Derivation of Laplace Transformation Formula

Modified Successive Approximations Method (LT-MSAM)

It quickly approaches the exact solution using a few iterations.

Vn+1 = Vn - λ∫{R(Vn-Vn-1)+(Gn-Gn-1)}ds

<p>V0 = f(x)</p> Signup and view all the answers

<p>To reach the exact solution with high efficiency using few iterations.</p> Signup and view all the answers

<p>N𝑉𝑛(𝑥,𝑡) = Gn(x, t) + O(t^n+1)</p> Signup and view all the answers

<p>Successive approximations and variational iteration methods</p> Signup and view all the answers

<p>Consider the partial differential equation $Lu(x,t)+Ru(x,t)+ Nu(x,t) = g(x,t)$.</p> Signup and view all the answers

<p>$U_n=B_n^0 + B_n^1t + B_n^2t^2 + ... + B_n^nt^n + B_n^{n+1}t^{n+1} + B_n^{n+2}t^{n+2} + (t^{n+3})$</p> Signup and view all the answers

<p>The method of successive approximations with Laplace transform is derived to solve the ATG system.</p> Signup and view all the answers

<p>To find solutions based on the previous derivations in order to solve the ATG system.</p> Signup and view all the answers

<p>Reduction in calculations, time, and effort through modifications and improvements.</p> Signup and view all the answers

<p>Understanding nonlinear partial differential equation models plays an important role in mathematics and physics as well as in applied fields such as the aerospace industry, oceanography, and meteorology.</p> Signup and view all the answers

<p>The successive approximation method, also called the Picard iteration method, is one of the important and classic methods used to solve partial differential equations, integral equations, and initial value problems. Combining this method with the Laplace transform facilitates the process of solving systems of nonlinear partial differential equations efficiently and with high accuracy.</p> Signup and view all the answers

<p>The variational iteration method is characterized by its effectiveness in dealing with linear and non-linear problems, as it provides an analytical solution to a wide range of problems in applied sciences. Moreover, it reduces the volume of calculations, and the process of integrating the Laplace transform with this method makes it easier to find the Lagrange multiplier. This means that using this method does not require special knowledge of integral and differential calculus, as this combination gives results with high accuracy and few repetitions.</p> Signup and view all the answers

<p>Combining the successive approximation method with the Laplace transform facilitates the process of solving systems of nonlinear partial differential equations efficiently and with high accuracy.</p> Signup and view all the answers

<p>The variational iteration method is characterized by its effectiveness in dealing with linear and non-linear problems, as it provides an analytical solution to a wide range of problems in applied sciences. Moreover, it reduces the volume of calculations, and the process of integrating the Laplace transform with this method makes it easier to find the Lagrange multiplier. This means that using this method does not require special knowledge of integral and differential calculus, as this combination gives results with high accuracy and few repetitions.</p> Signup and view all the answers