Extension of Laplace transform–homotopy perturbation method to solve nonlinear differential equations with variable coefficients defined with Robin boundary conditions

Abstract

This article proposes the application of Laplace transform–homotopy perturbation method with variable coefficients, in order to find analytical approximate solutions for nonlinear differential equations with variable coefficients. As case study, we present the oxygen diffusion problem in a spherical cell including nonlinear Michaelis–Menten uptake kinetics. It is noteworthy that this important problem introduces the Robin boundary conditions as an additional difficulty. In fact, after comparing figures between approximate and exact solutions, we will see that the proposed solutions are highly accurate. What is more, we will see that the square residual error of our solutions is 1.808511632 × 10⁻⁷ and 2.560574954 × 10⁻¹⁰ which confirms the accuracy of the proposed method, taking into account that we will just keep the first-order approximation.

Appendix

Laplace transform of $F(t)$ is denoted by $\Im \left\{ {F(t)} \right\}$ and is defined by the integral [1]

$$\Im \left\{ {F(t)} \right\} = f(s) = \int\limits_{0}^{\infty } {e^{ - st} } F(t){\text{d}}t.$$

(45)

Linearity of L.T. is an important property, which is expressed as

$$\Im \left\{ {c_{1} F_{1} (t) + c_{2} F_{2} (T)} \right\} = c_{1} f_{1} (s) + c_{2} f_{2} (s),$$

(46)

where $c_{1}$ and $c_{2}$ are constants, and we have denoted: $\Im \left\{ {F_{1} (t)} \right\} = f_{1} (s)$, $\Im \left\{ {F_{2} (t)} \right\} = f_{2} (s)$.

Some known properties of L.T. widely employed in this study are

$$1. \quad \Im \left\{ 1 \right\} = \frac{1}{s}\,(s > 0)$$

(47)

$$2. \quad \Im \left\{ {t^{n} } \right\} = \frac{n!}{{s^{n + 1} }}\,(s > 0)$$

(48)

$$\begin{aligned}3. \quad \Im \left\{ {F^{(n)} (t)} \right\} &= s^{n} f(s) - s^{n - 1} F(0) - s^{n - 2} F^{\prime}(0) - \cdots\\ & \quad- F^{(n - 1)} (0), \end{aligned}$$

(49)

where $F^{(n)} (t)$ denotes the nth derivative of $F(t)$ and $\Im \left\{ {F(t)} \right\} = f(s).$

$$\begin{aligned} 4. \quad \Im \left\{ {t^{n} F(t)} \right\} &= ( - 1)^{n} \frac{{{\text{d}}^{n} f(s)}}{{{\text{d}}t^{n} }},\\ &\quad\;n\;{\text{denotes}}\;{\text{a}}\;{\text{positive}}\;{\text{integer}}.\end{aligned}$$

(50)

If L.T. of $F(t)$ is $f(s)$, then $F(t)$ is called the inverse L.T. of $f(s)$ and is expressed by $F(t) = \Im^{ - 1} \left\{ {f(s)} \right\}$, where $\Im^{ - 1}$ is called the inverse L.T. operator.

From Eqs. (47) and (48), it is clear that

$$1 = \Im^{ - 1} \left( {\frac{1}{s}} \right),$$

(51)

$$t^{n} = \Im^{ - 1} \left( {\frac{n!}{{s^{n + 1} }}} \right),$$

(52)

and so on.

The following important result is obtained from (46) and denotes the linearity property of $\Im^{ - 1}$

$$\Im^{ - 1} \left\{ {c_{1} f_{1} (s) + c_{2} f_{2} (s)} \right\} = c_{1} F_{1} (t) + c_{2} F_{2} (T).$$

(53)