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As equações diferenciais ordinárias (EDOs) são a espinha dorsal da modelagem matemática em inúmeras áreas da ciência e engenharia. Elas descrevem como quantidades mudam ao longo do tempo ou espaço, capturando a dinâmica de fenômenos tão diversos quanto o movimento planetário, reações químicas, crescimento populacional, circuitos elétricos, sistemas mecânicos e a propagação de doenças. No entanto, encontrar soluções analíticas exatas para muitas EDOs, especialmente aquelas que surgem em problemas do mundo real, pode ser uma tarefa árdua ou mesmo impossível. É aqui que os métodos numéricos entram em cena, oferecendo ferramentas poderosas para aproximar as soluções dessas equações complexas. Dentre a vasta gama de técnicas numéricas disponíveis, os métodos de Runge-Kutta se destacam por sua eficiência e precisão na resolução de problemas de valor inicial (PVIs) para EDOs. Um PVI consiste em uma EDO acompanhada de uma condição inicial que especifica o valor da solução em um ponto particular. A ideia central dos métodos de Runge-Kutta é realizar múltiplas avaliações da função que define a EDO (a função que descreve a taxa de variação) dentro de cada passo de integração, combinando-as de forma ponderada para obter uma aproximação mais acurada da solução no próximo ponto.
Os métodos de Runge-Kutta são classificados como métodos de passo simples, o que significa que a aproximação da solução no próximo ponto temporal, digamos $y_{i+1}$, depende apenas da solução no ponto atual, $y_i$, e de informações sobre a derivada (a função $f(t,y))$ avaliada em diferentes pontos intermediários dentro do intervalo $[t_i, t_{i+1}]$. Isso contrasta com os métodos de passo múltiplo, que utilizam informações de vários pontos anteriores para calcular o próximo valor.
A "ordem" de um método Runge-Kutta está diretamente relacionada à sua precisão. De forma simplificada, um método de ordem $p$ consegue igualar os $p+1$ primeiros termos da expansão em série de Taylor da solução exata. Isso implica que o erro local de truncamento (o erro cometido em um único passo, assumindo que todos os valores anteriores são exatos) é proporcional a $h^{p+1}$, onde $h$ é o tamanho do passo. Portanto, métodos de alta ordem tendem a fornecer resultados mais precisos para um mesmo tamanho de passo, ou permitir o uso de passos maiores para alcançar uma precisão desejada, o que pode levar a uma maior eficiência computacional, especialmente em integrações longas.
Os métodos Runge-Kutta "explícitos" são aqueles em que o cálculo da solução no próximo ponto, $y_{i+1}$, pode ser feito diretamente a partir dos valores conhecidos no ponto atual, $y_i$, e das avaliações da função $f(t,y)$. Não há necessidade de resolver equações implícitas para $y_{i+1}$ a cada passo, o que simplifica a implementação computacional. A forma geral de um método Runge-Kutta explícito de s estágios pode ser escrita como:
\[ y_{i+1} = y_i + h \sum_{j=1}^{s} (b_j k_j)\]
onde $h$ é o tamanho do passo, e os $k_j$ são as avaliações intermediárias da função $f$:
$k_1 = f(t_i, y_i) $
$k_2 = f(t_i + c_2h, y_i + h(a_{21}k_1))$
$k_3 = f(t_i + c_3h, y_i + h(a_{31}k_1 + a_{32}k_2))$
...
$k_s = f(t_i + c_sh, y_i + h(a_{s1}k_1 + a_{s2}k_2 + ... + a_{s,s-1}k_{s-1})) $
Os coeficientes $b_j$, $c_j$ e $a_{ij}$ são constantes específicas que definem um método Runge-Kutta particular e determinam sua ordem e propriedades de estabilidade. Esses coeficientes são frequentemente apresentados de forma compacta em uma tabela conhecida como Tabela de Butcher. Um dos exemplos mais conhecidos e amplamente utilizados de método Runge-Kutta explícito é o método clássico de quarta ordem (RK4). Ele requer quatro avaliações da função $f$ por passo e alcança uma precisão de quarta ordem, o que significa que seu erro local é da ordem de $h^5$. Este método oferece um bom equilíbrio entre precisão e custo computacional para uma vasta gama de problemas.
Além do RK4, existem outros métodos Runge-Kutta explícitos de ordens ainda mais elevadas, como os de quinta, sexta ordem, sétima, oitava ordem ou ainda maiores. No entanto, a complexidade na derivação dos coeficientes e o número de avaliações da função por passo aumentam com a ordem. Para problemas que exigem altíssima precisão ou onde a avaliação da função $f$ é computacionalmente cara, a escolha da ordem do método Runge-Kutta torna-se uma consideração crucial.
Outra inovação importante na família Runge-Kutta são os métodos com controle de passo adaptativo, como os métodos de Runge-Kutta-Fehlberg (RKF45) ou Dormand-Prince (DOPRI5). Estes métodos utilizam pares de fórmulas Runge-Kutta explícitas de ordens diferentes (por exemplo, uma de quarta e outra de quinta ordem) para estimar o erro local em cada passo. Com base nessa estimativa de erro, o tamanho do passo $h$ pode ser ajustado dinamicamente: aumentado quando o erro é pequeno (para economizar cálculos) e diminuído quando o erro é grande (para manter a precisão desejada). Essa capacidade de adaptação torna esses métodos particularmente eficientes para problemas onde a solução varia rapidamente em algumas regiões e lentamente em outras.
Em resumo, os métodos Runge-Kutta explícitos de alta ordem representam uma classe de ferramentas numéricas versáteis e poderosas para a solução de equações diferenciais ordinárias. Sua capacidade de alcançar alta precisão com um esforço computacional razoável os tornou um pilar na simulação científica e na engenharia, permitindo-nos explorar e entender a dinâmica de sistemas complexos que, de outra forma, permaneceriam inacessíveis.
Nesta postagem vamos comparar os métodos RK de 6a., 8a. 10a. e 12a. ordens. Usamos o Scilab para fazer as simulações.
Referências:
Feagin, Terry. "A tenth-order Runge-Kutta method with error estimate." Proceedings of the IAENG Conf. on Scientific Computing, 2007.
academia.edu/83821799/The_Effectiveness_of_Runge_Kutta
coeficientes:
https://sce.uhcl.edu/rungekutta/
Resultados gráficos:
y_analitica = (1-exp(-t/2))./(1+exp(-2*t));
dy/dt = (4 e^(2 t))/(e^(2 t) + 1)^2
----------------------
y_analitica = exp(-t).*sin(4*t); dy/dt = 4*exp(-t).*cos(4*t) - y;
---------------- Código Scilab:
close(winsid()); clc;
// Definir a função da equação diferencial dy/dt = f(t, y)
function dydt=ff(t, y)
//dydt = 2*t.*exp(-t) - y;
//dydt=r*y*exp(-t/2)*(1-y) - y/2;
dydt = (exp((3*t)/2)*(-3 + 4*exp(t/2) + exp(2*t)))/(2*(1 + exp(2*t))^2);
//dydt = exp(-t) - y;
//dydt = 4*exp(-t).*cos(4*t) - y;
endfunction
////////
// t0: tempo inicial
// y0: valor inicial de y
// tf: tempo final
// h: tamanho do passo
t0 = 0;
y0 = 0;
tf = 10;
h = 0.25;
r = 2.5;
// Número de passos
N = ceil((tf - t0) / h);
// Inicializar vetores para armazenar os resultados
t = t0:h:(t0 + N*h);
y = zeros(1, N+1);
y6 = y;
y6(1) = y0;
y6i = y0;
for i = 1:N
ti = t(i);
y6i = y6(i);
// Estágios do RK6
k1 = h * ff(ti, y6i);
k2 = h * ff(ti + h/5, y6i + k1/5);
k3 = h * ff(ti + 2*h/5, y6i + 2*k2/5);
k4 = h * ff(ti + h/5, y6i + (11*k1 - 4*k2+5*k3)/60);
k5 = h * ff(ti + 3*h/5, y6i + (-3*k1+4*k2+3*k3+8*k4)/20);
k6 = h * ff(ti + 4*h/5, y6i + (-12*k2-8*k3+22*k4+10*k5)/15);
k7 = h * ff(ti + h, y6i + (51*k1+140*k2+225*k3-280*k4-...
120*k5+60*k6)/76);
// Atualizar y usando a combinação dos estágios
y6(i+1) = y6i + (19*k1+50*k3+75*k4+50*k5+75*k6+19*k7)/288;
end;
//The coefficients in exact form are:
c(1)=0;
c(2)=1/2,
c(3)=1/2,
c(4)=1/2+(1/14)*(21^(1/2)),
c(5)=c(4); //1/2+1/14*21^(1/2),
c(6)=1/2,
c(7)=1/2-(1/14)*(21^(1/2)),
c(8)=c(7); //1/2-1/14*21^(1/2),
c(9)=1/2,
c(10)=c(4); //1/2+1/14*21^(1/2),
c(11)=1,
a(2,1)=1/2,
a(3,1)=1/4,
a(3,2)=1/4,
a(4,1)=1/7,
a(4,2)=-1/14-(3/98)*(21^(1/2)),
a(4,3)=3/7+5/49*21^(1/2),
a(5,1)=11/84+1/84*21^(1/2),
a(5,2)=0,
a(5,3)=2/7+4/63*21^(1/2),
a(5,4)=1/12-1/252*21^(1/2),
a(6,1)=5/48+1/48*21^(1/2),
a(6,2)=0,
a(6,3)=1/4+1/36*21^(1/2),
a(6,4)=-77/120+7/180*21^(1/2),
a(6,5)=63/80-7/80*21^(1/2),
a(7,1)=5/21-1/42*21^(1/2),
a(7,2)=0,
a(7,3)=-48/35+92/315*21^(1/2),
a(7,4)=211/30-29/18*21^(1/2),
a(7,5)=-36/5+23/14*21^(1/2),
a(7,6)=9/5-13/35*21^(1/2),
a(8,1)=1/14,
a(8,2)=0,
a(8,3)=0,
a(8,4)=0,
a(8,5)=1/9-1/42*21^(1/2),
a(8,6)=13/63-1/21*21^(1/2),
a(8,7)=1/9,
a(9,1)=1/32,
a(9,2)=0,
a(9,3)=0,
a(9,4)=0,
a(9,5)=91/576-7/192*21^(1/2),
a(9,6)=11/72,
a(9,7)=-385/1152-25/384*21^(1/2),
a(9,8)=63/128+13/128*21^(1/2),
a(10,1)=1/14,
a(10,2)=0,
a(10,3)=0,
a(10,4)=0,
a(10,5)=1/9,
a(10,6)=-733/2205-1/15*21^(1/2),
a(10,7)=515/504+37/168*21^(1/2),
a(10,8)=-51/56-11/56*21^(1/2),
a(10,9)=132/245+4/35*21^(1/2),
a(11,1)=0,
a(11,2)=0,
a(11,3)=0,
a(11,4)=0,
a(11,5)=-7/3+7/18*21^(1/2),
a(11,6)=-2/5+28/45*21^(1/2),
a(11,7)=-91/24-53/72*21^(1/2),
a(11,8)=301/72+53/72*21^(1/2),
a(11,9)=28/45-28/45*21^(1/2),
a(11,10)=49/18-7/18*21^(1/2),
b(1)=1/20,
b(2)=0,
b(3)=0,
b(4)=0,
b(5)=0,
b(6)=0,
b(7)=0,
b(8)=49/180,
b(9)=16/45,
b(10)=49/180,
b(11)=1/20
// Inicializar vetores para armazenar os resultados
N = ceil((tf - t0) / h);
t = t0:h:(t0 + N*h);
y = zeros(1, N+1);
y8 = y;
y8(1) = y0;
for i = 1:N
ti = t(i);
y8i = y8(i);
k = 0*c;
k(1)=h*ff(ti, y8i)
for n=2:11
sk=0;
for p=1:(n-1)
sk = sk + a(n,p)*k(p);
end;
k(n) = h*ff(ti+c(n)*h, y8i + sk);
end
sb = 0;
for n=1:11
sb = sb + b(n)*k(n);
end
y8(i+1) = y8i + sb;
end;
//////////////
c(1) = 0;
c(2) = 0.1000000000000000000000000000000000000000000000000000000;
c(3) = 0.5393578408029817875324851978813024368572734497010090155055;
c(4) = 0.80903676120447268129872779682195365528591017455151352325825;
c(5) = 0.30903676120447268129872779682195365528591017455151352325825;
c(6) = 0.981074190219795268254879548310562080489056746118724882027805;
c(7) = 0.833333333333333333333333333333333333333333333333333333333333;
c(8) = 0.354017365856802376329264185948796742115824053807373968324184;
c(9) = 0.882527661964732346425501486979669075182867844268052119663791;
c(10) = 0.642615758240322548157075497020439535959501736363212695909875;
c(11) = 0.357384241759677451842924502979560464040498263636787304090125;
c(12) = 0.117472338035267653574498513020330924817132155731947880336209;
c(13) = 0.833333333333333333333333333333333333333333333333333333333333;
c(14) = 0.309036761204472681298727796821953655285910174551513523258250;
c(15) = 0.5393578408029817875324851978813024368572734497010090155055;
c(16) = 0.1000000000000000000000000000000000000000000000000000000000;
c(17) = 1.00000000000000000000000000000000000000000000000000000000000;
b(1) = 0.0333333333333333333333333333333333333333333333333333333333333;
b(2) = 0.025000000000000000000000000000000000000000000000000000000000;
b(3) = 0.0333333333333333333333333333333333333333333333333333333333333;
b(4) = 0;
b(5) = 0.050000000000000000000000000000000000000000000000000000000000;
b(6) = 0;
b(7) = 0.04000000000000000000000000000000000000000000000000000000000;
b(8) = 0;
b(9) = 0.189237478148923490158306404106012326238162346948625830327194;
b(10) = 0.277429188517743176508360262560654340428504319718040836339472;
b(11) = 0.277429188517743176508360262560654340428504319718040836339472;
b(12) = 0.189237478148923490158306404106012326238162346948625830327194;
b(13) = -0.0400000000000000000000000000000000000000000000000000000000000;
b(14) = -0.0500000000000000000000000000000000000000000000000000000000000;
b(15) = -0.0333333333333333333333333333333333333333333333333333333333333;
b(16) = -0.0250000000000000000000000000000000000000000000000000000000000;
b(17) = 0.0333333333333333333333333333333333333333333333333333333333333;
a(2,1) = 0.100000000000000000000000000000000000000000000000000000000000;
a(3,1) = -0.915176561375291440520015019275342154318951387664369720564660;
a(3,2) = 1.45453440217827322805250021715664459117622483736537873607016;
a(4,1) = 0.202259190301118170324681949205488413821477543637878380814562;
a(4,2) = 0.000000000000000000000000000000000000000000000000000000000000;
a(4,3) = 0.606777570903354510974045847616465241464432630913635142443687;
a(5,1) = 0.184024714708643575149100693471120664216774047979591417844635;
a(5,2) = 0.000000000000000000000000000000000000000000000000000000000000;
a(5,3) = 0.197966831227192369068141770510388793370637287463360401555746;
a(5,4) = -0.0729547847313632629185146671595558023015011608914382961421311
a(6,1) = 0.0879007340206681337319777094132125475918886824944548534041378
a(6,2) = 0.000000000000000000000000000000000000000000000000000000000000
a(6,3) = 0.000000000000000000000000000000000000000000000000000000000000
a(6,4) = 0.410459702520260645318174895920453426088035325902848695210406
a(6,5) = 0.482713753678866489204726942976896106809132737721421333413261
a(7,1) = 0.0859700504902460302188480225945808401411132615636600222593880
a(7,2) = 0.000000000000000000000000000000000000000000000000000000000000
a(7,3) = 0.000000000000000000000000000000000000000000000000000000000000
a(7,4) = 0.330885963040722183948884057658753173648240154838402033448632
a(7,5) = 0.489662957309450192844507011135898201178015478433790097210790
a(7,6) = -0.0731856375070850736789057580558988816340355615025188195854775
a(8,1) = 0.120930449125333720660378854927668953958938996999703678812621
a(8,2) = 0.000000000000000000000000000000000000000000000000000000000000
a(8,3) = 0.000000000000000000000000000000000000000000000000000000000000
a(8,4) = 0.000000000000000000000000000000000000000000000000000000000000
a(8,5) = 0.260124675758295622809007617838335174368108756484693361887839
a(8,6) = 0.0325402621549091330158899334391231259332716675992700000776101
a(8,7) = -0.0595780211817361001560122202563305121444953672762930724538856
a(9,1) = 0.110854379580391483508936171010218441909425780168656559807038
a(9,2) = 0.000000000000000000000000000000000000000000000000000000000000
a(9,3) = 0.000000000000000000000000000000000000000000000000000000000000
a(9,4) = 0.000000000000000000000000000000000000000000000000000000000000
a(9,5) = 0.000000000000000000000000000000000000000000000000000000000000
a(9,6) = -0.0605761488255005587620924953655516875526344415354339234619466
a(9,7) = 0.321763705601778390100898799049878904081404368603077129251110
a(9,8) = 0.510485725608063031577759012285123416744672137031752354067590
a(10,1) = 0.112054414752879004829715002761802363003717611158172229329393
a(10,2) = 0.000000000000000000000000000000000000000000000000000000000000
a(10,3) = 0.000000000000000000000000000000000000000000000000000000000000
a(10,4) = 0.000000000000000000000000000000000000000000000000000000000000
a(10,5) = 0.000000000000000000000000000000000000000000000000000000000000
a(10,6) = -0.144942775902865915672349828340980777181668499748506838876185
a(10,7) = -0.333269719096256706589705211415746871709467423992115497968724
a(10,8) = 0.499269229556880061353316843969978567860276816592673201240332
a(10,9) = 0.509504608929686104236098690045386253986643232352989602185060
a(11,1) = 0.113976783964185986138004186736901163890724752541486831640341
a(11,2) = 0.000000000000000000000000000000000000000000000000000000000000
a(11,3) = 0.000000000000000000000000000000000000000000000000000000000000
a(11,4) = 0.000000000000000000000000000000000000000000000000000000000000
a(11,5) = 0.000000000000000000000000000000000000000000000000000000000000
a(11,6) = -0.0768813364203356938586214289120895270821349023390922987406384
a(11,7) = 0.239527360324390649107711455271882373019741311201004119339563
a(11,8) = 0.397774662368094639047830462488952104564716416343454639902613
a(11,9) = 0.0107558956873607455550609147441477450257136782823280838547024
a(11,10) = -0.327769124164018874147061087350233395378262992392394071906457
a(12,1) = 0.0798314528280196046351426864486400322758737630423413945356284
a(12,2) = 0.000000000000000000000000000000000000000000000000000000000000
a(12,3) = 0.000000000000000000000000000000000000000000000000000000000000
a(12,4) = 0.000000000000000000000000000000000000000000000000000000000000
a(12,5) = 0.000000000000000000000000000000000000000000000000000000000000
a(12,6) = -0.0520329686800603076514949887612959068721311443881683526937298
a(12,7) = -0.0576954146168548881732784355283433509066159287152968723021864
a(12,8) = 0.194781915712104164976306262147382871156142921354409364738090
a(12,9) = 0.145384923188325069727524825977071194859203467568236523866582
a(12,10) = -0.0782942710351670777553986729725692447252077047239160551335016
a(12,11) = -0.114503299361098912184303164290554670970133218405658122674674
a(13,1) = 0.985115610164857280120041500306517278413646677314195559520529
a(13,2) = 0.000000000000000000000000000000000000000000000000000000000000
a(13,3) = 0.000000000000000000000000000000000000000000000000000000000000
a(13,4) = 0.330885963040722183948884057658753173648240154838402033448632
a(13,5) = 0.489662957309450192844507011135898201178015478433790097210790
a(13,6) = -1.37896486574843567582112720930751902353904327148559471526397
a(13,7) = -0.861164195027635666673916999665534573351026060987427093314412
a(13,8) = 5.78428813637537220022999785486578436006872789689499172601856
a(13,9) = 3.28807761985103566890460615937314805477268252903342356581925
a(13,10) = -2.38633905093136384013422325215527866148401465975954104585807
a(13,11) = -3.25479342483643918654589367587788726747711504674780680269911
a(13,12) = -2.16343541686422982353954211300054820889678036420109999154887
a(14,1) = 0.895080295771632891049613132336585138148156279241561345991710
a(14,2) = 0.000000000000000000000000000000000000000000000000000000000000
a(14,3) = 0.197966831227192369068141770510388793370637287463360401555746
a(14,4) = -0.0729547847313632629185146671595558023015011608914382961421311
a(14,5) = 0.0000000000000000000000000000000000000000000000000000000000000
a(14,6) = -0.851236239662007619739049371445966793289359722875702227166105
a(14,7) = 0.398320112318533301719718614174373643336480918103773904231856
a(14,8) = 3.63937263181035606029412920047090044132027387893977804176229
a(14,9) = 1.54822877039830322365301663075174564919981736348973496313065
a(14,10) = -2.12221714704053716026062427460427261025318461146260124401561
a(14,11) = -1.58350398545326172713384349625753212757269188934434237975291
a(14,12) = -1.71561608285936264922031819751349098912615880827551992973034
a(14,13) = -0.0244036405750127452135415444412216875465593598370910566069132
a(15,1) = -0.915176561375291440520015019275342154318951387664369720564660
a(15,2) = 1.45453440217827322805250021715664459117622483736537873607016
a(15,3) = 0.000000000000000000000000000000000000000000000000000000000000
a(15,4) = 0.000000000000000000000000000000000000000000000000000000000000
a(15,5) = -0.777333643644968233538931228575302137803351053629547286334469
a(15,6) = 0.000000000000000000000000000000000000000000000000000000000000
a(15,7) = -0.0910895662155176069593203555807484200111889091770101799647985
a(15,8) = 0.000000000000000000000000000000000000000000000000000000000000
a(15,9) = 0.000000000000000000000000000000000000000000000000000000000000
a(15,10) = 0.000000000000000000000000000000000000000000000000000000000000
a(15,11) = 0.000000000000000000000000000000000000000000000000000000000000
a(15,12) = 0.000000000000000000000000000000000000000000000000000000000000
a(15,13) = 0.0910895662155176069593203555807484200111889091770101799647985
a(15,14) = 0.777333643644968233538931228575302137803351053629547286334469
a(16,1) = 0.100000000000000000000000000000000000000000000000000000000000
a(16,2) = 0.000000000000000000000000000000000000000000000000000000000000
a(16,3) = -0.157178665799771163367058998273128921867183754126709419409654
a(16,4) = 0.000000000000000000000000000000000000000000000000000000000000
a(16,5) = 0.000000000000000000000000000000000000000000000000000000000000
a(16,6) = 0.000000000000000000000000000000000000000000000000000000000000
a(16,7) = 0.000000000000000000000000000000000000000000000000000000000000
a(16,8) = 0.000000000000000000000000000000000000000000000000000000000000
a(16,9) = 0.000000000000000000000000000000000000000000000000000000000000
a(16,10) = 0.000000000000000000000000000000000000000000000000000000000000
a(16,11) = 0.000000000000000000000000000000000000000000000000000000000000
a(16,12) = 0.000000000000000000000000000000000000000000000000000000000000
a(16,13) = 0.000000000000000000000000000000000000000000000000000000000000
a(16,14) = 0.000000000000000000000000000000000000000000000000000000000000
a(16,15) = 0.157178665799771163367058998273128921867183754126709419409654
a(17,1) = 0.181781300700095283888472062582262379650443831463199521664945
a(17,2) = 0.675000000000000000000000000000000000000000000000000000000000
a(17,3) = 0.342758159847189839942220553413850871742338734703958919937260
a(17,4) = 0.000000000000000000000000000000000000000000000000000000000000
a(17,5) = 0.259111214548322744512977076191767379267783684543182428778156
a(17,6) = -0.358278966717952089048961276721979397739750634673268802484271
a(17,7) = -1.04594895940883306095050068756409905131588123172378489286080
a(17,8) = 0.930327845415626983292300564432428777137601651182965794680397
a(17,9) = 1.77950959431708102446142106794824453926275743243327790536000
a(17,10) = 0.100000000000000000000000000000000000000000000000000000000000
a(17,11) = -0.282547569539044081612477785222287276408489375976211189952877
a(17,12) = -0.159327350119972549169261984373485859278031542127551931461821
a(17,13) = -0.145515894647001510860991961081084111308650130578626404945571
a(17,14) = -0.259111214548322744512977076191767379267783684543182428778156
a(17,15) = -0.342758159847189839942220553413850871742338734703958919937260
a(17,16) = -0.675000000000000000000000000000000000000000000000000000000000
// Inicializar vetores para armazenar os resultados
N = ceil((tf - t0) / h);
t = t0:h:(t0 + N*h);
y = zeros(1, N+1);
y10 = y;
y10(1) = y0;
for i = 1:N
ti = t(i);
y10i = y10(i);
k = 0*c;
k(1)=h*ff(ti, y10i)
for n=2:17
sk=0;
for p=1:(n-1)
sk = sk + a(n,p)*k(p);
end;
k(n) = h*ff(ti+c(n)*h, y10i + sk);
end
sb = 0;
for n=1:17
sb = sb + b(n)*k(n);
end
y10(i+1) = y10i + sb;
end;
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////
c = ck(:,2);
b = bk(:,2);
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24 7 0.229499544768994849582890233710555447073823569666506700662510
24 8 5.79046485790481979159831978177003471098279506036722411333192
24 9 0.418587511856506868874073759426596207226461447604248151080016
24 10 0.307039880222474002649653817490106690389251482313213999386651
24 11 -4.68700905350603332214256344683853248065574415794742040470287
24 12 3.13571665593802262152038152399873856554395436199962915429076
24 13 1.40134829710965720817510506275620441055845017313930508348898
24 14 -5.52931101439499023629010306005764336421276055777658156400910
24 15 -0.853138235508063349309546894974784906188927508039552519557498
24 16 0.103575780373610140411804607167772795518293914458500175573749
24 17 -0.140474416950600941142546901202132534870665923700034957196546
24 18 -0.418587511856506868874073759426596207226461447604248151080016
24 19 -0.229499544768994849582890233710555447073823569666506700662510
24 20 -0.348600717628329563206854421629657569274689947367847465753757
24 21 -0.291666666666666666666666666666666666666666666666666666666667
24 22 -0.421296296296296296296296296296296296296296296296296296296296
24 23 -0.787500000000000000000000000000000000000000000000000000000000];
aa = zeros(24,24);
for nn=1:max(size(ak))
px = ak(nn,1);
py = ak(nn,2)+1;
vv = ak(nn,3);
aa(px,py) = vv;
end
// Inicializar vetores para armazenar os resultados
// N = ceil((tf - t0) / h);
// t = t0:h:(t0 + N*h);
y = zeros(1, N+1);
y12 = y;
y12(1) = y0;
for i = 1:N
ti = t(i);
y12i = y12(i);
k = 0*c;
k(1)=h*ff(ti+c(1)*h, y12i);
k(2)=h*ff(ti+c(2)*h, y12i+aa(1,1)*k(1));
k(3)=h*ff(ti+c(3)*h, y12i+aa(2,1)*k(1)+aa(2,2)*k(2));
k(4)=h*ff(ti+c(4)*h, y12i+aa(3,1)*k(1)+aa(3,2)*k(2)+aa(3,3)*k(3));
for n=5:25
sk=0;
for p=1:(n-1)
sk = sk + aa(n-1,p)*k(p);
end;
k(n) = h*ff(ti+c(n)*h, y12i + sk);
end
sb = 0;
for n=1:max(size(b))
sb = sb + b(n)*k(n);
end
y12(i+1) = y12i + sb;
end;
// Solução analítica para comparação
// y_analitica = t.*t.*exp(-t);
// eqs = 'dy/dt = 2texp(-t) - y';
y_analitica = (1-exp(-t/2))./(1+exp(-2*t));
eqs = 'dy/dt = (4 e^(2 t))/(e^(2 t) + 1)^2';
// y_analitica = t.*exp(-t);
// eqs = 'dy/dt = exp(-t) - y';
//y_analitica = exp(-t).*sin(4*t);
//eqs = 'dy/dt = 4*exp(-t).*cos(4*t) - y';
plot(t,y_analitica, 'm',"LineWidth", 2);
plot(t,y6, 'b', "LineWidth", 2);
plot(t,y8, 'k', "LineWidth", 2);
plot(t,y10,'y', "LineWidth", 2);
plot(t,y12,'k', "LineWidth", 2);
title('Sinal e soluções numéricas (RK6, RK8, RK10 e RK12). Passo: '+string(h));
legend('Sinal','RK6','RK8','RK10','RK12',4);
//Erro e erro acumulado:
e6 = y_analitica - y6; E6a = sum(abs(e6));
e8 = y_analitica - y8; E8a = sum(abs(e8));
e10 = y_analitica - y10; E10a = sum(abs(e10));
e12 = y_analitica - y12; E12a = sum(abs(e12));
figure;
subplot(2,2,1); plot(e6); title('Erro RK-6. E6a = '+string(E6a)); xgrid;
subplot(2,2,2); plot(e8); title('Erro RK-8. E8a = '+string(E8a)); xgrid;
subplot(2,2,3); plot(e10); title('Erro RK-10. E10a = '+string(E10a)); xgrid;
subplot(2,2,4); plot(e12); title('Erro RK-12. E12a = '+string(E12a)); xgrid;
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