#### 3.1. An Approximate Solution of Linear Hypersingular Integral Equations with Second Order Singularity

Consider a one-dimensional hypersingular integral equation of the type

where

$f(t)=g(t)/(t-c)$ or

$f(t)=g(t)/((1-{t}^{2})(t-c)),\phantom{\rule{4pt}{0ex}}-1<c<1,g(t)\in C[-1,1].$Divide the interval $[-1,1]$ into two subintervals $[-1,c]$, $[c,1]$.

Let us fix a positive integer ${N}_{0}$. Put $h=2/{N}_{0},{N}_{1}=\lceil (1+c)/h\rceil ,{N}_{2}=\lceil (1-c)/h\rceil ,N={N}_{1}+{N}_{2}.$

Divide the interval $[-1,c]$ into ${N}_{1}$ subintervals at the points ${t}_{k}=-1+(c+1)k/{N}_{1},$ $k=0,1,\dots ,{N}_{1}$.

Divide the interval $[c,1]$ into ${N}_{2}$ subintervals at the points ${\tau}_{k}=c+(1-c)k/{N}_{2},$ $k=0,1,\dots ,{N}_{2}$.

Let us introduce the nodes ${\overline{t}}_{0}={t}_{0}+1/2{({N}_{1})}^{2},{\overline{t}}_{k}={t}_{k},k=1,2,\dots ,{N}_{1}-1,{\overline{t}}_{{N}_{1}}={t}_{{N}_{1}}-1/2{({N}_{1})}^{2};{\overline{\tau}}_{0}={\tau}_{0}+1/2{({N}_{2})}^{2},{\overline{\tau}}_{k}={\tau}_{k},k=1,2,\dots ,{N}_{2}-1,{\overline{\tau}}_{{N}_{2}}=1-1/2{({N}_{2})}^{2}$.

As an approximate solution of (

16), we shall seek in the form of a continuous function

where

${\phi}_{k}(t),k=0,1,\dots ,{N}_{1}$,

${\psi}_{k}(t),k=0,1,\dots ,{N}_{2}$ is a family of basis functions.

For nodes

${t}_{k},$ $k=1,\dots ,{N}_{1}-1,$ the corresponding basis elements are determined by

For boundary nodes

${t}_{k},$ $k=0$ and

$k={N}_{1}$ the corresponding basis elements are defined as

and

For nodes

${\tau}_{k},$ $k=0,1,\dots ,{N}_{2},$ the corresponding basis elements

${\psi}_{k},k=0,1,\dots ,{N}_{2},$ are determined in a the similar way: For nodes

${\tau}_{k},$ $k=1,\dots ,{N}_{2}-1,$ the corresponding basis elements are determined by

For boundary nodes

${\tau}_{k},$ $k=0$ and

$k={N}_{2}$ the corresponding basis elements are defined as

and

To simplify the description of computational scheme, we introduce the following notation:

- (1)
Unite the nodes ${t}_{k},k=0,1,\dots ,{N}_{1}$ and ${\tau}_{l},l=0,1,\dots ,{N}_{2}$, denoting them by ${v}_{i},i=0,1,\dots ,{N}^{\ast},{N}^{\ast}={N}_{1}+{N}_{2};$

- (2)
Unite the nodes ${\overline{t}}_{k},k=0,1,\dots ,{N}_{1}$ and ${\overline{\tau}}_{l},l=0,1,\dots ,{N}_{2}$, denoting them by ${\overline{v}}_{i},i=0,1,\dots ,{N}^{\ast}+1;$

- (3)
Denote the family of basis functions $\left\{{\phi}_{k}\right\},k=0,1,\dots ,{N}_{1}$ , $\left\{{\psi}_{l}\right\},l=0,1,\dots ,{N}_{2}$ by $\left\{{\zeta}_{j}\right\},j=0,1,\dots ,{N}^{\ast}+1$;

- (4)
Denote by $\left\{{\gamma}_{k}\right\},k=0,1,\dots ,{N}^{\ast}+1,$ unknowns $\left\{{\alpha}_{i}\right\},i=0,1,\dots ,{N}_{1}$ , $\left\{{\beta}_{j}\right\},j=0,1,\dots ,{N}_{2}$.

Here ${v}_{i}={t}_{i},i=0,1,\dots ,{N}_{1}$ , ${v}_{{N}_{1}+i}={\tau}_{i},i=1,2,\dots ,{N}_{2}$,

${\gamma}_{i}={\alpha}_{i},i=0,1,\dots ,{N}_{1}$ , ${\gamma}_{{N}_{1}+1+i}={\beta}_{i},i=0,1,\dots ,{N}_{2},$

${\zeta}_{i}={\phi}_{i},i=0,1,\dots ,{N}_{1}$ , ${\zeta}_{{N}_{1}+1+i}={\psi}_{i},i=0,1,\dots ,{N}_{2}.$

Let us recall that the points ${t}_{{N}_{1}}$ and ${\tau}_{0}$ coincide.

Applying the collocation method on the knots

${\overline{v}}_{k},k=0,1,\dots ,{N}^{\ast}+1$ to the Equation (

16), we obtain the following system of algebraic equations for finding unknown coefficients

$\left\{{\gamma}_{k}\right\}$ of the polygon (

17)

$k=0,1,\dots ,{N}^{\ast}+1.$Using the definition of hypersingular integrals, we receive:

Here

$\sum {}_{l}^{\prime}$,

$\sum {}_{l}^{\u2033}$,

$\sum {}_{l}^{\u2034}$,

$\sum {}_{l}^{\u2034\prime}$ indicates a summation over

$l\ne {N}_{1}$,

$l\ne {N}_{1}+1$,

$l\ne k(1\le k\le {N}_{1}-1)$,

$l\ne k({N}_{1}+2\le k\le {N}^{\ast}-1)$, respectively. Detailed calculations are given in [

23].

We can rewrite the system (

24) as

Here $\sum {}^{\prime}$, $\sum {}^{\u2033}$, $\sum {}^{\u2034}$ indicates a summation over $l\ne k,$ $l\ne {N}_{1},$ $l\ne {N}_{1}+1,$ respectively.

The system (

37)–(

42) is equivalent to the system

Here $\sum {}^{\prime}$, $\sum {}^{\u2033}$, $\sum {}^{\u2034}$ indicates a summation over $l\ne k$, $l\ne {N}_{1}$, $l\ne {N}_{1}+1,$ respectively.

Let us write the system (

43)–(

48) in a matrix form

where

$D=\left\{{d}_{kl}\right\}$,

$k,l=0,1,\dots ,{N}^{\ast}+1$,

$X=({x}_{0},{x}_{1},\dots ,{x}_{{N}^{\ast}+1})$,

$F=({f}_{0},{f}_{1},\dots ,{f}_{{N}^{\ast}+1})$. The values

$\left\{{d}_{kl}\right\}$,

$\left\{{x}_{k}\right\}$, and

$\left\{{f}_{k}\right\}$ are obvious.

The diagonal elements in the left-hand side of the system of Equations (

43)–(

48) have the following forms

$k=1,2,\dots ,{N}_{1}-1,$$k={N}_{1}+2,\dots ,{N}^{\ast},$The cubic logarithmic norm of the matrix

D is equal to

From (

25)–(

36) it follows that for sufficiently large

N ${\Lambda}_{2}(D)<0$ occurs. By Theorem 2 it is clear that the system (

43)–(

48) (and (

37)–(

42)) has a unique solution

${x}_{N}^{\ast}(t)$ and

$\parallel {D}^{-1}\parallel \le 1/|{\Lambda}_{2}(D)|.$Let

${x}^{\ast}(t)$ and

${x}_{N}^{\ast}$ be solutions of (

16 ) and (

37)–(

42), respectivety.

We recall the following definitions.

**Definition** **3.** The class ${W}^{r}(M,[a,b]),$ $r=1,2,\dots ,$ consists of all functions $f\in C([a,b]),$ which have an absolutely continuous derivative ${f}^{(r-1)}(x)$ and piecewise derivative ${f}^{(r)}(x)$ with $|{f}^{(r)}(x)|\le M.$

**Definition** **4.** Denote by ${W}^{r}(f:{f}_{1},{f}_{2};M,c),r=1,2,\dots ,$ a set of functions $f(x),x\in [a,b],$ such that $f(x)={f}_{1}(x),x\in [a,c),f(x)={f}_{2}(x),x\in (c,b]$, where ${f}_{1}(x)\in {W}^{r}(M,[a,c]),{f}_{2}(x)\in {W}^{r}(M,[c,b]),{f}_{1}(c)\ne {f}_{2}(c),c\in (a,b).$

Repeating the proof presented in [

24] we see that the approximation of

$f(t)\in {W}^{1}((f:{f}_{1},{f}_{2};M,c))$ by piecewise linear functions constructed on the basis

${\zeta}_{k}(t),\phantom{\rule{4pt}{0ex}}k=0,1,\dots ,{N}^{\ast}+1$, has the error

$\frac{C}{N}max(\omega ({f}_{1}^{(1)},\frac{1}{N}),\omega ({f}_{1}^{(1)},\frac{1}{N}))$ for

$f(t)\in {W}^{1}((f:{f}_{1},{f}_{2};M,c))$, and

$\frac{C}{{N}^{2}}$ for

$f(t)\in {W}^{2}((f:{f}_{1},{f}_{2};M,c))$.

In this paper, we denote the constants that do not depend on N by C.

Let ${x}^{\ast}(t)\in {W}^{2}(({x}^{\ast}:{x}_{1}^{\ast},{x}_{2}^{\ast};M,c))$, and $\parallel {x}_{1}^{\ast (1)}{(t)\parallel}_{C}\le {M}_{1},t\in [a,c]$, $\parallel {x}_{2}^{\ast (1)}{(t)\parallel}_{C}\le {M}_{2},t\in [c,b]$, $M=max({M}_{1},{M}_{2}),0<M<\infty $, where M is a bounded constant.

Repeating the arguments given in [

24], we arrive at the following statement.

**Theorem** **6.** Let the following conditions be fulfilled:

- (1)
Equation (16) has the unique solution ${x}^{\ast}(t)\in {W}^{2}({x}_{1}^{\ast},{x}_{2}^{\ast};M,c),-1<c<1,M=const.$ - (2)
For all $t\in [-1,1]$ the function $h(t,t)\ne 0.$

- (3)
${\Lambda}_{2}(D)<0$.

Then the system of Equations (37)–(42) has a unique solution ${x}_{N}^{\ast}(t)$ and the following estimate holds: $\left|\right|{x}^{\ast}-{x}_{N}^{\ast}{\left|\right|}_{1}\le C{N}^{-1}lnN$. #### 3.2. Nonlinear Hypersingular Integral Equations

Consider the nonlinear hypersingular integral equation:

The approximate solution of the Equation (

49) we shall seek as a continuous function (

17) with the coefficients

${\gamma}_{k}$. The coefficients

${\gamma}_{k}$ are determined by the following system of nonlinear algebraic equations

**Remark** **4.** Note that the set ${\gamma}_{k},k=0,1,\dots ,{N}^{\ast}+1,$ is union of sets ${\alpha}_{k},k=0,1,\dots ,{N}_{1},$ and ${\beta}_{k},k=0,1,\dots ,{N}_{2}.$

By computing the hypersingular integrals in (

50), we can rewrite the system of Equation (

50) as

Here $\sum {}^{\prime}$, $\sum {}^{\u2033}$, $\sum {}^{\u2034}$ indicate summations over $l\ne k,$ $l\ne {N}_{1},$ $l\ne {N}_{1}+1,$ respectively.

The Frechet derivative on a vector

$({\overline{\alpha}}_{0},{\overline{\alpha}}_{1},\cdots ,{\overline{\alpha}}_{{N}^{\ast}+1})$ in the space

${R}_{{N}^{\ast}+1}$ is equal to

Here $\sum {}^{\prime}$, $\sum {}^{\u2033}$ $\sum {}^{\u2034}$ indicate summations over $l\ne k,$ $l\ne {N}_{1},$ $l\ne {N}_{1}+1,$ respectively.

The notation ${h}^{\prime}{}_{3}(t,\tau ,u)=\frac{\delta h(t,\tau ,u)}{\delta u}$ is used here.

Let the Equation (

49) has the unique solution

${x}^{\ast}(t)$ inside the ball

$B({x}^{\ast},\delta ).$ We shall assume that the Frechet derivative (

57) in the ball

${R}_{{N}^{\ast}+1}({x}^{\ast},\delta )$ satisfies the conditions of Theorem 5. Thus, according to statements of the Theorem 5, the solution of the system of differential equations

converges to the solution of the Equation (

49).

Thus, we have proven the following statement.

**Theorem** **7.** Let the following conditions hold:

- (1)
Equation (49) has a unique solution ${x}^{\ast}(t)$ inside some ball $B({x}^{\ast},\delta ),{x}^{\ast}\in {W}^{2}({x}^{\ast}:{x}_{1}^{\ast},{x}_{2}^{\ast};M,c);$ - (2)
The Frechet derivative (57) in the ball ${R}_{{N}^{\ast}+1}({x}^{\ast},\delta )$ satisfies the conditions of Theorem 5.

Then the system of Equations (51)–(56) has a unique solution inside the ball $B({x}^{\ast},\delta )$, and the solution of Equation (58) converges to this solution. The effectiveness of the presented algorithms is illustrated by solving two hypersingular integral equations modeling aerodynamics problems.

**Example** **1.** Let us illustrate the effectiveness of continuous method by solving the following linear hypersingular equationwhere $f({\gamma}_{1},{\gamma}_{2},t)$ is the given right-hand side of the equation: The exact solution of the equation is $x(t)=({x}_{1}(t),{x}_{2}(t));{x}_{i}(t)={a}_{i}+{\gamma}_{i}t,i=1,2$.

To solve the Equation (

59) numerically we use the continuous method for solving operator equations and arrive to the following evolution equation

Nodes ${v}_{k},{\overline{v}}_{k},k=0,1,\dots ,{N}^{\ast}+1,$ have been entered above.

In

Figure 1 we show the trajectories of the exact solution of the Equation (

59); its approximate solution, received with continuous method; and values of error.

Here ${a}_{1}=1,{a}_{2}=1.5,{\gamma}_{1}=0.5,{\gamma}_{2}=0.3$.

**Example** **2.** Let us illustrate the effectiveness of the continuous method for the solutions of nonlinear hypersingular equationswhere $f({\gamma}_{1},{\gamma}_{2},t)$ is the given right-hand side of the equation: The exact solution of the equation is $x(t)=({x}_{1}(t),{x}_{2}(t));{x}_{i}(t)={a}_{i}+{\gamma}_{i}t,i=1,2$.

It easy to see that, if

$x(t)$ is a solution of the Equation (

60), then functions

$-x(t)$,

$|x(t)|$ and

$-|x(t)|$ are solutions of this equation too.

To solve the Equation (

60) numerically we use the continuous method and receive the following evolution equation

$k=0,1,\dots ,{N}^{\ast}+1$.

At first, we take

${\alpha}_{k}(0)=0.0$ as an initial condition in order to demonstrate applicability of our method in cases of the Newton–Kantorovich method, the minimal residual method and other numerical methods; using in their construction the derivative of nonlinear operator is not applicable. Indeed, in this case the Frechet derivative (

57) is not only degenerate—and, therefore, not invertable—but is an identical zero.

In

Figure 2 we put

${a}_{1}=1,{a}_{2}=1.4,{\gamma}_{1}=0.5,{\gamma}_{2}=-0.4$.

In

Figure 2 we show the trajectories of the exact solution of the Equation (

60), its approximate solution, received with continuous method and values of error.

The exact solution at

$t=0$ has a jump discontinuity of

$h=0.4$. The slopes of the exact solution also change at

$t=0$. In

Figure 2 we demonstrate that the numerical solution approximates the exact one at

$[-1,0)$ well. At

$t=0$ the approximate solution has a jump

$\tilde{h}=0.15$.