Nonlinear integral equations

Nonlinear integral equations are a vital area of study in mathematics, offering applications in diverse fields such as physics, engineering, and biology. These equations, characterized by the presence of nonlinear terms involving the unknown function, are inherently more complex than their linear counterparts. Despite these challenges, recent advancements in analytical and numerical methods have made it possible to solve certain classes of nonlinear integral equations, though not all.

Unlike linear integral equations, the solutions to nonlinear integral equations are often not unique. However, under specific conditions, the existence of a unique solution can be ensured. Furthermore, there is a profound connection between integral equations and differential equations, as many nonlinear integral equations can be derived from differential equations. This relationship not only highlights their significance but also guides the development of solution techniques.

In general, nonlinear integral equations can take the form of Volterra or Fredholm integral equations, defined as follows:

1. Nonlinear Volterra Integral Equation:

   $$u(x) = f(x) + \lambda \int_0^x K(x, t) F(u(t)) \, dt$$

2. Nonlinear Fredholm Integral Equation:

   $$u(x) = f(x) + \lambda \int_a^b K(x, t) F(u(t)) \, dt$$

Here, $u(x)$ is the unknown function, $f(x)$ is a known function, $K(x, t)$ is the kernel, and $F(u(t))$ is a nonlinear function. If $F(u(t)) = u^n(t)$ for $n \geq 2$, the equation is nonlinear. Conversely, when $F(u(t)) = u(t)$, the equation reduces to a linear form.

For instance:

• A Volterra integral equation: $$u(x) = x + \lambda \int_0^x (x – t) u^2(t) \, dt$$
• A Fredholm integral equation: $$u(x) = x + \lambda \int_0^1 \cos(x) u^3(t) \, dt$$

These examples demonstrate the nonlinear nature of the equations, where the dependence on $u(t)$ introduces additional complexities in finding solutions.

The method of successive approximations, also known as the iterative method or Picard’s method (in the context of differential equations), is a mathematical technique used to approximate solutions to equations or problems where finding an exact solution is challenging. This method involves iteratively refining guesses or estimates of the solution to converge to the correct answer. It is widely used in solving equations, fixed-point problems, and certain types of differential equations.

Core Idea

The method relies on the idea of generating a sequence of approximations $(x_0, x_1, x_2, \dots)$ such that:

where $g(x)$ is a function that maps a previous guess $x_n$ to the next guess $x_{n+1}$.

Under certain conditions (e.g., if $g(x)$ is a contraction mapping), the sequence will converge to a fixed point $x^*$, which is a solution to the equation $g(x) = x$.

Mathematical Concepts Involved

1. Fixed-Point Theory:

   • A fixed point of a function $g(x)$ is a value $x^*$ such that $g(x^*) = x^*$.
   • The method of successive approximations is a tool to find these fixed points.

2. Contraction Mapping:

   • A function $g(x)$ is a contraction on a domain $D$ if there exists a constant $L$, with $0 \leq L < 1$, such that: $$|g(x) – g(y)| \leq L |x – y| \quad \forall x, y \in D.$$
   • If $g(x)$ is a contraction, the Banach fixed-point theorem guarantees that the sequence $(x_n)$ converges to a unique fixed point.

3. Initial Approximation:

   • The method starts with an initial guess $x_0$. The choice of $x_0$ can affect the speed of convergence but not the final solution if the method converges.

4. Error Analysis:

   • The error at the $n$-th step can often be bounded as: $$|x_n – x^*| \leq \frac{L^n}{1 – L} |x_1 – x_0|,$$ indicating exponential decay of the error when $L < 1$.

5. Iterative Form:

   • For specific types of problems, like differential equations, the method can be expressed as: $$y_{n+1}(x) = y_0(x) + \int_{x_0}^x f(t, y_n(t)) \, dt,$$ where $y_n(x)$ is the $n$-th approximation.

Applications

1. Solving Nonlinear Equations: For $f(x) = 0$, rewrite it as $x = g(x)$, then apply the iterative formula:

   $$x_{n+1} = g(x_n).$$

2. Differential Equations (Picard’s Iteration): Solve initial value problems of the form:

   $$y'(x) = f(x, y), \quad y(x_0) = y_0.$$

3. Numerical Methods: Used in numerical root-finding techniques like the Newton-Raphson method.

Examples

1. Solving a Nonlinear Equation

Solve $x^2 – 2 = 0$ using successive approximations.

1. Rewrite as $x = g(x)$, e.g., $g(x) = \frac{2}{x}$.
2. Start with an initial guess $x_0 = 1$.
3. Iteratively compute: $$x_{n+1} = \frac{2}{x_n}.$$ Example iterations:
   • $x_1 = \frac{2}{1} = 2$,
   • $x_2 = \frac{2}{2} = 1$,
   • $x_3 = \frac{2}{1} = 2$.

Here, the sequence oscillates. A better function $g(x)$ (e.g., $g(x) = \frac{x + 2/x}{2}$) would ensure convergence.

2. Solving a Differential Equation (Picard’s Method)

Solve $y'(x) = x + y$, $y(0) = 1$.

1. Start with $y_0(x) = 1$.
2. Define the iterative formula: $$y_{n+1}(x) = 1 + \int_0^x (t + y_n(t)) \, dt.$$
3. Compute:
   • $y_1(x) = 1 + \int_0^x (t + 1) \, dt = 1 + \left[\frac{t^2}{2} + t\right]_0^x = 1 + \frac{x^2}{2} + x$,
   • $y_2(x) = 1 + \int_0^x \left(t + \frac{t^2}{2} + t + 1\right) \, dt = 1 + \int_0^x \left(\frac{t^2}{2} + 2t + 1\right) dt = 1 + \frac{x^3}{6} + x^2 + x$.

The sequence converges to the exact solution $y(x) = e^x – x – 1$.

Advantages

1. Simple to implement.
2. Provides successive refinements for greater accuracy.
3. Can handle nonlinear problems effectively.

Limitations

1. Requires careful selection of the function $g(x)$ for convergence.
2. Convergence can be slow for some functions.
3. May diverge if the contraction condition is not satisfied.

Picard’s method is an iterative technique used to solve initial value problems for differential equations. The process involves converting the differential equation into an equivalent integral equation, then iteratively refining the solution by substituting the approximation from the previous step into the integral equation.

General Formulation

The first-order initial value problem:

can be transformed into an equivalent integral equation:

Picard’s method starts with an initial approximation, $u_0(x) = b$, and generates successive approximations:

where $n = 0, 1, 2, \dots$.

Under suitable conditions (e.g., $f(x, u)$ is continuous and satisfies a Lipschitz condition), the sequence of approximations $\{u_n(x)\}$ converges to the unique solution $u(x)$.

Example 4.1

Problem

Solve the initial value problem:

Solution

Transform into the integral equation:

• Zeroth Approximation:

  $$u_0(x) = 0.$$

• First Approximation: Substitute $u_0(x) = 0$ into the integral equation:

  $$u_1(x) = \int_0^x t \, dt = \frac{x^2}{2}.$$

• Second Approximation: Substitute $u_1(x) = \frac{x^2}{2}$ into the integral equation:

  $$u_2(x) = \int_0^x \left(t + \left(\frac{t^2}{2}\right)^2\right) \, dt = \frac{x^2}{2} + \frac{x^5}{20}.$$

• Third Approximation: Substitute $u_2(x) = \frac{x^2}{2} + \frac{x^5}{20}$ into the integral equation:

  $$u_3(x) = \int_0^x \left(t + \left(\frac{t^2}{2} + \frac{t^5}{20}\right)^2\right) \, dt.$$

  After expanding and integrating:

  $$u_3(x) = \frac{x^2}{2} + \frac{x^5}{20} + \frac{x^8}{160} + \frac{x^{11}}{4400}.$$

Continuing this process, the solution takes the form of a power series:

Example 4.2

Problem

Solve the coupled system:

with initial conditions $u(0) = 1$ and $v(0) = \frac{1}{2}$.

Solution

Transform into integral equations:

• First Approximation: Assume $u_0(x) = 1$ and $v_0(x) = \frac{1}{2}$. Substitute:

  $$u_1(x) = 1 + \int_0^x \frac{1}{2} \, dt = 1 + \frac{x}{2},$$ $$v_1(x) = \frac{1}{2} + \int_0^x t^3 \left(1 + \frac{1}{2}\right) \, dt = \frac{1}{2} + \frac{3x^4}{8}.$$

• Second Approximation: Substitute $u_1(x)$ and $v_1(x)$:

  $$u_2(x) = 1 + \int_0^x \left(\frac{1}{2} + \frac{3t^4}{8}\right) \, dt = 1 + \frac{x}{2} + \frac{3x^5}{40},$$ $$v_2(x) = \frac{1}{2} + \int_0^x t^3 \left(1 + \frac{t}{2} + \frac{3t^5}{40} + \frac{3t^4}{8}\right) \, dt.$$

Example 4.3

Problem

Solve the second-order nonlinear differential equation:

with $u(0) = 1$ and $u'(0) = 1$.

Solution

Convert to a system:

Integral equations:

• First Approximation:

  $$u_1(x) = 1 + \int_0^x 1 \, dt = 1 + x, \quad v_1(x) = 1 + \int_0^x 1 \, dt = 1 + x.$$

• Second Approximation: Substitute $u_1(x)$ and $v_1(x)$:

  $$u_2(x) = 1 + \int_0^x (1 + t) \, dt = 1 + x + \frac{x^2}{2},$$ $$v_2(x) = 1 + \int_0^x \frac{(1 + t)^2}{1 + t} \, dt = 1 + x + \frac{x^2}{2}.$$

Continuing, the solution converges to:

The Adomian decomposition method

References

1. Polyanin, A. D., & Manzhirov, A. V. (2008). Handbook of Integral Equations. CRC Press.
2. Brunner, H. (2004). Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press.
3. Atkinson, K. E. (1997). The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press.