is hard, it would be nice to have a direct formula for the inverse transform of a product. Maybe that way the transformation of f can be avoided. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization The Inverse Laplace Transform of a Product 1. Solving initial value problems ay ′′ + by ′ + cy = f with Laplace transforms leads to a transform Y = F · R ( s )+ ··· . 2. If the Laplace transform F of f is not easily computed or if the inverse transform of the product is hard, it would be nice to have a direct formula for the inverse transform of a product. Maybe that way the transformation of f can be avoided. 3. The convolution of the functions f ( t ) and g ( t ) is � t f ∗ g ( t ) = 0 f ( τ ) g ( t − τ ) d τ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization The Inverse Laplace Transform of a Product 1. Solving initial value problems ay ′′ + by ′ + cy = f with Laplace transforms leads to a transform Y = F · R ( s )+ ··· . 2. If the Laplace transform F of f is not easily computed or if the inverse transform of the product is hard, it would be nice to have a direct formula for the inverse transform of a product. Maybe that way the transformation of f can be avoided. 3. The convolution of the functions f ( t ) and g ( t ) is � t f ∗ g ( t ) = 0 f ( τ ) g ( t − τ ) d τ and L ( f ∗ g ) = FG . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions

Transforms and New Formulas Using Convolutions An Example Double Check Visualization The Inverse Laplace Transform of a Product 1. Solving initial value problems ay ′′ + by ′ + cy = f with Laplace transforms leads to a transform Y = F · R ( s )+ ··· . 2. If the Laplace transform F of f is not easily computed or if the inverse transform of the product is hard, it would be nice to have a direct formula for the inverse transform of a product. Maybe that way the transformation of f can be avoided. 3. The convolution of the functions f ( t ) and g ( t ) is � t f ∗ g ( t ) = 0 f ( τ ) g ( t − τ ) d τ and L ( f ∗ g ) = FG . 4. So it is possible to avoid transforming the forcing term, but the price we pay is that the solution is represented as an integral. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Laplace Transforms and Convolutions