Numerical Approximation Methods Quiz

Differential equations are prominent in applications of engineering, physics, and even models related to chemical, biological, and social processes.

The concept of the differential equation began to develop more strongly at the beginning of the sixteenth century.

Mathematical modeling is a powerful tool for testing hypotheses, confirming experiments, and simulating the dynamics of complex systems as well as helping to understand the mechanistic underpinnings of complex systems in a relatively quick time without the huge costs of laboratory experiments and biological changes, particularly in oncology.

The tumor is a mass of tissue formed by the division of cells at an accelerated rate.

Solving mathematical models is a complex task, and it involves a diverse array of techniques, including Adomian decomposition methods, successive approximation, variational iteration, homotopy analysis, and several other methods.

Integral transformations, such as the Laplace transform, are powerful tools used in solving differential equations.

It transforms differential equations and integral equations into polynomials, facilitating their solution.

It breaks down mathematical equations into fundamental patterns and provides approximations to solutions through the Adomian expansion.

The initial approximation is known as the "zero approximation".

The variational iteration method is widely used to tackle a wide range of nonlinear equations.

It employs a continuous transformation parameter to create convergent series solutions.

Integral transformations, like the Laplace transform, can convert differential equations and integral equations into more solvable forms.

Equation (1.2) represents the integration of both sides of equation (1.1) with respect to s on the interval (0, t), which gives a solution for u(x, t) in terms of an integral.

The series of solutions given by equations (1.3) to (1.7) represent successive approximations to the exact solution of equation (1.1). Each approximation is obtained by substituting the previous approximation into the integral in equation (1.2).

The limit expression states that as the number of iterations (n) approaches infinity, the approximation u_n(x, t) converges to the exact solution u(x, t) of the original differential equation (1.1) within the specified domain.

Equation (1.8) represents a general differential equation of the form L(x, t) + N(v(x, t)) = h(x, t), where L is a linear operator, N is a nonlinear operator, and h(x, t) is an inhomogeneous term.

The variational iteration method (VIM) is introduced as a technique for solving the general differential equation given in equation (1.8), which includes both linear and nonlinear terms, as well as an inhomogeneous term.

The iterative approximation process described by equations (1.3) to (1.7) is an application of the variational iteration method (VIM) to the specific differential equation (1.1). The VIM provides a general framework for constructing successive approximations that converge to the solution of a given differential equation.