Search
- https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
- https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/05%3A_Laplace_Transforms/5.05%3A_The_Convolution_TheoremFinally, we consider the convolution of two functions. Often, we are faced with having the product of two Laplace transforms that we know and we seek the inverse transform of the product.
- https://math.libretexts.org/Courses/Monroe_Community_College/MTH_225_Differential_Equations/08%3A_Laplace_Transforms/8.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/05%3A_The_Laplace_Transform/5.03%3A_ConvolutionThis page discusses the use of inverse Laplace transforms and convolution in solving ordinary differential and Volterra integral equations. It highlights the simplification of computations through the...This page discusses the use of inverse Laplace transforms and convolution in solving ordinary differential and Volterra integral equations. It highlights the simplification of computations through these methods. An example is provided where a differential equation involving an integral is transformed into the frequency domain, resulting in the expression \( X(s) = \dfrac{s-1}{s^2-2} \). The final solution is obtained as \( x(t) = \cosh(\sqrt{2}\, t) - \dfrac{1}{\sqrt{2}} \sinh(\sqrt{2}\,t) \).
- https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Reed)/07%3A_Laplace_Transforms/7.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
- https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/09%3A_Transform_Techniques_in_Physics/9.06%3A_The_Convolution_OperationIn the list of properties of the Fourier transform, we defined the convolution of two functions, f(x) and g(x) to be the integral (f∗g)(x). In some sense one is looking at a sum of the overlaps of on...In the list of properties of the Fourier transform, we defined the convolution of two functions, f(x) and g(x) to be the integral (f∗g)(x). In some sense one is looking at a sum of the overlaps of one of the functions and all of the shifted versions of the other function. The German word for convolution is faltung, which means "folding" and in old texts this is referred to as the Faltung Theorem. In this section we will look into the convolution operation and its Fourier transform.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/05%3A_The_Laplace_TransformThe Laplace transform can also be used to solve differential equations and reduces a linear differential equation to an algebraic equation, which can then be solved by the formal rules of algebra.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_420%3A_Differential_Equations_(Breitenbach)/09%3A_Laplace_Transforms/9.06%3A_Convolution\[\begin{aligned}{\cal L}|\sin t|&={1\over 1-e^{-s\pi}}\int_0^\pi e^{-st}|\sin t|\,dt \\[5pt] &={1\over 1-e^{-s\pi}}\int_0^\pi e^{-st}\sin t\,dt\\[5pt] &={1\over 1-e^{-s\pi}}\left(\int_0^\pi e^{-st}\s...\[\begin{aligned}{\cal L}|\sin t|&={1\over 1-e^{-s\pi}}\int_0^\pi e^{-st}|\sin t|\,dt \\[5pt] &={1\over 1-e^{-s\pi}}\int_0^\pi e^{-st}\sin t\,dt\\[5pt] &={1\over 1-e^{-s\pi}}\left(\int_0^\pi e^{-st}\sin t\,dt+\int_\pi^\infty e^{-st}(0)\,dt\right)\\[5pt] &={1\over 1-e^{-s\pi}}{\cal L}\left(\sin t-\sin t {\cal U}(t-\pi)\right)\\[5pt] &={1\over 1-e^{-s\pi}}{\cal L}\left(\sin t+\sin (t-\pi){\cal U}(t-\pi)\right)\\[5pt] &={1\over 1-e^{-s\pi}}\left({1\over s^2+1}+{e^{-\pi s}\over s^2+1}\right)\\[5pt]…
- https://math.libretexts.org/Courses/Red_Rocks_Community_College/MAT_2561_Differential_Equations_with_Engineering_Applications/08%3A_Laplace_Transforms/8.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/05%3A_The_Laplace_Transform/5.03%3A_Convolution/5.3E%3A_Exercises_for_Section_5.3This page covers exercises on convolution computations relevant to differential equations and integral solutions. It includes tasks like finding convolutions, solving linear differential equations wit...This page covers exercises on convolution computations relevant to differential equations and integral solutions. It includes tasks like finding convolutions, solving linear differential equations with initial conditions, and using convolution to derive answers with specific functions. The findings are presented through integrals or function forms, emphasizing the application of convolution in addressing differential equations and Laplace transforms.
- https://math.libretexts.org/Courses/Mission_College/Math_4B%3A_Differential_Equations_(Kravets)/07%3A_Laplace_Transforms/7.06%3A_ConvolutionThis section deals with the convolution theorem, an important theoretical property of the Laplace transform.
